PHENOMENOLOGY OF SUPERSOLIDS

By

CHI-DEUK YOO

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT

OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2009

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c© 2009 Chi-Deuk Yoo

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To my family

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ACKNOWLEDGMENTS

I am greatly indebted to my advisor, Professor Alan T. Dorsey, for his guidance,

encouragement, and patience he demonstrated throughout my work. Without his support

this work would not have been possible.

I would like to thank Professor P. J. Hirschfeld, Professor Y.-S. Lee, Professor K.

Matchev, Professor M. W. Meisel, Professor S. R. Phillpot, and Professor A. Roitberg

for valuable discussion and support. I also thank Professor M. H. W. Chan of the

Pennsylvania State University and Professor H. Kojima of the Rutgers University for

sharing their valuable experimental data.

Finally I thank my parents, Hae-Seun Yoo and Hee-Sook Yoo, for their unflagging

interest, support, and encouragement. Most of all I want to thank my dearest wife, Sae-il,

and children, Dan and Seul, for standing beside me with endless love and trust.

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TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

CHAPTER

1 INTRODUCTION TO SUPERSOLIDS . . . . . . . . . . . . . . . . . . . . . . . 12

1.1 History of Supersolids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121.2 Torsional Oscillator Experiments on 4He Solid by Kim and Chan . . . . . . 131.3 Recent Theoretical and Experimental Works . . . . . . . . . . . . . . . . . 15

2 VISCOELASTIC SOLIDS: ALTERNATIVE EXPLANATION OF NCRI . . . . 28

2.1 Equation of Motion for the Torsional Oscillator . . . . . . . . . . . . . . . 282.2 Properties of Viscoelastic Solids under Oscillatory Motion . . . . . . . . . 30

2.2.1 Infinite Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.2.2 Finite Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332.2.3 Infinite Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3 Torsional Oscillator with Viscoelastic Solids . . . . . . . . . . . . . . . . . 342.4 Possible Connection between Anomalies in Shear Modulus and NCRI . . . 37

3 NON-DISSIPATIVE HYDRODYNAMICS OF A MODEL SUPERSOLID . . . . 49

3.1 Variational Principle in Supersolids . . . . . . . . . . . . . . . . . . . . . . 493.1.1 Introduction to the Variational Principle in Continuum Mechanics . 493.1.2 Isotropic Supersolids . . . . . . . . . . . . . . . . . . . . . . . . . . 523.1.3 Anisotropic Supersolids . . . . . . . . . . . . . . . . . . . . . . . . . 583.1.4 Quadratic Lagrangian Density of Supersolids . . . . . . . . . . . . . 62

3.2 Collective Modes and the Density-Density Correlation Function . . . . . . 66

4 DISSIPATIVE HYDRODYNAMICS OF A MODEL SUPERSOLID . . . . . . . 72

4.1 Andreev and Lifshitz Hydrodynamics of Supersolids . . . . . . . . . . . . . 734.2 Density-Density Correlation Function and its Detection . . . . . . . . . . . 77

4.2.1 Normal Fluids and Superfluids . . . . . . . . . . . . . . . . . . . . . 784.2.2 Normal Solids and Supersolids . . . . . . . . . . . . . . . . . . . . . 79

5 DYNAMICS OF TOPOLOGICAL DEFECTS IN SUPERSOLIDS . . . . . . . . 85

5.1 Vortex Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.2 Dislocation Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

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6 CONCLUSION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

APPENDIX

A CALCULATION OF BACK ACTION TERMS . . . . . . . . . . . . . . . . . . 105

A.1 Infinite Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105A.2 Finite Cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106A.3 Infinite Annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

B VARIATIONAL PRINCIPLE IN SUPERSOLIDS WITH THE ROTATIONALVELOCITY OF SUPER COMPONENTS . . . . . . . . . . . . . . . . . . . . . 111

C STATIC CORRELATION FUNCTIONS OF ISOTROPIC SUPERSOLIDS . . . 114

D KUBO FUNCTIONS AND CORRELATION FUNCTIONS . . . . . . . . . . . 115

E CALCULATION OF THE DENSITY-DENSITY CORRELATION FUNCTION 117

F DERIVATION OF AN EFFECTIVE ACTION FOR EDGE DISLOCATIONS . 119

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

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LIST OF TABLES

Table page

2-1 Fitting parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

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LIST OF FIGURES

Figure page

1-1 Resonant period change in temperature . . . . . . . . . . . . . . . . . . . . . . . 20

1-2 Resonant period change in temperature for various concentration of 3Heimpurities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1-3 Resonant period and the amplitude of oscillation of annular cell (panel A) andof blocked annular cell (panel B) as a function of temperature . . . . . . . . . . 22

1-4 Annealing effect in the resonant period . . . . . . . . . . . . . . . . . . . . . . . 23

1-5 Annealing effect in the inverse of Q-factor . . . . . . . . . . . . . . . . . . . . . 24

1-6 Specific heat peaks of 4He solid with different concentrations of 3He impurities . 25

1-7 Shear modulus of 4He solid as a function of temperature . . . . . . . . . . . . . 26

1-8 Shear modulus change for various concentrations of 3He impurities . . . . . . . . 27

2-1 Schematic illustration of TO and geometry of a torsion cell . . . . . . . . . . . . 39

2-2 Effective moment of inertia of an infinite cylinder of viscoelastic solid as afunction of the driving frequency ω . . . . . . . . . . . . . . . . . . . . . . . . . 40

2-3 Effective damping coefficient of an infinite cylinder of viscoelastic solid as afunction of the driving frequency ω . . . . . . . . . . . . . . . . . . . . . . . . . 40

2-4 Displacement vector in a half cycle for ω ¿ 1/τE. . . . . . . . . . . . . . . . . . 41

2-5 Displacement vector in a half cycle for ω = 3/τE. . . . . . . . . . . . . . . . . . 41

2-6 Displacement vector in a half cycle for ω = 4/τE. . . . . . . . . . . . . . . . . . 42

2-7 Effective moment of inertia of a finite cylinder of viscoelastic solid as a functionof the driving frequency ω with τ/τE = 1/100 . . . . . . . . . . . . . . . . . . . 42

2-8 Effective damping coefficient of a finite cylinder of viscoelastic solid as afunction of the driving frequency ω with τ/τE = 1/100 . . . . . . . . . . . . . . 43

2-9 Effective moment of inertia of an infinite annulus of viscoelastic solid as afunction of the driving frequency ω with τ/τE = 1/1000 . . . . . . . . . . . . . . 43

2-10 Effective damping coefficient of an infinite annulus of viscoelastic solid as afunction of the driving frequency ω with τ/τE = 1/1000 . . . . . . . . . . . . . . 44

2-11 F (x) of finite cylinder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2-12 F (x) of infinite annulus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

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2-13 Resonant period of the blocked capillary sample of BeCu TO . . . . . . . . . . . 45

2-14 Inverse of Q-factor of the blocked capillary sample of BeCu TO . . . . . . . . . 46

2-15 Resonant period of the annealed blocked capillary sample of BeCu TO . . . . . 46

2-16 Inverse of Q-factor of the annealed blocked capillary sample of BeCu TO . . . . 47

2-17 Resonant period of the constant temperature sample of BeCu TO . . . . . . . . 47

2-18 Inverse of Q-factor of the constant temperature sample of BeCu TO . . . . . . . 48

4-1 Brillouin spectrum of liquid argon . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4-2 Brillouin spectra of 4He superfluid . . . . . . . . . . . . . . . . . . . . . . . . . . 82

4-3 Density-density correlation functions of isothermal and isotropic normal solidsand supersolids) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4-4 Splitting of the Rayleigh peak due to the defect diffusion mode of a normalsolid into the Brillouin doublet of the second sound modes . . . . . . . . . . . . 84

5-1 Cut for an edge dislocation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy

PHENOMENOLOGY OF SUPERSOLIDS

By

Chi-Deuk Yoo

May 2009

Chair: Alan T. DorseyMajor: Physics

We investigate the phenomenological properties of supersolids - materials that

simultaneously display both crystalline order and superfluidity. To explain the recent

observation in the torsional oscillator experiments on 4He solid by Kim and Chan we

adopt a viscoelastic solid model which is characterized by a frequency-dependent complex

shear modulus. In this model, we found that a characteristic time scale which accounts for

dissipation in solids grows rapidly as the temperature is reduced, and results in a decrease

in the resonant period and a peak in the inverse of Q-factor. We also briefly discuss the

possible relation between the torsional oscillator results and the anomalous increase of

shear modulus obtained by Day and Beamish.

In a related study, we employ a variational principle together with Galilean covariance

and thermodynamic relations to obtain the non-dissipative hydrodynamics and an

effective Lagrangian density for supersolids. We study the mode structure of supersolids

by calculating the second and fourth sound speeds due to defect propagation. We also

calculate the density-density correlation function of a model supersolid using the

hydrodynamics of Andreev and Lifshitz, and propose a light scattering experiment to

measure the density-density correlation function (which is related to the intensity of

scattered light). We find that the central Rayleigh peak of the defect diffusion mode of a

normal solid in the density-density correlation function splits into an additional Brillouin

doublet due to the longitudinal second sound modes in supersolid phase.

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Finally, we study the dynamics of vortices and dislocations in supersolids by using

the derived Lagrangian for supersolids. We obtain the effective actions for vortices and

dislocations in two-dimensional isotropic supersolids emphasizing the differences from the

dynamics in superfluids and solids. As a result we obtain the frequency-dependent inertial

masses for slowly moving vortices and dislocations.

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CHAPTER 1INTRODUCTION TO SUPERSOLIDS

1.1 History of Supersolids

After Kapitza [1], and Allen and Misener [2] simultaneously discovered the

superfluidity of He II at a temperature around 2.17 K in 1938, there were theoretical

speculations about the coexistence of crystalline order and superfluidity in matter. In 1956

Penrose and Onsager studied the possibility of such a supersolid phase of matter using the

density matrix formalism, and concluded that supersolids could not exist [3]. However, in

1969 several novel theoretical proposals for a supersolid phase were made. Andreev and

Lifshitz proposed the possibility of a condensation of zero-point defects in a 4He solid.

Every solid contains defects: vacancies, interstitials, and so on. Classically these defects

are considered to be objects localized at the lattice sites. However, at low temperatures,

due to quantum fluctuations, defects in 4He solid vibrate from the lattice sites and become

mobile. They called these quantum excitations “zero-point defectons”. They generalized

the two-fluid model developed by Landau for superfluids to solids with defects, and

obtained a new collective mode (the propagation of defects) at zero temperature [4].

One year later, Chester suggested that a system of interacting bosons can exhibit

both crystalline order and Bose-Einstein condensation at the same time [5]. Also, Leggett

suggested that the “non-classical rotational inertia” (NCRI) of liquid He II may be

observed in the solid phase. NCRI of He II can be explained by the two fluid model.

When a vessel containing He II is rotated, the absence of viscosity of the superfluid part

causes only the normal part to be dragged by the wall of container. Thus, the effective

moment of inertia of He II is less than that of normal helium liquid. Leggett predicted an

supersolid fraction ρs/ρ of 3×10−4, and a simple and direct experiment of rotating a solid

was suggested to detect it [6].

Thereafter, Saslow [7] and Liu [8] improved Andreev and Lifshitz’s calculation of

hydrodynamic modes. In Ref. [9], the supersolid fraction for a fcc lattice as a function

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of the ratio between the localization parameter and the lattice constant was obtained.

Fernandez et al. obtained another upper bound for supersolid fraction for the hcp lattice

of 0.3 [10]. Other feasible systems that can exhibit supersolidity were also proposed

theoretically: vortex crystals in type-II superconductors [11], Wigner crystals formed by

excitons in electron-hole bilayers [12], and cold atoms in optical lattices [13].

However, experimental searches for possible signatures of supersolidity of solid 4He

were not successful prior to 2004. These include sound speed experiments [14], mass flow

experiments [15], and torsional oscillator (TO) experiments [16, 17]. The early experiments

searching for the supersolid phase are summarized in Ref. [18].

1.2 Torsional Oscillator Experiments on 4He Solid by Kim and Chan

In 2004, Kim and Chan reported two TO experiments that may have shown the

superfluid phase of 4He solid [19, 20]. Both experiments observed drops of resonant period

in the solid phase of 4He, which might be an indication of the NCRI proposed by Leggett.

The TO used by Kim and Chan consist of a Be-Cu torsion bob and a Be-Cu torsional

rod which also was used to introduce 4He into the torsion bob. In Ref. [19], the torsion

bob contained a porous medium (Vycor glass), while in Ref. [20] the experiment was

performed with bulk 4He confined in an annular channel. They used pressures of 62 bar

and 51 bar for 4He in porous media and bulk 4He, respectively. They then electrically

drove the oscillator and measured the resonant period at a fixed temperature. The

characteristic dependence of resonant period on the temperature is shown in Figs. 1-1

and 1-2A. The drop of resonant period occurred below the critical temperature around

Tc = 175 mK for in porous media and Tc = 250 mK for bulk helium.

The resonant period of a TO without dissipation is given by

P = 2π√

I/k, (1–1)

where I is the moment of inertia of the torsion bob with 4He, and k is the torsional spring

constant. Using Eq. 1–1 Kim and Chan interpreted their results of the decrease in the

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resonant period as a change in the moment of inertia of the 4He solid, assuming that the

torsion constant k remained constant. This would mean that a part of the mass of 4He

solid is decoupled from the oscillatory motion, indicating the supersolid phase. The NCRI

fraction (NCRIF) which is defined as the ratio of the superfluid density ρs to the total

density ρ of a bose solid, is related to the relative change in the total moment of inertia

[21],

ρs(T )

ρ=

I(T0)− I(T )

I(T0)− Iempty

=P (T0)− P (T )

P (T0)− Pempty

, (1–2)

where T0 is the onset temperature, and Iempty the moment of inertia of the empty TO. The

largest observed NCRIFs are about 0.5 % and 1.7% for 4He in Vycor glass and bulk 4He,

respectively.

Kim and Chan also performed several control experiments to support their

interpretation. First, they investigated on the effect of the critical velocity, and found

that, in both experiments, the drop in period decreases with increasing rim velocity. Kim

and Chan estimated the critical velocity, at which the NCRI disappears, to be 300 µm/s

for 4He in the Vycor glass and 420 µm/s for bulk 4He (Figs. 1-1 and 1-3A). Second, they

repeated the same experiment with solid 3He, which is a fermionic solid; consequently, no

Bose condensation is possible. With solid 3He they did not observed any change in the

resonant period. However, an important and interesting feature is found that increasing

the concentration of 3He impurities in solid 4He increases the onset temperature and

broadens the change in the resonant period (Fig. 1-2). Third, taking advantage of the cell

geometry of the bulk 4He sample they inserted a barrier into the annulus channel around

which superflow is blocked, and found that the resonant period is significantly reduced

(Fig. 1-3B). Kim and Chan concluded that this is due to the disruption of superflow

around the annulus. Finally, they measured the amplitude, which is related to dissipation,

and observed a broad minimum over the range of temperatures where the resonant

period dropped. The minimum in the amplitude has the same trend as the rim velocity,

Fig. 1-3A.

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Two years after the initial reports several groups replicated the TO result [22–26].

Rittner and Reppy reported first the annealing effect on TO results [22] and the sample

preparation effect [24]. Both signatures of TO experiments - a drop in the resonant period

and a peak in the inverse of Q-factor - disappeared by annealing the 4He solid (Figs.

1-4 and 1-5). In Ref. [24] they reported the effects of quenching the sample by cooling

it rapidly, the effect of which is to make a solid of poor quality with a large number of

defects. They reported NCRIFs as high as 20%. Based on their results, Rittner and

Reppy suggested that extended defects such as dislocations and grain boundaries play an

important role in understanding the TO results.

On the other hand, Aoki et al. studied the frequency dependence of the NCRI using

a double torsional oscillator. They added another dummy cell concentrically above the

torsion bob with solid 4He. This allowed them to investigate the NCRI of the same sample

with two different frequencies: the resonant frequency of the out-of-phase mode was a little

more than a twice that of the in-phase mode. They found no frequency dependence to

the onset temperature of NCRI. In addition to this, Aoki et al. found a hysteresis that

depends on the rim velocity: at T = 19 mK the NCRIF increased upon lowering the rim

velocity, but saturated at the maximum value as the rim velocity was again increased.

1.3 Recent Theoretical and Experimental Works

Kim and Chan’s results have revived both theoretical and experimental interest in

supersolids [27]. Reviews on both recent theoretical and experimental works can be found

in Refs. [28] and [21]. Saslow has suggested that one should use a three-fluid model,

instead of the two-fluid model of Landau, to correctly describe the supersolid. In his

model the mass density and the mass current contain an additional term consisting of the

lattice velocity and the lattice mass density [29]. Ceperley and Bernu calculated exchange

frequencies in perfect bulk hcp 4He using the Path Integral Monte Carlo (PIMC) method

and concluded that superfluidity would not be observed in a perfect crystal [30]. Prokof’ev

and Svistunov have found a similar conclusion that zero-point defects were necessary

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for 4He solid to be a supersolid by using a coarse-graining procedure [31]. However,

Boninsegni et al. obtained a large activation energy of 13 K for vacancies and of 23 K

for interstitials, and suggested that point defects are unlikely to be present in the low

temperature range of experimental conditions [32].

On the experimental side, several results that are unfavorable to the supersolid

interpretation are reported. Day et al. performed experiments of mass flow through small

capillaries with solid 4He in Vycor glass [33] and in bulk [34]. One expects a persistent

mass flow in a supersolid; however, they detected no mass flow in either experiments. On

the contrary, a mass flow in solid 4He was detected by Sasaki et al. [35] and by Ray and

Hallock [36]. Sasaki et al. observed a flow in 4He solid on the melting curve with grain

boundaries (a poor quality crystal). For a single crystal (good quality crystal) no flow

was detected. Initially they suggested that the flow took place through grain boundaries,

but further experiments showed that mass could flow along the channel between a grain

boundary and a wall [37]. Ray and Hallock injected superfluid through one line into a

cell filled with solid 4He, and detected a change on the other line. They have found a

mass flow in solid 4He at pressure off the melting curve [36]. Similarly, superfluidity in

grain boundaries [38] and in screw dislocations [39] was studied using PIMC simulations.

Pollet et al. found that superfluid is formed within grain boundaries in solid 4He at

temperature around 0.5 K. Boninsegni et al. found superfludity along the core of a screw

dislocation in a 4He solid at zero temperature. On the other hand, there are two neutron

scattering experiments to measure the condensate fraction [40] and to detect changes in

the Debye-Waller factor [41]. Neither experiment showed any evidence for the existence of

a supersolid phase in solid 4He.

It is well known that the transition of 4He from the normal fluid to superfluid is

a second order phase transition accompanied with a λ-anomaly in the specific heat at

the transition temperature, e.g. see Ref. [42]. Dorsey et al. suggested that there should

be a λ-anomaly in the specific heat if the supersolid transition is of second order [43].

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They also found that the λ-anomaly will be smeared out due to the inhomogeneity of

solid. Therefore, finding a feature (possibly a cusp) in the specific heat would support

the supersolid interpretation of the observed NCRI. Following the TO experiments,

Clark and Chan carried out a measurement of the specific heat in solid 4He [44]. They

measured down to a temperature of about 80 mK, and did not observed any signature.

In contrast to this null result, Lin et al. reported a peak in the specific heat at about

T = 75 mK [45]. In this second experiment a silicon sample cell was used instead of an

aluminum cell; silicon has a smaller heat capacity and higher thermal conductivity at

low temperatures than aluminum. Thus, they could measure the specific heat down to

a temperature about 30 mK, and they observed deviations from the T 3 Debye specific

heat. Figure 1-6 shows the observed peaks in specific heat of solid 4He with various

concentrations of 3He impurities after subtracting the contributions of the empty cell,

phonons, and 3He impurities. In addition, the constant specific heat term, which might

be due to the mobility of 3He impurities, was also found in the 10 p.p.m. and 30 p.p.m

samples (the inset of Fig. 1-6). It is found that the height of peaks is about 20 µJ mol−1

K−1, and does not depend on concentration of 3He impurities. Lin et al. estimated the

NCRIF to be about 0.06 % which is comparable to one of their TO results [45]. Finally

they concluded that the observed peaks in specific heat measurements are another possible

signature of the supersolid phase transition, in addition to their TO results.

Another interesting and important experiment on solid 4He was performed by Day

and Beamish [46]. They measured directly the shear modulus at low frequencies and

amplitudes using two piezoelectric transducers filled with solid 4He. One transducer was

used to apply a shear stress while the other detects the induced shear deformation. They

found that the shear modulus of solid 4He increased by about 10% upon lowering the

temperature (Fig. 1-7). Day and Beamish explained their observation using the adsorption

(desorption) of 3He impurities into (from) a dislocation network. The adsorbed 3He

impurities pin dislocations at low temperatures, increasing the shear modulus. If 3He

17

impurities evaporate from dislocations by thermal fluctuations, the dislocations become

mobile, reducing the shear modulus. The anomalous increase in the shear modulus of 4He

solid manifested surprisingly similar behaviors to the TO results, such as the dependence

on the maximum amplitude of applied shear stress and on the concentration of 3He

impurities. In Fig. 1-8 the reduced changes of the shear modulus for different values of 3He

impurity concentration are shown as a function of temperature, and in comparison with

the reduced NCRI. Moreover, Day and Beamish also studied the resonance in the cavity.

They monitored the resonant frequency and the Q-factor, and found similar behavior to

the TO experiments: the resonant frequency increased as the temperature was lowered,

accompanied with a peak in the inverse Q-factor. The two measurements are very similar,

suggesting that they are closely related. This would mean that dislocations and grain

boundaries present in solids might be responsible for both the shear anomaly and the

NCRI [47].

On the other hand, several alternative explanations for the observed NCRI of 4He

solid have been proposed as well. Dash and Wettlaufer gave an argument that there exists

a thin layer of liquid helium between the wall and the helium solid, and they showed

that the slippage between them could be responsible for the NCRI [48]. Nussinov at el.

proposed a glass model for the 4He solid [49]. In their model, solid 4He was assumed to be

in glassy phase at low temperatures, and they studied its effect on the TO experiments.

Remarkably they could find the reasonable agreement with the experiment of Ref. [22].

Finally, Huse and Khandker proposed a phenomenological two-fluid model for a supersolid

with a temperature dependent coupling constant [50] to explain the TO results.

To summarize, the existence of a supersolid phase in solid 4He has remained

controversial, both theoretically and experimentally. The explanation for the observed

TO experiments is not complete, opening possibilities ranging from a supersolid transition

to a possibly already known mechanical effect such as dislocation unbinding. In this work

we will focus on

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• finding an alternative model to explain the TO result, such as a viscoelastic solidmodel;

• deriving the hydrodynamic equations for a supersolid, and obtaining thehydrodynamic mode and new sound speeds;

• proposing another method to detect the supersolid phase. We believe that lightscattering measurements can give some important information about detecting thesecond sound modes;

• studying the dynamics of vortices and dislocations in a supersolid.

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Figure 1-1. Resonant period change in temperature. The empty cell data and film dataare shifted up by 4,260 ns and 3290 ns, respectively. P ∗ = 971, 000 ns.Reprinted by permission from Macmillan Publishers Ltd: Nature [E. Kim andM. H. W. Chan, Nature 427, 225 (2004)], copyright (2004).

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Figure 1-2. Resonant period change in temperature for various concentration of 3Heimpurities. P ∗ = 971, 000 ns. Reprinted by permission from MacmillanPublishers Ltd: Nature [E. Kim and M. H. W. Chan, Nature 427, 225 (2004)],copyright (2004).

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Figure 1-3. Resonant period and the amplitude of oscillation of annular cell (panel A)and of blocked annular cell (panel B) as a function of temperature. τ ∗ is theresonant period at T = 300 mK. From E. Kim and M. H. W. Chan, Science305, 1941 (2004). Reprinted with permission from AAAS. Copyright (2004) byAAAS.

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Figure 1-4. Annealing effect in the resonant period. P ∗ = 5.428053 ms. Reprinted figure 3with permission from A. S. C. Rittner and J. D. Reppy, Phys. Rev. Lett. 97,165301 (2006). Copyright (2006) by the American Physical Society(http://link.aps.org/doi/10.1103/PhysRevLett.97.165301).

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Figure 1-5. Annealing effect in the inverse of Q-factor. Reprinted figure 4 with permissionfrom A. S. C. Rittner and J. D. Reppy, Phys. Rev. Lett. 97, 165301 (2006).Copyright (2006) by the American Physical Society(http://link.aps.org/doi/10.1103/PhysRevLett.97.165301).

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Figure 1-6. Specific heat peaks of 4He solid with different concentrations of 3Heimpurities. Open squares are of 10 p.p.m., blue triangles 0.3 p.p.m., and redcircles 1 p.p.b. The inset shows the data of the 10 p.p.m. sample beforesubtracting a constant term (dotted line) of 59 µJ mol−1 K−1. Reprinted bypermission from Macmillan Publisher Ltd: Nature [X. Lin, A. C. Clark, andM. H. W. Chan, Nature 449, 1025 (2007)], copyright (2007).

25

Figure 1-7. Shear modulus of 4He solid as a function of temperature. Data are shifted forbetter clarity. Reprinted by permission from Macmillan Publisher Ltd: Nature[J. Day and J. Beamish, Nature 450, 853 (2007)], copyright (2007).

26

Figure 1-8. Shear modulus change for various concentrations of 3He impurities. Reprintedby permission from Macmillan Publisher Ltd: Nature [J. Day and J. Beamish,Nature 450, 853 (2007)], copyright (2007).

27

CHAPTER 2VISCOELASTIC SOLIDS: ALTERNATIVE EXPLANATION OF NCRI

2.1 Equation of Motion for the Torsional Oscillator

In this section we investigate alternatives to the supersolid explanation for the

observed decrease in the resonant period of TO experiments. We believe that it is

worthwhile to speculate on these possibilities for several reasons. First, the NCRI is

accompanied by an increase in the damping, signaled by a peak in the inverse of quality

factor. A true supersolid transition in three dimensions will not involve any change in

damping. Second, the supersolid interpretation of the TO results is not supported by other

experiments as described in Chapter 1. These include the lack of pressure driven mass

flow, an anomalous increase of shear modulus with its dependence on the concentration

of 3He impurities similar to NCRIF [46], and annealing effects of NCRI related to sample

preparation [22, 24]. These effects suggest that the elastic properties of solid 4He must be

well understood because they might be responsible for the NCRI.

At the first step of this chapter, we start describing the dynamics of an empty TO.

Let us assume that the torsion bob is completely rigid. The rigid body motion of the

torsion bob results in a constant moment of inertia Iosc. The torsion rod is assumed to

be massless with a spring constant k that provides a restoring force proportional to the

angular displacement. Then the mode frequency of the undamped harmonic oscillator is

ωempty =√

k/Iosc. Let Mext(t) be the external torque applied on the torsion bob producing

angular displacements φ(t) from the equilibrium position. Since the motion of the TO is

along the axis of the torsion bob, φ(t) is sufficient to describe the dynamics of the TO. If

the motion is damped, the equation of motion for small φ of the torsion bob is

(Iosc

d2

dt2+ γosc

d

dt+ k

)φ(t) = Mext(t), (2–1)

where γosc is the dissipation coefficient. With damping the resonant frequencies become

ω = ±√

ω2empty −

γ2osc

4I2osc

. (2–2)

28

Now let’s fill the torsion cell with solid 4He. If solid 4He is perfectly rigid, it is

expected that the total moment of inertia of the TO will be increased by the moment of

inertia of solid 4He (IHe) with a no-slip boundary condition between the solid 4He and the

torsion cell. In this case, the mode frequency of undamped harmonic oscillator becomes

ω0 =

√k

Itot

, (2–3)

where Itot = Iosc + IHe. Therefore, the principal effect of loading solid 4He in the torsion

cell is to simply increase the total moment of inertia by a constant IHe. However, if we

assume that solid 4He is not completely rigid, we need to be careful in describing the

dynamics of the TO with solid 4He. In fact, angular displacements of the torsion bob

induce a shear stress in the solid 4He, and drag it along into motion. The generated stress

causes elastic shear deformations to propagate with a finite velocity throughout the solid

4He. The deviation from the rigid body motion results in an effective moment of inertia

which depends upon the driving frequency. Moreover, damping processes present in every

solid produce an effective frequency-dependent damping coefficient.

As discussed in Chapter 1, the modification of Eq. 2–1 for a TO with solid 4He was

made first by Nussinov et al. [49] by adding the back action torque M(t) due to solid 4He:

(Iosc

d2

dt2+ γosc

d

dt+ k

)φ(t) = Mext(t)−M(t). (2–4)

The motion of solid 4He affects the dynamics of the TO by exerting back again an torque

M(t) on the torsion bob. Following Nussinov et al. [49] the torque exerted by the solid

4He on the TO is taken to be related to the angular displacement φ through a linear

response function as

M(t) = −∫

dt′g(t, t′)φ(t′). (2–5)

The response function g(t) is referred to as the “back action term” by Nussinov et al. [49].

Let us assume time translational symmetry so that g(t, t′) depends on t − t′. Fourier

29

transforming Eq. 2–5, we obtain

g(ω) = −M(ω)/φ(ω). (2–6)

Additionally, we assume that the response of φ(t) to the external torque Mext(t) is linear

and related by the susceptibility χ(t); in Fourier space, we have

χ−1(ω) = −Ioscω2 − iγoscω + k − g(ω), (2–7)

where we have used Eq. 2–6. The zeros of the susceptibility, χ−1(ω) = 0, give the resonant

period

P =2π

<[ω], (2–8)

and the quality factor of the TO

Q−1 = −2=[ω]

<[ω]. (2–9)

Therefore, if the back action g(ω), and therefore the torque, depend on the temperature

T , the back action term contains all the information of the dynamics of the TO filled with

solid 4He. In the following section we calculate the back action terms by modeling solid

4He as a viscoelastic solid.

2.2 Properties of Viscoelastic Solids under Oscillatory Motion

In this section we study the general dynamical properties of a viscoelastic solid under

shear oscillation. A viscoelastic solid is a material that possesses both elastic and viscous

properties: its response to external disturbances becomes liquid-like or solid-like depending

upon the perturbing frequency. For simplicity we only consider isotropic viscoelastic solids

without pressure gradients.

In TO experiments, the shapes of torsion cells vary from a simple cylindrical cell

to complicated ones such as a blocked annulus. In this work we consider, for simplicity,

torsion cells with cylindrical symmetry. In Fig. 2-1 we show the geometry of a finite

cylinder torsion cell of radius R and height h. When a shear stress is applied to a

viscoelastic solid, due to the cylindrical symmetry, the only displacement is in the

30

azimuthal direction (uφ), and its magnitude depends on the distances from and along

the axis of oscillation. In this geometry the divergence of the displacement field vanishes.

For large driving frequencies, viscoelastic solids respond as viscous fluids to the shear

stress, and the Navier-Stokes equation for the velocity field vφ(t) is suitable to describe its

dynamics

ρ∂tvφ = η

[∂2

r +1

r∂r − 1

r2+ ∂2

z

]vφ, (2–10)

where ρ is the mass density and η is the shear viscosity. By contrast, the elastodynamic

equation for the displacement uφ(t) describes the solid-like dynamics for small frequencies

ρ∂2t uφ = µ

[∂2

r +1

r∂r − 1

r2+ ∂2

z

]uφ, (2–11)

where µ is the shear modulus. Equation 2–11 predicts the transverse sound speed of

cT =√

ρ/µ. Identifying vφ(t) = ∂tuφ(t), and combining these two equations together we

have the Voigt model, also called the Kelvin model, of viscoelastic solids [51, 52]

ρ∂2t uφ = (µ + η∂t)

[∂2

r +1

r∂r − 1

r2+ ∂2

z

]uφ. (2–12)

From Eq. 2–12 we also identify a relaxation time defined as τ = η/µ. When ωτ ¿1, a viscoelastic solid responds elastically whereas for ωτ À 1 it responds viscously.

Equation 2–12 can be generalized to a viscoelastic solid model of a form like Eq. 2–11 with

a frequency dependent complex shear modulus µ(ω) with the following properties: for

small ω

limω→0

µ(ω) = µ0, (2–13)

limω→0

=[µ(ω)]

ω= −η0, (2–14)

and for large ω

limω→∞

<[µ(ω)] = µ∞, (2–15)

limω→∞

µ(ω)

ω= −iη∞. (2–16)

The shear modulus of the Voigt model of viscoelastic solids is µ(ω) = µ− iωη.

31

We study the dynamical properties of viscoelastic solids by solving Eq. 2–12 for

the displacement field, given an oscillatory boundary condition. From the obtained

displacement field we calculate the shear stresses (σφr and/or σφz) on the boundary, and

the torque M(t). Then the effective description of a system in oscillatory motion can be

investigated by analyzing the effective moment of inertia

Ieff(ω) = − 1

ω2φ0

<[M(t) exp(iωt)

], (2–17)

and the effective damping coefficient

γeff(ω) = − 1

ωφ0

=[M(t) exp(iωt)

], (2–18)

where φ0 is the initial angular displacement. The viscoelastic solid model predicts that, for

small ω, γeff(ω) vanishes and Ieff(ω) becomes the moment of inertia of rigid body IRB. In

the following we present the results of calculation for three different geometries: infinite

cylinder, finite cylinder and infinite annulus (Appendix A for details).

2.2.1 Infinite Cylinder

When an infinite cylinder of viscoelastic solid of radius R is oscillating with a

frequency ω, its effective moment of inertia and the effective damping coefficient are

Ieff(ω) = IRB<[H

(τEω√

1− iωτ

)], (2–19)

γeff(ω) = ωIRB=[H

(τEω√

1− iωτ

)], (2–20)

where IRB = πρR4h/2,

H(x) =4J2(x)

xJ1(x), (2–21)

and a characteristic time τE = R/cT . In Figs. 2-2 and 2-3 we show the effective moment

of inertia and the effective damping coefficients, respectively. There are elastic resonances

that appear as peaks in the damping coefficients and an effective moment of inertia that

increases and then decreases to a negative value. To understand the negative effective

32

moment of inertia consider the displacement field with τ = 0,

uφ = Rφ0J1(ωr/cT )

J1(ωτE)exp(−iωt). (2–22)

The resonant frequencies are given by the zeros of J1(ωτE) in Eq. 2–22. In Fig. 2-4 we

show how the normalized displacement vector evolves during a half cycle when ω ¿ 1/τE.

In such a limit, the motion becomes purely like a rigid body motion (straight lines in Fig.

2-4). Increasing ω from 0, the deviation of the amplitude of displacement field from the

rigid body motion becomes apparent, developing an effectively larger out-of-phase motion

with the applied shear stress (Fig. 2-5). As a result, elastic solids have an effectively

larger moment of inertia than the rigid body. When ω passes through the first resonant

frequency, the direction of the displacement field changes. Figure 2-5 is a schematic

showing the in-phase motion, thus the apparent moment of inertia becomes negative. We

estimate the first resonance frequency ω1 ' 3.83cT /R = 0.2 MHz for cT ∼ 300 m/s and

R ∼ 0.5 cm. In experimental conditions, driving frequencies (∼ 1 KHz) are well below the

first resonance frequency to observe elastic resonances. The effect of viscosity (for large τ)

is to smear out the resonances.

2.2.2 Finite Cylinder

The effective moment of inertia for a finite cylinder of viscoelastic solid of radius R

and height h is

Icyleff (ω)

IRB

=8

π2

∞∑m=1

<{[

1

(2m− 1)2− π2R2

h2α2m

]H(iαm)

}+

8R2

h2

∞∑m=1

<{

1

α2m

}, (2–23)

where αm =√

(2m− 1)2π2R2/h2 − ω2τ 2E/(1− iωτ) and IRB = πρR4h/2. On the other

hand, its effective damping coefficient is

γfin cyleff (ω)

ωIRB

=8

π2

∞∑m=1

={[

1

(2m− 1)2− π2R2

h2α2m

]H(iαm)

}+

8R2

h2

∞∑m=1

={

1

α2m

}. (2–24)

Figures 2-7 and 2-8 illustrate the effective moment of inertia and the effective damping

coefficient of finite cylinder, respectively. The presence of the top and bottom of the finite

33

cylinder makes the mode structure of the elastic oscillation more complicated because

there is one more degree of freedom along the z-axis. Consequently, the position of the

resonant frequencies changes for different dimensions of the cylinder. Increasing the height

h or the radius R results in decreasing the resonant frequency.

2.2.3 Infinite Annulus

In the case of an infinite annular viscoelastic solid of inner radius Ri and outer radius

R, the effective moment of inertia is

I inf anneff (ω) = <

{2πρhR2

qE

[AJ2(qER) + BN2(qER)

]}

−<{

2πρhR2i

qE

[AJ2(qERi) + BN2(qERi)

]}, (2–25)

and the effective dampting coefficient is

γinf anneff (ω) = =

{2πρhR2ω

qE

[AJ2(qER) + BN2(qER)

]}

−={

2πρhR2i ω

qE

[AJ2(qERi) + BN2(qERi)

]}, (2–26)

where IRB = πρh(R4 −R4i )/2 and

A =RiN1(qER)−RN1(qERi)

J1(qERi)N1(qER)− J1(qER)N1(qERi), (2–27)

B =RiJ1(qER)−RJ1(qERi)

N1(qERi)J1(qER)−N1(qER)J1(qERi). (2–28)

In Figs. 2-9 and 2-10 we show the effective moment of inertia and the effective damping

coefficients of the infinite annulus of viscoelastic solid, respectively. We observe the same

trend of resonant frequencies as in the previous cases that resonant frequencies decrease as

the gap of the annulus is increased.

2.3 Torsional Oscillator with Viscoelastic Solids

Now we study the TO dynamics by modeling solid 4He as a viscoelastic solid.

Our approach is to assume that solid 4He at low temperatures, in the range that TO

34

experiments are done, becomes an isotropic viscoelastic solid which has only two complex

elastic moduli dependent on frequency. This is a simplification: solid 4He has a hcp

crystalline structure, and has five independent elastic moduli.

In Section 2.2, we investigated the dynamical response of an isotropic viscoelastic

solid to oscillatory disturbances by studying its effective moment of inertia and effective

damping coefficient. Our main conclusion is that both quantities change with the driving

frequency ω and a relaxation time τ . Hence, the resonant frequency and the quality factor

of the TO depend on the relaxation time as well.

As explained in Section 2.1, to get the resonant period and the quality factor for each

case, we need to find the poles of Eq. 2–7 employing the back action term of a viscoelastic

solid with a given geometry [Appendix A for the calculation of g(ω)]. But under all

experimental conditions, the wavelength of the transverse sound mode is much larger than

the typical dimensions of torsion cells (|qER|, |qEh| ¿ 1). In this regime, the back action

terms, Eqs. A–8, A–29 and A–38, can be cast into a single form

g(ω) = ω2IHe +ρR2ω4IHeF (h/R)

24µ(1− iωτ), (2–29)

where a function F (x) is defined such that for an infinite cylinder F inf cyl = 1, for an

infinite annulus

F inf ann =1

R2

(R−Ri)2(R + Ri)

2

R2 + R2i

, (2–30)

and for a finite cylinder

F fin cyl(x) = −192x2

π4

∞∑m=1

1

(2m− 1)4H

((2m− 1)π

xi

), (2–31)

where H(x) ≡ H(x) − 1. In Figs. 2-11 and 2-11 we show F inf ann(R/Ri) and F fin cyl(h/R),

respectively. Note that 0 ≤ F inf ann, F fin cyl ≤ 1.

Now we are in a position to calculate the resonant period and the inverse of the

quality factor of the TO. With the back action term, Eq. 2–29, the susceptibility of

35

Eq. 2–7 reduces to

χ−1(ω) ' −Itotω2 − iγoscω + k − ρR2ω4IHeF (h/R)

24µ(1− iωτ). (2–32)

For simplicity we assume that there is no damping from the oscillator (γosc = 0). In fact,

this assumption is acceptable because for a high quality TO (Q ∼ 106) the damping of the

oscillator (γosc) can be neglected. Since IHe/Iosc ∼ 10−3 under experimental conditions, the

contribution due to viscoelasticity can be treated as a perturbation from the rigid body

motion, whose resonant frequency is ω0 given in Eq. 2–3. We expand the poles about ω0

such that χ−1(ω = ω0 + ω1) = 0 with ω1 ¿ ω0. Then we obtain

ω1 = −ρR2IHeF (h/R)

48µItot

ω20

(1− iω0τ). (2–33)

In the viscoelastic solid model, the resonant period and the inverse of quality factor of TO

are, using Eqs. 2–8 and 2–9,

∆P ≡ P − P0 ' πρR2ω0IHeF (h/R)

24µItot

1

1 + τ 2ω20

, (2–34)

and

∆Q−1 ' ρR2ω20IHeF (h/R)

24µItot

ω0τ

1 + τ 2ω20

, (2–35)

where P0 = 2π/ω0. Equations 2–34 and 2–35 are our central results in this chapter.

To interpret these results, first notice that when τ passes through 1/ω0, there would

be a peak in Q−1 and a drop in the resonant period whose sizes are given by

∆Q−1max = ∆P/P0 =

ρR2ω20IHeF (h/R)

48µItot

. (2–36)

Therefore, the viscoelastic model predicts that ∆Q−1max/(∆P/P0) = 1. However, the TO

experiments showed that the maximum dissipation ∆Q−1max is lower than (∆P/P0). The

typical experimental value varies around 0.1 [24, 53] reaching up to 0.65 [22] and down

to 0.01 [54]. This implies that we can only fit either the peak in ∆Q−1max or the decrease

in ∆P taking the shear modulus as a fitting parameter. Other quantities such as the

36

radius and moment of inertia are controllable by the experimental setup. Second, since

the sizes of F fin cyl and F inf ann are less than the unity (Figs. 2-11 and 2-12), the reduction

of dimension of the system from the infinite cylinder geometry results in decreased period

shifts and sizes of peak in ∆Q−1. Consequently, in fitting the experimental data, the shear

modulus (fitting parameter) of the finite cylinder turns out to be greater than that of

the infinite cylinder by a factor of 1/F . Third, following Nussinov et al. [49], we take the

relaxation time as

τ = τ0 exp

(E0

kBT

), (2–37)

and use τ0 and E0 as fitting parameters. As the temperature is lowered below E0/kB, the

relaxation time τ becomes larger than 1/ω0, and solid 4He behaves like a viscous fluid. In

Figs. 2-13 - 2-18 we show change in the resonant period and in the inverse of quality factor

measured in Clark et al. [53] and their fitting using Eqs. 2–34 and 2–35, respectively. We

have chosen to fit ∆Q−1max using the infinite cylinder model. In Table 2-1 we list the shear

modulus, the activation energy E0, and τ0 used in fitting. As mentioned earlier, the change

in the resonant period of the viscoelastic model only accounts for about 10% of what is

actually observed. This might imply that the unexplained part is caused by a supersolid

transition. Moreover, it is useful to study the dynamical effect of dislocations on the

shear modulus of normal solids because the motion of dislocations can change the elastic

properties of regular crystals, and determine the characteristic time of the viscoelastic

solid model (e.g., Ref. [55]).

2.4 Possible Connection between Anomalies in Shear Modulus and NCRI

In this section, we now briefly discuss a possible relation between TO experiments and

shear modulus experiments. In Section 2.3, we have studied how the viscoelastic model

explains the observed signatures in the resonant period and Q−1 of TO experiments. We

have shown that the relaxation time τ that decreases exponentially in temperature is

responsible for TO results with a constant shear modulus of solid 4He. However, Day

and Beamish reported about 10% increase of shear modulus at the same temperature

37

range that TO experiments are performed [46]. We focus only on elasticity because in the

range of the employed frequency to measure the shear modulus the viscosity of solid 4He

could not be probed simultaneously. As we discussed in Chapter 1, Day and Beamish also

investigated the resonance effect in the cavity where the apparatus to measure the shear

modulus was embedded. They found behaviors that resemble the NCRI. However, the

relation between the dissipation peak observed in the cavity resonance and the viscosity

present in the gap between two transducers is not yet clear.

Let us examine first the effective moment of inertia of an infinite cylinder. For small

ω, we have I inf cyl(µ) ' IRB[1 + (ρω2R2/24µ)] without viscosity. The second term is the

correction to the rigid body value due to a finite shear modulus. Since the correction term

in I inf cyl(µ) is inversely proportional to µ, at a fixed frequency I inf cyl(µ) decreases as µ(ω)

increases. Therefore an increase of shear modulus will enhance the change in resonant

period. This could be a connection between the increasing shear modulus observed by Day

and Beamish and NCRI.

We calculate the actual change in the resonant period induced by an increase of shear

modulus. The resonant period of TO for an elastic solid of a finite shear modulus can be

easily obtained by setting τ to zero in Eq. 2–34. The result is

P (µ) = P0

[1 +

ρR2ω20IHeF (h/R)

48µItot

]. (2–38)

One can verify this result finding the zeros of Eq. 2–32 without γ and τ . Let’s consider a

small change in shear modulus ∆µ from µ0. Then the fractional change in P (µ) due to ∆µ

becomes

∆P

P (µ0)' −ρR2ω2

0IHeF (h/R)

48µItot

∆µ

µ0

. (2–39)

For Ref. [24], Eq. 2–39 predicts only 0.8×10−5 % decrease change in the resonant period

whereas the measured change is about 2.6×10−3 %. Therefore, we find that an increasing

shear modulus accompanies a decrease in the resonant period of a TO; however, the elastic

solid model does not account for all the change observed in experiments.

38

Figure 2-1. Schematic illustration of TO and geometry of a torsion cell.

39

-8

-6

-4

-2

0

2

4

6

8

10

0 2 4 6 8 10 12 14 16 18 20

I eff

/ IR

B

ω τE

τ / τE = 1τ / τE = 5

τ / τE = 1/10τ / τE = 1/100

Figure 2-2. Effective moment of inertia of an infinite cylinder of viscoelastic solid as afunction of the driving frequency ω.

0

10

20

30

40

50

60

0 2 4 6 8 10 12 14 16 18 20

γ eff

τ E /

I RB

ω τE

τ / τE = 1τ / τE = 5

τ / τE = 1/10τ / τE = 1/100

Figure 2-3. Effective damping coefficient of an infinite cylinder of viscoelastic solid as afunction of the driving frequency ω.

40

-1

-0.5

0

0.5

1

0 0.2 0.4 0.6 0.8 1

u(r)

/ R

φ0

r / R

t = 0t = π / 4 ωt = π / 2ω

t = 3π / 4ωt = π / ω

Figure 2-4. Displacement vector in a half cycle for ω ¿ 1/τE.

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

0 0.2 0.4 0.6 0.8 1

u(r)

/ R

φ0

r / R

t = 0t = π / 4 ωt = π / 2ω

t = 3π / 4ωt = π / ω

Figure 2-5. Displacement vector in a half cycle for ω = 3/τE.

41

-10

-8

-6

-4

-2

0

2

4

6

8

10

0 0.2 0.4 0.6 0.8 1

u(r)

/ R

φ0

r / R

t = 0t = π / 4 ωt = π / 2ω

t = 3π / 4ωt = π / ω

Figure 2-6. Displacement vector in a half cycle for ω = 4/τE.

-4

-3

-2

-1

0

1

2

3

4

5

6

0 1 2 3 4 5 6 7 8 9 10

I eff

/ IR

B

ω τE

h / R = 0.5h / R = 1h / R = 2

Figure 2-7. Effective moment of inertia of a finite cylinder of viscoelastic solid as afunction of the driving frequency ω with τ/τE = 1/100.

42

0

5

10

15

20

25

30

35

40

45

0 1 2 3 4 5 6 7 8 9 10

γ eff

τ E /

I RB

ω τE

h / R = 0.5h / R = 1h / R = 2

Figure 2-8. Effective damping coefficient of a finite cylinder of viscoelastic solid as afunction of the driving frequency ω with τ/τE = 1/100.

-60

-40

-20

0

20

40

60

0 10 20 30 40 50 60 70 80 90 100

I eff

/ IR

B

ω τE

Ri / R = 0.9Ri / R = 0.7 Ri / R = 0.5

Figure 2-9. Effective moment of inertia of an infinite annulus of viscoelastic solid as afunction of the driving frequency ω with τ/τE = 1/1000.

43

0

100

200

300

400

500

600

700

800

900

0 10 20 30 40 50 60 70 80 90 100

γ eff

τ E /

I RB

ω τE

Ri / R = 0.9Ri / R = 0.7Ri / R = 0.5

Figure 2-10. Effective damping coefficient of an infinite annulus of viscoelastic solid as afunction of the driving frequency ω with τ/τE = 1/1000.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 2 4 6 8 10 12 14 16 18 20

F

h/R

Figure 2-11. F (x) of finite cylinder.

44

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

F

Ri / R

Figure 2-12. F (x) of infinite annulus.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

5

10

15

∆ P

[ns]

T [K]

Clark et al.infinite cylinder modelfinite cylinder model

Figure 2-13. Resonant period of the blocked capillary sample of BeCu TO. Experimentaldata were adapted with permission from A. C. Clark, J. T. West, and M. H.W. Chan, Phys. Rev. Lett. 99, 135302 (2007). Copyright (2007) by theAmerican Physical Society(http://link.aps.org/doi/10.1103/PhysRevLett.99.135302).

45

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

∆ (Q

−1 )

× 10

6

T [K]

Clark et al.infinite cylinder modelfinite cylinder model

Figure 2-14. Inverse of Q-factor of the blocked capillary sample of BeCu TO.Experimental data were adapted with permission from A. C. Clark, J. T.West, and M. H. W. Chan, Phys. Rev. Lett. 99, 135302 (2007). Copyright(2007) by the American Physical Society(http://link.aps.org/doi/10.1103/PhysRevLett.99.135302).

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

1

2

3

4

5

6

7

8

9

10

∆ P

[ns]

T [K]

Clark et al.infinite cylinder modelfinite cylinder model

Figure 2-15. Resonant period of the annealed blocked capillary sample of BeCu TO.Experimental data were adapted with permission from A. C. Clark, J. T.West, and M. H. W. Chan, Phys. Rev. Lett. 99, 135302 (2007). Copyright(2007) by the American Physical Society(http://link.aps.org/doi/10.1103/PhysRevLett.99.135302).

46

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

1

2

3

4

5

6

7

8

9

10

∆ (Q

−1 )

× 10

7

T [K]

Clark et al.infinite cylinder modelfinite cylinder model

Figure 2-16. Inverse of Q-factor of the annealed blocked capillary sample of BeCu TO.Experimental data were adapted with permission from A. C. Clark, J. T.West, and M. H. W. Chan, Phys. Rev. Lett. 99, 135302 (2007). Copyright(2007) by the American Physical Society(http://link.aps.org/doi/10.1103/PhysRevLett.99.135302).

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

1

2

3

4

5

6

∆ P

[ns]

T [K]

Clark et al.infinite cylinder modelfinite cylinder model

Figure 2-17. Resonant period of the constant temperature sample of BeCu TO.Experimental data were adapted with permission from A. C. Clark, J. T.West, and M. H. W. Chan, Phys. Rev. Lett. 99, 135302 (2007). Copyright(2007) by the American Physical Society(http://link.aps.org/doi/10.1103/PhysRevLett.99.135302).

47

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

0.5

1

1.5

2

2.5

3

3.5

4

∆ (Q

−1 )

× 10

7

T [K]

Clark et al.infinite cylinder modelfinite cylinder model

Figure 2-18. Inverse of Q-factor of the constant temperature sample of BeCu TO.Experimental data were adapted with permission from A. C. Clark, J. T.West, and M. H. W. Chan, Phys. Rev. Lett. 99, 135302 (2007). Copyright(2007) by the American Physical Society(http://link.aps.org/doi/10.1103/PhysRevLett.99.135302).

Table 2-1. Fitting parameters.

sample µ [g/cm s2] τ0 [µs] E0/kB [mK]blocked capillary 0.34× 108 2.86 158

annealed blocked capillary 0.75× 108 0.21 395constant temperature 1.53× 108 3.72 166

48

CHAPTER 3NON-DISSIPATIVE HYDRODYNAMICS OF A MODEL SUPERSOLID

3.1 Variational Principle in Supersolids

In this chapter we investigate the hydrodynamics of a supersolid whose properties

are governed by conservation laws and broken symmetries. As we discussed in Chapter 1,

the hydrodynamic equations of motion for supersolids are first derived by Andreev and

Lifshitz [4]. In their model point defects, such as interstitials and vacancies, in bosonic

solids undergoes a Bose-Einstein condensation and a crystal becomes a supersolid. The

hydrodynamics of supersolids was studied further by Saslow [7] and Liu [8].

We use the variational principle to derive a Lagrangian for the supersolid and the

non-dissipative hydrodynamic equations of motion. The variational principle was often

used in the literature to obtain the hydrodynamics of various continuum systems: normal

fluids [56–58], superfluids [56, 59–63], normal solids [57, 64], liquid crystals [65], and so on.

Let us start this section by giving a plain example of variational principle applied to ideal

fluids to show the simplicity of the method.

3.1.1 Introduction to the Variational Principle in Continuum Mechanics

We write a Lagrangian density L for isentropic ideal fluids, which are irrotational,

inviscid, and incompressible, in the Eulerian description1 as follows

LIF =1

2ρv2 − UIF(ρ), (3–1)

where ρ is the mass density, v the velocity field, and UIF the internal energy density. The

internal energy density satisfies the thermodynamic relation

dUIF = µdρ, (3–2)

1 The Eulerian description of the motion of continuum medium is a field descriptionwith a coordinate system fixed in space. In this Eulerian description, the properties offlow are functions of both space and time. An alternative is the Lagrangian description inwhich the motion of individual particles is traced as a function of time.

49

where µ the chemical potential per unit mass. All the dynamical variables are taken to be

functions of the position x and time t.

The variational principle states that the equations of motion can be derived by

minimizing the action S of a Lagrangian density L

S =

∫dt

∫dx L, (3–3)

with respect to all the dynamical variables i.e., ρ and v in this example. However,

the variation of the action with the Lagrangian density in Eq. 3–1 with respect to the

dynamical variables ρ and v does not provide us with the right equations of motion - they

are not independent and restricted by side conditions such as conservation laws. The easy

way to see this is to take the variation of the action of Eq. 3–1 with respect to v: the

resulting equation of motion is a trivial and irrelevant one (v = 0). In order to overcome

this problem, the side constraints must be included into the Lagrangian density. As one

knows, for fluids the total mass is conserved, and this conservation law is expressed in the

continuity equation

∂tρ + ∂i(ρvi) = 0. (3–4)

Since we are considering isentropic ideal fluids, the mass conservation is the only

side condition to be taken into account. In fact the momentum is conserved as well.

Nonetheless, the momentum conservation law is not a side condition, but the byproduct

of the variational principle. The continuity equation is incorporated into the Lagrangian

density Eq. 3–1 by using a Lagrange multiplier φ:

LIF =1

2ρv2 − UIF(ρ) + φ

[∂tρ + ∂i(ρvi)

]. (3–5)

Then the equations of motion are obtained by taking variations of the action with respect

to ρ and v. With Eq. 3–5 we obtain

ρvi − ρ∂iφ = 0, (3–6)

50

1

2v2 − ∂UIF

∂ρ− Dφ

Dt= 0, (3–7)

where D/Dt ≡ ∂t + vi∂i. The other trivial equation of motion is the continuity equation

which one obtains by taking the variation with respect to the Lagrange multiplier φ.

Equation 3–6 implies that

vi = ∂iφ, (3–8)

and this confirms that there is no vorticity (∇ × vs = 0) for ideal fluids, as expected.

We can express Eq. 3–7 in more familiar form by taking its gradient. Since in thermal

equilibrium the change in the chemical potential per unit mass µ is related to the change

in the pressure P by the Gibbs-Duhem relation as follows

ρdµ = dP, (3–9)

we obtain

ρDvi

Dt= −∂iP. (3–10)

Equation 3–10 is the Euler equation for ideal fluids without external forces. Consequently,

we derived the hydrodynamics describing ideal fluids using the variational principle: the

continuity equation and the Euler equation.

As we showed in the analysis above, the variational principle provides us with

the Lagrangian density for a continuum system; Eq. 3–5 is the Lagrangian density for

isentropic ideal fluids. Moreover, as long as the boundary contributions are negligible in

the action, the analysis above is equivalent to that with a Lagrangian density

LIF = −ρ∂tφ− 1

2ρ(∂iφ)2 − UIF(ρ). (3–11)

We have integrated by parts the action of Eq. 3–5, and replaced the velocity with φ

using Eq. 3–8. This Lagrangian density also can be derived from the time-dependent

Gross-Pitaevskii Lagrangian density for the superfluid which is

LSF =i~2

[ψ∗∂tψ − ψ∂tψ∗]− 1

2m(i~∂j)ψ

∗(−i~∂j)ψ − µψ∗ψ − g

2(ψ∗ψ)2, (3–12)

51

where ~ = h/2π with h the Planck constant, and m the mass of the superfluid component.

Taking ψ =√

neiθ with the number density n, and the velocity vi = (~/m)∂iθ, the

Lagrangian density Eq. 3–12 can be written as follows:

LSF =i~2

∂tn− ~n∂tθ − ~2

2mn(∂iθ)

2 − ~2

8mn(∂in)2 − µn− g

2n2. (3–13)

Since the first term is a total derivative of an analytical field, it does not contribute to the

dynamics and can be neglected. Hence we can identify that ρ = mn and φ = (~/m)θ;

therefore, we have shown that the time-dependent Gross-Pitaevskii Lagrangian density is

the same as the Lagrangian density for isentropic ideal fluids Eq. 3–11 with a particular

form of the internal potential energy density.

3.1.2 Isotropic Supersolids

Let us now turn on the variational principle applied for supersolids which have

both crystalline order and superfluidity. In this part we start with the simple case of an

isotropic supersolid whose Lagrangian density in the Eulerian description is given by

LSS =1

2ρsvs

2 +1

2(ρ− ρs)vn

2 − USS(ρ, ρs, s, Rij), (3–14)

where ρs is the density of super-components, ρ the total density, vs the velocity of

super-components, vn the velocity of normal components, s the entropy density, and

Rij ≡ ∂Rj/∂xi, (3–15)

the deformation tensor with R and x being Lagrangian and Eulerian coordinates,

respectively. The first two terms in the Lagrangian density are the kinetic energy

densities of the super-component and normal components, and the last term is the

internal potential energy density. Because of the isotropy, the Lagrangian density of

an isotropic supersolid is very similar to the Lagrangian density of superfluid, e.g.

the Lagrangian density of a superfluid used by Zisel (Eq. 2.1 in Ref. [59]). The only

exception is that USS depends on the deformation tensor. In contrast to fluids, the elastic

52

energy which is proportional to the Eulerian strain tensor (δij − RikRjk)/2 arises for

solids. In linear elasticity, the Eulerian strain tensor turns into a more familiar form

(δij −RikRjk)/2 = (∂iuj + ∂jui)/2− ∂iuk∂juk/2 with the displacement vector u.

Given the Lagrangian density, Eq. 3–14, the total energy density for supersolids is

defined as the sum of the kinetic energy densities and the potential energy density

ESS =1

2ρsvs

2 +1

2(ρ− ρs)vn

2 + USS(ρ, ρs, s, Rij). (3–16)

This total energy density can be related to the energy density ε measured in the frame

where the super-component is at rest as follows:

ESS =1

2ρvs

2 + (ρ− ρs)(vni − vsi)vsi + ε. (3–17)

As ε is a Galilean invariant, it must depend on Galilean invariant quantities [66]. Similar

to the two-fluid model for superfluid [66], ε has a thermodynamic relation [4]

dε = Tds + µdρ− λikdRik + (vni − vsi)d[(ρ− ρs)(vni − vsi)], (3–18)

with λik the stress tensor. We now can obtain the thermodynamic relation for ESS by

differentiating Eq. 3–17 and replacing Eq. 3–18 for dε. The result is

dESS = Tds− λikdRik − (vni − vsi)vnidρs +

[µ +

1

2(2vn

2 − 2vnivsi + vs2)

]dρ

+ρsvsidvsi + (ρ− ρs)vnidvni. (3–19)

This thermodynamic relation was used by Saslow [7] and by Liu [8] in deriving the

hydrodynamics of a supersolid provided that the chemical potential per unit mass is given

by

µSaslow, Liu = µ− vnivsi + vs2/2. (3–20)

Finally from Eq. 3–16 and Eq. 3–19, we get the thermodynamic relation for USS:

dUSS = Tds +

[µ +

1

2(vni − vsi)

2

]dρ− 1

2(vni − vsi)

2dρs − λikdRik. (3–21)

53

Note that the difference between a superfluid and an isotropic supersolid is the dependence

of the potential energy on the deformation tensor Rij.

As illustrated in Section 3.1.1, the dynamical variables of a supersolid in the

Lagrangian density, Eq. 3–14, are not independent each other. The side conditions

relating the dynamical variables must be included in the Lagrangian density in order to

derive the correct equations of motion. There are two important conservation laws for a

supersolid: the mass conservation law and the entropy conservation law. Once again the

momentum conservation law is not a side condition to be imposed, but a consequence of

the variational principle. The conservation of mass is expressed by the continuity equation

∂tρ + ∂i

[ρsvsi + (ρ− ρs)vni

]= 0, (3–22)

with the total current

ji = ρsvsi + (ρ− ρs)vni. (3–23)

For the entropy conservation we have

∂ts + ∂i(svni) = 0. (3–24)

One should note that in the equation for the conservation of entropy, Eq. 3–24, only the

velocity of the normal component are involved because the entropy is carried solely by the

normal component. In addition to the two conservation laws above there is one more side

condition, called Lin’s constraint, to be included [56]:

DnRi

Dt= 0, (3–25)

where Dn/Dt ≡ ∂t + vni∂i. Lin’s constraint for solids is an expression of the fact that the

Lagrangian coordinates, (i.e., the initial positions of particles) do not change along the

paths of the normal component. The same condition is also used for an isentropic normal

fluid to generate vorticity whereas the entropy conservation equation produces vorticity for

the normal fluid [56]. Let’s incorporate these constraints into the Lagrangian density using

54

Lagrangian multipliers φ, α and β. Then we have

LSS =1

2ρsvs

2 +1

2(ρ− ρs)vn

2 − U(ρ, ρs, s, Rij) + φ

{∂tρ + ∂i

[ρsvsi + (ρ− ρs)vni

]}

+α

[∂ts + ∂i(svni)

]+ βi

[∂t(sRi) + ∂j(sRivnj)

], (3–26)

where we have used the Lin’s constraint combined with the conservation equation of

entropy making it in a form of continuity equation. It is also possible to use the continuity

equation, instead of the conservation equation of entropy, combining the Lin’s constraint.

We are now in a position to get the equations of motion. We take the variations of

the action with respect to all the dynamical variables. We obtain

• δρ1

2vn

2 − ∂USS

∂ρ− Dnφ

Dt= 0; (3–27)

• δρs1

2vs

2 − 1

2vn

2 − ∂USS

∂ρs

− (vsi − vni)∂iφ = 0; (3–28)

• δsDnα

Dt+ Ri

Dnβi

Dt+

∂USS

∂s= 0; (3–29)

• δvsi

ρsvsi − ρs∂iφ = 0; (3–30)

• δvni

(ρ− ρs)vni − (ρ− ρs)∂iφ− s∂iα− sRj∂iβj = 0; (3–31)

• δRi

sDnβi

Dt− ∂j

(∂USS

∂Rji

)= 0. (3–32)

Obviously the variations with respect to the Lagrange multipliers recover the imposed

constraints, Eqs. 3–22 through 3–25. In the following we show that the derived equations

of motion can be rearranged to recover the hydrodynamics of a supersolid developed by

Andreev and Lifshitz [4], by Saslow [7], and by Liu [8].

55

We obtain first the Clebsch potential representation [57] for the velocity of the

super-component, from Eq. 3–30,

vsi = ∂iφ, (3–33)

and the velocity of the normal component, from Eq. 3–31,

vni − vsi =s

ρ− ρs

(∂iα + Rj∂iβj

). (3–34)

With these representations we find first that

∇× vs = 0. (3–35)

Consequently the velocity of the super-component is irrotational as one expects for a

superfluid without vortices: vs is only longitudinal. On the other hand, we also find that

vorticity can be generated for vn:

∇× vn = ∇(

s

ρ− ρs

)×∇α + ∇

(sRi

ρ− ρs

)×∇βi. (3–36)

In fact it is also possible to include systematically the transverse part of vs by introducing

a second Lin’s constraint (Appendix B).

The use of Eq. 3–33 in Eq. 3–28 corroborates one of the thermodynamic relations

given in Eq. 3–21,

∂USS

∂ρs

= −1

2(vni − vsi)

2. (3–37)

Taking the gradient of Eq. 3–27 and using the Clebsh representation of vs, Eq. 3–33, we

get the Josephson equation for vs

Dsvsi

Dt= −∂i

(∂USS

∂ρ

)− ∂i

(∂USS

∂ρs

)

= −∂iµ, (3–38)

where Ds/Dt ≡ ∂t + vsi∂i. In deriving Eq. 3–38, we have used Eq. 3–37 and vsj∂ivsj =

vsj∂jvsi because vs is irrotational.

56

The Euler equation for vn can also be derived. We take Dn/Dt of Eq. 3–31, and use

the derived equations of motion to eliminate the Clebsch potentials of the relative velocity.

The resulting equation is the Euler equation for vn:

(ρ− ρs)Dnvni

Dt= −(ρ− ρs)∂i

(∂USS

∂ρ

)− (ρ− ρs)∂i

(∂USS

∂ρs

)− ∂j

(∂USS

∂Rjk

)Rik

−s∂i

(∂USS

∂s

)− 1

2(ρ− ρs)∂i(vnj − vsj)

2 −[∂tρs + ∂j(ρsvsj)

]vsi

−{

∂t(ρ− ρs) + ∂j

[(ρ− ρs)vnj

]}vni

= −(ρ− ρs)∂iµ− ∂jλjkRik − s∂iT − 1

2(ρ− ρs)∂i(vnj − vsj)

2

−{

∂t(ρ− ρs) + ∂j

[(ρ− ρs)vnj

]}(vni − vsi), (3–39)

where we have used an useful identity of the convective derivative,

D(a∂ib)

Dt= ∂ib

Da

Dt+ a∂i

(Db

Dt

)− a∂jb∂ivj. (3–40)

Finally the Josephson equation and the Euler equation can be put together into the

momentum conservation equation,

∂t

[ρsvsi + (ρ− ρs)vni

]= −∂k

{ρsvsivsk + (ρ− ρs)vnivnk

−[ε− Ts− µρ− ρn(vnj − vsj)

2

]δik − λkjRij

},

(3–41)

because λjk∂iRjk + ∂jλjk∂iRk = ∂j(λjkRik). In deriving Eq. 3–41 we have used the mass

conservation equation and the thermodynamic equation for ε.

As shown in the example of ideal fluids, the Lagrangian density for the isotropic

supersolid can also be reduced in a compact form by using the Clebsh representation of vs.

Neglecting the boundary contributions, the Lagrangian density Eq. 3–26 is equivalent to

57

the following Lagrangian density:

LSS =1

2ρsvs

2 +1

2(ρ− ρs)vn

2 − USS(ρ, ρs, s, Rij)− ρ∂tφ−[ρsvsi + (ρ− ρs)vni

]∂iφ

−sDnα

Dt− sRi

Dnβi

Dt. (3–42)

The next step is to use Eqs. 3–29 through 3–31 to replace vs, α and β. Then we obtain

the Lagrangian density for an isotropic supersolid

LSS = −ρ∂tφ− 1

2ρ(∂iφ)2 +

1

2(ρ− ρs) (vni − ∂iφ)2 − f(ρ, ρs, T, Rij), (3–43)

where f = USS − sT . This Lagrangian density used in the derivation of the hydrodynamics

of a supersolid has many other applications, including the calculation of correlation

functions and collective modes, and the study of the dynamics of dislocations and vortices.

3.1.3 Anisotropic Supersolids

Having completed the variational principle in an isotropic supersolid, let us extend

the analysis to an anisotropic supersolid. Because of the anisotropy, the density of the

super-component becomes a tensor ρsij which is another distinction of solids from fluids.

Then the Lagrangian density for an anisotropic supersolid in Eulerian description is given

by

LSS =1

2ρsijvsivsj +

1

2(ρδij − ρsij)vnivnj − USS(ρ, ρsij, s, Rij). (3–44)

The internal potential energy satisfies the thermodynamic relation

dUSS = Tds +

[µ +

1

2(vni − vsi)

2

]dρ− λikdRik − 1

2(vni − vsi)(vnj − vsj)dρsij. (3–45)

The derivation of this thermodynamic relation is identical to the calculation of Eq. 3–21

for isotropic supersolids.

The constraints to be imposed for the anisotropic supersolid are

• conservation of mass

∂tρ + ∂i

[ρsijvsj + (ρδij − ρsij)vnj

]= 0; (3–46)

58

• conservation of entropy∂ts + ∂i(svni) = 0; (3–47)

• Lin’s constraintDnRi

Dt= 0. (3–48)

We incorporate these constraints into the Lagrangian density using Lagrange multipliers φ,

α and β. Then we have

LSS =1

2ρsijvsivsj +

1

2(ρδij − ρsij)vnivnj − USS(ρ, ρsij, s, Rij) + α

[∂ts + ∂i(svni)

]

+φ

{∂tρ + ∂i

[ρsijvsj + (ρδij − ρsij)vnj

]}+ βi

[∂t(sRi) + ∂j(sRivnj)

]. (3–49)

In the above Lagrangian density Lin’s constraint was introduced after combining it with

the equation of the entropy conservation. We calculate the equations of motion as follow

• δρ1

2vn

2 − ∂USS

∂ρ− ∂tφ− vni∂iφ = 0; (3–50)

• δρsij

1

2vsivsj −

1

2vnivnj −

∂USS

∂ρsij

− 1

2(vsj − vnj)∂iφ− 1

2(vsi − vni)∂jφ = 0; (3–51)

• δsDnα

Dt+ Ri

Dnβi

Dt+

∂USS

∂s= 0; (3–52)

• δvsi

ρsij(vsj − ∂jφ) = 0; (3–53)

• δvni

∂iα + Rj∂iβj =1

s(ρδij − ρsij)(vnj − vsj); (3–54)

• δRi

Dnβi

Dt− 1

s∂j

(∂USS

∂Rji

)= 0. (3–55)

Additionally, the variations with respect to the Lagrange multipliers just reproduce the

imposed side conditions, Eqs. 3–46 through 3–48. First, the variation with respect to

vsi leads again to Eq. 3–33. Next, the derivation of the Josephson equation, the Euler

59

equation for vn, and the momentum conservation equation for an anisotropic supersolid is

analogous to that for an isotropic supersolid. The results are: the Josephson equation

∂tvsi = −∂iµ− 1

2∂ivs

2, (3–56)

the Euler equation

Dn

Dt

[(ρδij − ρsij)(vnj − vsj)

]= −s∂i

(∂USS

∂s

)− ∂iRj∂k

(∂USS

∂Rkj

)

−(ρδij − ρsij)(vnj − vsj)∂kvnk

−(ρδjk − ρsjk)(vnk − vsk)∂ivnj, (3–57)

and the equation of the conservation of momentum

∂t

[(ρδij − ρsij)vnj + ρsijvsj

]− ∂j

(Rikλjk

)+ ∂j

(ρvsjvsi + vsipj + vnjpi

)

− ∂i

[ε− Ts− µρ− (vnj − vsj)pj

]= 0,

(3–58)

where

pi ≡ (ρδij − ρsij)(vnj − vsj). (3–59)

Note that this momentum conservation equation agrees with the non-dissipative

momentum current derived by Andreev and Lifshitz (Eq. 12 in Ref. [4]) except for the

nonlinear strain term that they neglected. Moreover, the derived momentum conservation

equation also can be mapped into Eq. 4.16 of Saslow [7] by taking vs as a Galilean

velocity, and Eq. 3.40 of Liu [8] with the vanishing super thermal current with the

chemical potential given by Eq. 3–20.

Similar to the isotropic supersolid case, the equivalent Lagrangian density of

anisotropic supersolids to Eq. 3–49 is

LSS = −ρ∂tφ− 1

2ρsij∂iφ∂jφ +

1

2(ρδij − ρsij)vnivnj

−(ρδij − ρsij)vnj∂iφ− f(ρ, ρsij, T, Rij), (3–60)

60

where f ≡ USS − Ts. There are several works that have derived a Lagrangian density for

supersolid at zero temperature, that employ different methods. Son derived the Andreev

and Lifshitz nondissipative hydrodynamics and a Lagrangian density by using Galilean

invariance and symmetry arguments [67]. First, we can also make the connection to the

Lagrangian density derived by Son [67] by replacing vn with the displacement vector R

and the strain tensor Rij using the inverted Lin’s constraint,

vni = −R−1ji ∂tRj, (3–61)

where R−1ji ≡ ∂xi/∂Rj and RijR

−1jk = δik. Josserand et al. used the homogenization method

starting from the time-dependent Gross-Pitaevskii equation [68]. On the other hand, Ye

proposed a Lagrangian density for a supersolid introducing an arbitrary phenomenological

coupling constant between elasticity and superfluidity in the Gross-Pitaevskii equation

[69].

At this point it is worthwhile to speculate about the possible connection between the

derived Lagrangian density and the time-dependent Gross-Pitaevskii Lagrangian density

coupled to an elastic field. In extending the superfluid Gross-Pitaevskii equation Eq. 3–12,

to a supersolid one must require Galilean invariance. The covariant form of the gradient of

the order parameter wave function is

− i~∂iψ → −i~∂iψ −m∂tujψ + i~∂kuj∂kψ, (3–62)

where m is the mass of the bosonic particle. However, the use of the covariant gradient

results in the coupling constant ρs between vs and vn while from the interaction term

in the Lagrangian density, Eq. 3–60, the coupling constant is −(ρδij − ρsij) which is set

by conservation laws and Galilean invariance. Unfortunately the connection between the

variational principle and the Gross-Pitaevskii Lagrangian density is still unclear.

61

3.1.4 Quadratic Lagrangian Density of Supersolids

The quadratic Lagrangian is interesting because it gives us the linearized

hydrodynamics from which collective modes can be obtained. We study the dynamics

of a supersolid by considering small fluctuations from a thermal equilibrium state (or

at T = 0). Unless we specify otherwise, in the remaining part of this chapter thermal

fluctuations are excluded. We first consider small fluctuations in densities, denoted with δ,

from constant equilibrium values, subscripted by zero, as

ρ = ρ0 + δρ, (3–63)

and

ρsij = ρs0ij + δρsij. (3–64)

For the velocity of the super-component, we take a variation in the Clebsch potential φ so

that we have its non vanishing gradient and the time derivative: ∂iφ and ∂tφ. In addition,

we assume that the Lagrangian coordinates differ from the Euler coordinates by a small

displacement vector field u:

R = x− u. (3–65)

The deformation tensor then becomes

Rij = δij − wij, (3–66)

where wij ≡ ∂iuj. From the inverted Lin’s constraint, Eq. 3–61, we obtain the linear

relation between the velocity of the normal component and the displacement vector,

vni = ∂tui. (3–67)

Therefore, we found that in linear elasticity the velocity of the normal component is given

by the time derivative of the displacement vector. It becomes clear that Lin’s constraint

is the hydrodynamic equation for the elastic variables that arises from the translational

broken symmetries of solids rather than the conservation of the initial positions.

62

We now expand the Lagrangian density up to second order in small fluctuations δρ,

δρsij, and u:

LquadSS = −ρ0∂tφ− λ0ijwij − µ0δρ− δρ∂tφ− 1

2ρ0(∂iφ)2 − ∂µ

∂wij

∣∣∣∣ρ

δρwij − 1

2

∂µ

∂ρ

∣∣∣∣wij

(δρ)2

+1

2ρn0ij (∂tui − ∂iφ) (∂tuj − ∂jφ)− 1

2

∂λij

∂wlk

∣∣∣∣ρ

wijwlk, (3–68)

where ρn0ij ≡ ρ0ij − ρs0ij, and we have used

∂f

∂ρ= µ +

1

2(vni − vsi)

2 ' µ, (3–69)

and

∂f

∂wij

= λij. (3–70)

In the expansion we have dropped constants which do not contribute to the equations of

motion. Additionally the terms proportional to ∂f/∂ρsij are neglected because they are of

higher order:

∂f

∂ρsij

=∂USS

∂ρsij

= −1

2(vni − vsi)

2 ' 0. (3–71)

As a result, the dependence on δρsij is absent in the quadratic expansion of the

Lagrangian density, Eq. 3–68. Often the first two terms in Eq. 3–68 are neglected because

they are total derivatives, and do not contribute to the equations of motion as long as

boundary contributions are not important. This is true as long as topological defects,

such as vortices or dislocations, are not present in the supersolid. In Chapter 5 we will

show that the first term is responsible for the Magnus force acting on vortices [70] while

the second term gives the Peach-Koehler force on a dislocation [71, 72]. Since we are not

considering any topological defects in this chapter, we neglect these terms for now.

The linearized equations of motion are obtained taking the variations of the action of

Eq. 3–68 with respect to δρ, φ, and ui:

• δ(δρ)

δρ =∂ρ

∂wij

∣∣∣∣µ

wij − ∂ρ

∂µ

∣∣∣∣wij

(∂tφ + µ0) (3–72)

63

• δφ∂tδρ + ρn0ij∂i(∂tuj) + ρs0ij∂i∂jφ = 0 (3–73)

• δui

ρn0ij∂2t uj − ∂λji

∂wlk

∣∣∣∣ρ

∂jwlk − ∂µ

∂wji

∣∣∣∣ρ

∂jδρ− ρn0ij∂t∂jφ = 0 (3–74)

where we have used the identity

∂x

∂y

∣∣∣∣z

= −∂z

∂y

∣∣∣∣x

∂x

∂z

∣∣∣∣y

. (3–75)

Note that Eq. 3–73 is the linearized continuity equation. The other two equations of

motion are equivalent to Eq. 19 of Andreev and Lifshitz [4] when the Josephson equation,

Eq. 3–56, (∂tφ = −µ0 − δµ) is used.

Density fluctuations in solids are caused by fluctuations in either the lattice

displacement or the net defect density defined as [73, 74]

ρ∆ = ρi − ρv, (3–76)

where ρi is the density of interstitials and ρv the density of vacancies. The defect density is

conserved as long as surface effects are ignored; vacancies and interstitial are created and

destroyed by pairs in the bulk, but can be created or destroyed individually by migrating

to the surface. Now we can take δρ∆ as the independent variable instead of δρ. Since

δµ =∂µ

∂wij

∣∣∣∣ρ∆

wij +∂µ

∂ρ∆

∣∣∣∣wij

δρ∆, (3–77)

Eq. 3–72 reduces to

δρ =∂ρ

∂wij

∣∣∣∣ρ∆

wij +∂ρ

∂ρ∆

∣∣∣∣wij

δρ∆, (3–78)

where we have used Eq. 3–75 and

∂x

∂y

∣∣∣∣0

=∂x

∂y

∣∣∣∣z

+∂x

∂z

∣∣∣∣y

∂z

∂y

∣∣∣∣0

. (3–79)

In the higher order expansion of the Lagrangian density, the terms proportional to the

superfluid density fluctuation must be considered in Eq. 3–78. Following Zippelius et al.

64

[73], we can identify

∂ρ

∂wij

∣∣∣∣ρ∆

= −ρ0δij, (3–80)

and

∂ρ

∂ρ∆

∣∣∣∣wij

= 1. (3–81)

The same relation was used by Ostlund et al. when they studied the hydrodynamics of

an anisotropic normal solid [75]. Taking the time derivative of Eq. 3–78 and using the

continuity equation Eq. 3–73, we get the linearized conservation equation of the defect

density

∂tδρ∆ = ρs0ij(∂twij − ∂i∂jφ), (3–82)

with the defect current proportional to the density of the super-component

j∆i = ρs0ij(∂tuj − ∂jφ). (3–83)

We found a very important feature: the defect current only arises if superfluidity is present

with non-vanishing relative velocity.

Replacing Eqs. 3–72 and 3–82, into Eq. 3–74 and the time derivative of Eq. 3–73, we

get the following equations of motion

∂2t δρ∆ − ρs0ij

∂µ

∂ρ∆

∣∣∣∣wij

∂i∂jδρ∆ − ρs0ij∂i∂2t uj − ρs0ij

∂µ

∂wlk

∣∣∣∣ρ∆

∂i∂jwlk = 0, (3–84)

ρn0ij∂2t uj −

(∂µ

∂wji

∣∣∣∣ρ

− ρn0ij

∂µ

∂ρ∆

∣∣∣∣wij

)∂jδρ∆

−(

∂λji

∂wlk

∣∣∣∣ρ

− ρn0ij

∂µ

∂wlk

∣∣∣∣ρ∆

− ρ0∂µ

∂wij

∣∣∣∣ρ

δlk

)∂jwlk = 0.

(3–85)

In the case where u = 0, we get the dispersion relation of the fourth sound modes, from

Eq. 3–84,

ω2 = ρs0ij

∂µ

∂ρ∆

∣∣∣∣wij

qiqj, (3–86)

which is just the result obtained by Andreev and Lifshitz [4], since (∂/∂ρ∆)wij= (∂/∂ρ)wij

.

On the other hand, when defect fluctuations are absent (δρ∆ = 0), Eqs. 3–84 and 3–85

65

combine to become

ρ0∂2t ui =

∂λji

∂wlk

∣∣∣∣ρ

∂jwlk − ρ0∂µ

∂wlk

∣∣∣∣ρ∆

∂iwlk − ρ0∂µ

∂wij

∣∣∣∣ρ

∂jwkk. (3–87)

Therefore, there are no additional sound modes without defects.

Another interesting case is vanishing the superfluid density, i.e., normal solids. In

such a case, we get the same sound speed as Eq. 3–87 in the case in which the defect

density fluctuations vanish because the second term in Eq. 3–85 does not contribute to the

sound speed, but rather to the diffusion. Moreover, the conservation equation for defects,

Eq. 3–82, implies that ∂tδρ∆ = 0 when ρsij = 0, which agrees with the dissipationless

description of supersolid: a defect current arises only when dissipation is taken into

account [73]. In Chapter 4 we study the dissipative hydrodynamics and calculate the

defect diffusion coefficient.

To conclude this section we rewrite the Lagrangian density, Eq. 3–68, in terms of the

defect fluctuation δρ∆ using Eq. 3–78:

LquadSS = −ρ0∂tθ − λ0ijwij + ρ0wii∂tθ − ρn0ij∂tui∂jθ − δρ∆∂tθ − 1

2ρs0ij∂iθ∂jθ

− ∂µ

∂wij

∣∣∣∣ρ

δρ∆wij + ρ0∂µ

∂ρ∆

∣∣∣∣wij

wiiδρ∆ − 1

2

∂µ

∂ρ∆

∣∣∣∣wij

δρ∆2 − 1

2

∂λji

∂wlk

∣∣∣∣ρ

wijwlk

+ρ0∂µ

∂wij

∣∣∣∣ρ

wijwkk − 1

2ρ2

0

∂µ

∂ρ

∣∣∣∣wij

w2ii +

1

2ρn0ij∂tui∂tuj, (3–88)

where we have introduced θ = φ + µ0t. In the following section, we study this Lagrangian

density considering a two-dimensional isotropic supersolid to calculate the hydrodynamic

modes and the density-density correlation function.

3.2 Collective Modes and the Density-Density Correlation Function

In Section 3.1.4 we have taken into account defect density fluctuations, and recovered

the fourth sound speed of defects obtained by Andreev and Lifshitz [4]. In this section

we calculate the second sound speed and the density-density correlation function which

contains additional information on the second sound modes.

66

Hydrodynamic modes of a system can be inferred by simply counting the number of

conservation laws and broken symmetries of a system [76]. We start by enumerating the

conservation laws and broken symmetries of a three-dimensional supersolid. Since thermal

fluctuations are not considered, the energy conservation law can be omitted; therefore,

there are only conservation laws of mass and three components of momentum. In addition

to these, there are three broken translational symmetries, and one broken gauge symmetry

due to the Bose-Einstein condensation. Thus, we have eight conservation laws and the

broken symmetries, and therefore eight hydrodynamic modes for the three-dimensional

supersolid without thermal fluctuations. The corresponding hydrodynamic modes are two

pairs of transverse propagating modes, one pair of longitudinal propagating modes, and

one pair of longitudinal second sound modes. The appearance of these longitudinal second

sound modes is another definite signature of a supersolid, in addition to the λ-anomaly in

the specific heat [43], the NCRI [6], and so on.

In order to investigate the appearance of the second sound modes in a supersolid we

consider a two-dimensional isotropic supersolid. The reduction by one spatial dimension

results in only one pair of transverse propagating modes; the longitudinal part remains

intact. Note that the indices on the densities are lost because of the isotropy. We have the

following thermodynamic relations:

∂λji

∂wlk

∣∣∣∣ρ

= λδjiδlk + µ(δilδjk + δikδjl), (3–89)

∂µ

∂wij

∣∣∣∣ρ

= γδij, (3–90)

∂µ

∂ρ∆

∣∣∣∣wij

=∂µ

∂ρ

∣∣∣∣wij

=1

ρ20χ

, (3–91)

where χ is the isothermal compressibility at constant strain, γ is a phenomenological

coupling constant between the strain and the density, and λ and µ are the bare Lame

coefficients at constant density. Then, the Euclidean Lagrangian (with the imaginary time,

τ = it) of Eq. 3–88 by using the above representations of the thermodynamic relations

67

becomes

LquadE = ρ0wii∂τθ − iρ0uii∂τθ + ρn0∂τui∂iθ + iδρ∆∂τθ +

1

2ρs0(∂iθ)

2 + γδρ∆uii

− 1

ρ0χuiiδρ∆ +

1

2

1

ρ20χ

δρ∆2 +

1

2

(λ +

1

χ− 4ρ0γ

)u2

ii + µu2ij

+1

2ρn0(∂τui)

2, (3–92)

where uij = (wij + wji)/2, and we have neglected the linear terms which do not affect

the dynamics because they are the total derivatives of analytical variables. When Fourier

transformed Eq. 3–92 becomes

LquadE =

1

2

(δρ∆(Qn) θ(Qn) uL(Qn)

)A

δρ∆(−Qn)

θ(−Qn)

uL(−Qn)

+1

2

(ρn0ω

2n + µq2

)uT (Qn)uT (−Qn), (3–93)

where Qn ≡ (q, ωn) with Matsubara frequencies ωn, uL = (q · u)/q with q = |q|,uT = u− (uL/q)q, and

A =

1ρ20χ

−ωn −iq(γ − 1

ρ0χ

)

ωn q2ρs0 iωnqρs0

iq(γ − 1

ρ0χ

)iωnqρs0 ρn0ω

2n + q2

(λ + 1

χ− 2ρ0γ

)

, (3–94)

where λ ≡ λ + 2µ. As we discussed earlier, there is one pair of transverse sound modes

whose sound speed is

cT =

õ

ρn0

. (3–95)

In addition, we get one pair of longitudinal first sound modes and another pair of

longitudinal second sound modes. Their sound speeds are

c2L =

λ

2ρn0

+1

2ρ0χ− γ +

1

2

√(λ

ρn0

+1

ρ0χ− 2γ

)2

− 4ρs0

ρn0

[λ

χρ20

− γ2

], (3–96)

68

and

c22 =

λ

2ρn0

+1

2ρ0χ− γ − 1

2

√(λ

ρn0

+1

ρ0χ− 2γ

)2

− 4ρs0

ρn0

[λ

χρ20

− γ2

]. (3–97)

When the superfluid density vanishes, i.e., for a normal solid, the second sound speed

c2 vanishes. Therefore, the normal solid has only one pair of transverse sound modes and

one pair of longitudinal sound modes whose sound speeds become

cT =

õ

ρ0

, (3–98)

and

c2NS =

1

ρ0

(λ + 2µ +

1

χ

)− 2γ, (3–99)

which agree with the results obtained by Zippelius et al. [73] after identifying λ =

λZippelius + 2γZippelius + 1/χZippelius, µ = µZippelius, γ = (γZippelius + 1/χZippelius)/ρ0, and

χ = χZippelius. Moreover, we recover the sound speed of the normal fluid by assuming that

the Lame coefficients and the coupling constant γ vanish.

On the other hand, the correlation functions can be easily derived from Eq. 3–93. We

obtain

〈δρ∆(Qn)δρ∆(−Qn)〉 = q2ρs0

χ

ω2nχρ0 + (λχ + 1− 2ρ0γχ)q2

∆A

, (3–100)

〈δρ∆(Qn)uL(−Qn)〉 = iqρs0

ρ0χ

χρ0ω2n + (1− ρ0γχ)q2

∆A

, (3–101)

and

〈uL(Qn)uL(−Qn)〉 =1

ρ20χ

ρ20χω2

n + ρs0q2

∆A

, (3–102)

where

∆A = ρn0ω4n +

[λ + ρn0

(1

ρ0χ− 2γ

)]q2ω2

n − ρs0

(γ2 − λ

χρ20

)q4. (3–103)

69

Since the density fluctuation is related to the defect density fluctuation and the strain

tensor by Eq. 3–78, the density-density correlation function becomes

〈δρ(Qn)δρ(−Qn)〉 = A

(1

iωn − cLq− 1

iωn + cLq

)+ B

(1

iωn − c2q− 1

iωn + c2q

), (3–104)

where

A = −qρ0ρn0c

2L − ρs0λ

2cLρn0(c2L − c2

2), (3–105)

B = −qρ0ρn0c

22 − ρs0λ

2c2ρn0(c22 − c2

L). (3–106)

We can now perform the analytic continuation (iωn = ω + iδ) from 〈δρ(Qn)δρ(−Qn)〉 to

〈δρ(Q)δρ(−Q)〉, resulting in

〈δρ(Q)δρ(−Q)〉 = A

(1

ω − cLq + iδ− 1

ω + cLq + iδ

)+ B

(1

ω − c2q + iδ− 1

ω + c2q + iδ

),

(3–107)

where Q ≡ (q, ω).

Then, the response function can be obtained by taking the imaginary part of the

density-density correlation function Eq. 3–107:

χ′′ρρ(q, ω) = −πA

[δ(ω − cLq)− δ(ω + cLq)

]− πB

[δ(ω − c2q)− δ(ω + c2q)

], (3–108)

where we have used the identity

1

ω′ − ω − iε= P

1

ω′ − ω+ iπδ (ω − ω′) . (3–109)

It is easy to show that the response function satisfies the sum rules: the thermodynamic

sum rule ∫ ∞

−∞

dω

π

χ′′ρρ(q, ω)

ω= χρρ(q), (3–110)

where χρρ(q) is the static density-density correlation function (Appendix C), and the

f-sum rule ∫ ∞

−∞

dω

πωχ′′ρρ(q, ω) = ρ0q

2. (3–111)

70

We close this chapter with a summary of our main results. Based on the conservation

laws and the broken symmetries present in a supersolid, we derived the Lagrangian density

and the non-dissipative hydrodynamics, which agrees with the work done by Andreev

and Lifshitz [4], using the variational principle. Starting from the derived non-dissipative

hydrodynamics we calculated both the fourth sound speed and the second sound speed.

In addition, we obtained the density-density correlation function of a model supersolid

using the Lagrangian density, and found that each pair of longitudinal propagating modes

produces singularities in the density-density correlation function and a pair of δ-function

peaks in the response function.

71

CHAPTER 4DISSIPATIVE HYDRODYNAMICS OF A MODEL SUPERSOLID

In Chapter 3 we obtained the non-dissipative hydrodynamics for a supersolid using

the variational principle, and showed that the equations of motion are those which

Andreev and Lifshitz had derived without dissipation [4]. Using the linearized equations of

motion we found that for a two-dimensional supersolid there are in total six propagating

modes: one pair of transverse sound modes, one pair of longitudinal first sound modes,

and one pair of longitudinal second sound modes. The longitudinal propagating modes

appear in the density-density correlation function as double delta-function peaks located

at ω = ±cq with a sound speed c.

When dissipation is taken into account, the mode structure of the system changes:

due to dissipation some diffusive modes appear, e.g., a thermal diffusion mode arises

due to viscosity. On the other hand, dissipation damps the sound mode. The dispersion

relation of a propagating mode in the limit of small q is given by

ω = ±cq − iD

2q2, (4–1)

where D is the attenuation coefficient.

For a three dimensional normal solid there are five conservation laws: mass, energy

and three components of momentum. In addition, the translational symmetries are

broken in three different directions, and there are eight hydrodynamic modes for a three

dimensional normal solid. The corresponding hydrodynamic modes are four transverse

sound modes, two longitudinal first sound modes, one thermal diffusion mode and one

defect diffusion mode [76].

In contrast, as we discussed in Chapter 3 there are nine hydrodynamic modes for

a three-dimensional supersolid because the additional broken gauge symmetry due to

the Bose condensation. Because of this broken gauge symmetry one of the diffusion

modes of the normal solid becomes a pair of propagating longitudinal second sound

72

modes. The defect-mediated supersolid proposed by Andreev and Lifshitz suggests

that the defect diffusion mode of the normal solid becomes the second sound modes in

the supersolid phase. In this chapter we investigate the mode structure of a supersolid

including dissipation, obtain the sound speeds and their attenuation constants, and

describe a light scattering experiment to observe the conversion of the defect diffusion

mode of normal solid into the second sound modes of supersolid.

4.1 Andreev and Lifshitz Hydrodynamics of Supersolids

The hydrodynamics for the supersolid with dissipation was first derived by Andreev

and Lifshitz [4]. Their derivation starts by identifying nine hydrodynamic variables,

and writing down the conservation equations for the conserved densities and additional

equations for the broken symmetry variables. The mass, the momentum and the energy

are conserved:

∂tρ + ∂iji = 0, (4–2)

∂tji + ∂kΠik = 0, (4–3)

∂tE + ∂iQi = 0, (4–4)

where Πik is the stress tensor and Q the energy current. In addition to these conservation

laws, there are equations of motion for the broken symmetry variables. First, from the

broken translational symmetry in the three spatial directions we have

∂tui − Ji = 0, (4–5)

where J is an arbitrary function to be determined. For a perfect solid, i.e., solid without

any defects, the density change is linked to the lattice fluctuation (δρ ∼ ∇ · u). However,

Andreev and Lifshitz pointed out that the density fluctuation is independent of the lattice

displacement due to defects; therefore, the density becomes a separate hydrodynamic

variable. Since the gauge symmetry of the Bose-Einstein condensate wave function is

73

broken, we have

∂tvsi + ∂iΦ = 0. (4–6)

The currents involved in the conservation laws and the arbitrary functions J and Φ are to

be determined. In addition to these eight hydrodynamic equations, there is an additional

equation which is the entropy production equation

∂ts + ∂i

(svni +

qi

T

)=

R

T, (4–7)

where q is the heat current and R is the positive definite dissipation function. The

redundancy condition, Eq. 4–7, combined with the thermodynamics and Galilean

covariance allows us to obtain the constitutive relations that relate currents, J, and Φ

to hydrodynamic variables. The resulting reversible currents are

ΠRij = −

[ε− Ts− µρ− (vnk − vsk)pk

]δij + ρvsjvsi + vsipj + vnjpi − λji − λjkwik, (4–8)

JRk = vni − wkivnk, (4–9)

ΦR = µ +1

2vs

2. (4–10)

On the other hand, using the positiveness of R we obtain the dissipative currents:

qi = −κik∂kT − Tαik∂lλlk, (4–11)

ΦD = −Λ∂i [pi − ρ (vni − vsi)]− Σki∂ivnk, (4–12)

JDk = Γkl∂mλml + αkl∂lT, (4–13)

ΠDki = −ηkilm∂lvnm − Σki∂l [pl − ρ (vnl − vsl)] , (4–14)

where κik is the thermal conductivity tensor, ηiklm, Λ, and Σik are the viscosity tensors,

αik is the thermodiffusion coefficient tensor for the defects, and Γik is the defect diffusion

coefficient tensor. Note that the dissipative currents are linked to the quantities of

opposite time-reversal properties. The reversible current has the same time-reversal

74

properties as the hydrodynamic variables whereas the dissipative current has opposite

time-reversal properties.

We are now in a position to investigate the mode structure of a supersolid with

dissipation. Since we are interested in the collective motion of defects rather than thermal

diffusion, we will neglect thermal fluctuations (αik = κik = 0). Also we consider, for

simplicity, an isotropic two-dimensional supersolid. As we discussed in Chapter 3, the

reduction of dimension for a solid with isotropy results in removing a pair of transverse

propagating modes. Due to isotropy, we have for the viscosity coefficients

ηiklm = ζδikδlm + η

(δilδkm + δimδkl − 2

3δikδlm

), (4–15)

Σik = Σδik, (4–16)

Γik = Γδik. (4–17)

Then the hydrodynamic equations for a two-dimensional isotropic supersolid, including the

nonlinear term which was neglected by Andreev and Lifshitz, are

∂tρ + ∂iji = 0, (4–18)

∂tji + ∂kΠRik − ζ∂i∂kvnk − η∂2vni + Σ∂i∂k

[ρs(vnk − vsk)

]= 0, (4–19)

∂tui − JRi − Γ∂kλki = 0, (4–20)

∂tvsi + ∂iΦR + Λ∂i∂k

[ρs(vnk − vsk)

]− Σ∂i∂kvnk = 0. (4–21)

We consider fluctuations from the equilibrium values, Eqs. 3–63 through 3–66 and

δµ =∂µ

∂ρ

∣∣∣∣wij

δρ +∂µ

∂wij

∣∣∣∣ρ

wij

=1

ρ20χ

δρ + γwii, (4–22)

75

δλij =∂λij

∂ρ

∣∣∣∣wij

δρ +∂λij

∂wlk

∣∣∣∣ρ

wlk

= γδijδρ + λδijwkk + µ(wij + wji), (4–23)

where we have used the representations Eqs. 3–89 through 3–91 and the Maxwell relation

(∂µ/∂wlm)ρ = (∂λlm/∂ρ)wij, and linearized Eqs. 4–18 through 4–21. We next divide the

linearized hydrodynamic equations into the transverse and longitudinal parts. First, the

Fourier transformed equations of motion for the transverse part are

ρn0∂tvnT − µ∂2uT − η∂2vnT = 0, (4–24)

∂tuT − vnT − µΓ∂2uT = 0. (4–25)

Consequently, we get the transverse sound speed obtained in the previous section with the

attenuation constant DT = η + ρn0µΓ. Second, the longitudinal equations of motion are

∂tδρ + ρs0∂vs + ρn0∂vnL = 0, (4–26)

ρn0∂tvnL+

(ρn0

ρ20χ− γ

)δρ−

(λ−ρn0γ

)∂2uL−

(ζ−2ρs0σ−ρs

20Λ

)∂2vnL−ρs0σ∂2vs = 0, (4–27)

∂tuL − vnL − γΓ∂δρ− λΓ∂2uL = 0, (4–28)

∂tvs +1

ρ20χ

∂δρ + γ∂2uL − σ∂2vnL − ρs0Λ∂2vs = 0, (4–29)

where σ ≡ Σ − ρs0Λ, and ζ ≡ ζ + η. Sound speeds and the attenuation constants can

be obtained by calculating the dispersion relations after Fourier transforming Eqs. 4–26

through 4–29. We Laplace-Fourier transform them instead for later purposes. After

Laplace-Fourier transforming we find

C(z, q)

δρ(q, z)

vnL(q, z)

uL(q, z)

vs(q, z)

=

δρ(q)

vnL(q)

uL(q)

vs(q),

, (4–30)

76

where C(z, q) is a 4×4 matrix given by

−iz iqρn0 0 iqρs

iq(

1ρ20χ− γ

ρn0

)−iz + q2 1

ρn0

(ζ − 2ρs0σ − ρs

20Λ

)q2 1

ρn0(λ− ρn0γ) q2 ρs0

ρn0σ

−iqγΓ −1 −iz + q2λΓ 0

iq 1ρ20χ

q2σ −q2γ −iz + q2ρs0Λ

.

(4–31)

From Eq. 4–31 we obtain two sound speeds cL, Eq. 3–96 and c2, Eq. 3–97, with two

attenuation constants, respectively,

DL = − 1

ρn0(c2L − c2

2)

(c2Ln1 + n2

), (4–32)

D2 =1

ρn0(c2L − c2

2)

(c22n1 + n2

), (4–33)

where

n1 ≡ ζ − 2ρs0σ + ρn0Γλ + ρs0(ρn0 − ρs0)Λ, (4–34)

n2 ≡ 1

ρ20χ

{2ρ0ρs0(ρ0χγ − 1)σ + ρ0ρn0(λ− ρ2

0χγ2)Γ

+ ρs0ζ + ρ0ρs0

[ρn0 − ρs0 + ρ0χ(λ− 2γρn0)

]Λ

}.

(4–35)

Now we can see that when ρs = 0, i.e. a normal solid, the second sound modes disappear

but there is the defect diffusion mode with the diffusion constant

D2 =λ− ρ2

0χγ2

ρ0χc2L

Γ, (4–36)

which agrees with the result obtained by Zippelius et al. [73].

4.2 Density-Density Correlation Function and its Detection

Light scattering measures fluctuations in the local dielectric constant, which is a

function of the local mass density. The intensity of the scattered light is related to the

density-density correlation function [77]

I(q, ω) ∝ S(q, ω) = 〈δρ(q, ω), δρ(−q, ω)〉 . (4–37)

77

Thus, light scattering provides a direct measure of the density-density correlation function

which can be calculated from the hydrodynamic equations using the Laplace transform

technique [74, 78]. We start this section by reviewing the light scattering experiments

done on normal fluids and on the He II superfluid. We then calculate the density-density

correlation function for a model supersolid, and describe a light scattering experiment for

detecting the supersolid transition.

4.2.1 Normal Fluids and Superfluids

The density-density correlation function for a normal fluid was first calculated by

Landau and Placzek [79]:

[χ′′ρρ(~q, ω)

ω

]

NF

= ρ∂ρ

∂P

∣∣∣∣T

(1− cv

cP

)Dthq

2

ω2 + (Dthq2)2+ ρ

∂ρ

∂P

∣∣∣∣T

cv

cP

c2NFq4DNF

(ω2 − c2NFq2)

2+ (DNFq2ω)2

+ ρ∂ρ

∂P

∣∣∣∣T

(cv

cP

− 1

)(ω2 − c2

NFq2) q2Dth

(ω2 − c2NFq2)

2+ (DNFq2ω)2

,

(4–38)

where cv is the volume specific heat at constant volume, cP the volume specific heat at

constant, the pressure. The thermal diffusion constant is

Dth =κ

cP

, (4–39)

with κ the thermal conductivity. The sound speed of normal fluid is c2NF = (∂P/∂ρ)s with

an attenuation constant

DNF =1

ρ

(4

3η + ζ

)+

κ

cP

(cP

cv

− 1

), (4–40)

where η is the shear viscosity and ζ the bulk viscosity. The first term in Eq. 4–38 is

a Lorentzian of width 2Dthq2 located at ω = 0 due to the thermal diffusion: for each

diffusion mode a Lorentzian peak (the Rayleigh peak) at the center appears. The

remaining two terms are contributions from the sound modes. The second term in

Eq. 4–38 is a pair of Lorentzians of width DNFq2 located at ±cNFq (the Brillouin doublet).

The last term does not contribute much in the correlation function; however, its existence

78

is important for the sum rules [74, 78]. Figure 4-1 shows the spectrum of liquid argon at

T = 84.97 K measured by Fleury and Boon [80].

The density-density correlation function for a superfluid was obtained by Hohenberg

and Martin using the two-fluid hydrodynamics [81]. The first sound contribution to the

density-density correlation function is [81, 82]

[χ′′ρρ(~q, ω)

ω

]1

SF

=2D2

1c2NFq4

(ω2 − c2NFq2)2 + (D1q2ω)2

S1(q), (4–41)

where S1(q) = kBTρ2KT /γ with the isothermal compressibility KT = 1/ρ(∂ρ/∂T )P , and

D1 =1

ρ

(4

3η + ζ

). (4–42)

The contribution of the second sound is [81, 82]

[χ′′ρρ(~q, ω)

ω

]2

SF

=2q2[Dζc

22q

2 + Dthω2]

(ω2 − c22q

2)2 + (Dζ + Dth)2q4ω2S2(q), (4–43)

where S2(q) = (cP /cv − 1)S1(q) and

Dζ =ρs

ρn

(ρζ3 − ζ1 − ζ4 + D1

), (4–44)

where ζ1, ζ2, ζ3 and ζ4(= ζ1) are the coefficients of second viscosity, and η is the coefficient

of first viscosity. For temperatures above Tλ, we have ρs = 0, and Eq. 4–43 reduces to

the first term in Eq. 4–38; therefore, the Rayleigh peak of the thermal diffusion mode in

normal fluid splits into a Brillouin doublet of the second sound modes. This splitting was

observed in the light scattering experiments by Winterling et al. [83] and by Tarvin et al.

[84] (Fig. 4-2).

4.2.2 Normal Solids and Supersolids

If the supersolid transition is of second order similar to the superfluid transition, it

is natural to ask if the same observation of the light scattering repeats in the supersolid

transition. The most effective way to answer this question is to obtain the density-density

correlation function using the hydrodynamics for supersolids, Eq. 4–30. In order to obtain

79

the density-density correlation function, we calculate first the density-density Kubo

function (Appendix D) from Eq. 4–31. The expression for the density-density Kubo

function can be found in Appendix E, Eq. E–1. Then the correlation function χ′′ρρ(q, ω)

can be obtained by taking the real part of Eq. E–1:

[χ′′ρρ(q, ω)

ω

]

SS

= − iq4c2LDLI1(q)

(ω2 − c2Lq2)2 + (ωq2DL)2

+(ω2 − c2

Lq2)I3(q)

(ω2 − c2Lq2)2 + (ωq2DL)2

− iq4c22D2I2(q)

(ω2 − c22q

2)2 + (ωq2D2)2+

(ω2 − c22q

2)I4(q)

(ω2 − c22q

2)2 + (ωq2D2)2, (4–45)

where I1(q), I2(q), I3(q), and I4(q) are given in Appendix E. The first and third terms in

Eq. 4–45 give two Brillouin doublets centered at ω = ±cLq, and ω = ±c2q with width

DLq2 and D2q2, respectively. The second and fourth terms in Eq. 4–45 are negligible near

the Brillouin doublets. The obtained density-density correlation function satisfies both the

thermodynamic sum rule, Eq. 3–110, and f-sum rule, Eq. 3–111.

On the other hand, we can show that when dissipation is neglected, the second and

fourth terms in Eq. 4–45 vanish, and delta functions are obtained from the remaining two

terms by taking the limit DL, D2 → 0. Therefore, the susceptibility, Eq. 4–45, reduces to

the non-dissipative density-density correlation function, Eq. 3–108.

Moreover, for the normal solid (ρs = 0), the second term in Eq. 4–45 vanishes so that

there is only one Brillouin doublet due to the longitudinal first sound modes. At the same

time, the fourth term in Eq. 4–45 becomes a Lorentzian centered at ω = 0 which is the

Rayleigh peak due to the defect diffusion. Therefore, we conclude that the same splitting

as in the transition from a normal fluid to a superfluid occurs in the transition from a

normal solid to a supersolid.

We now study qualitatively the density-density correlation function Eq. 4–45. Let us

set first γ = Λ = Σ = 0 for simplicity. Then for small ρs, the density-density correlation

80

function reduces to

c4NS

ρ0D∆

χ′′ρρ(q, ω)

ω'

[1− 2

(x− 1)2

x2

ρs0

ρ0

]q4∆c2

LDL/c2NSD∆

(ω2∆ − c2

Lq2∆/c2

NS)2 + (ω∆q2

∆DL/D∆)2

−{

(x− 1)2

x2− 2

[(x− 1)3(x− 3)

x3− x− 1

x2y

]ρs0

ρ0

}q2∆(ω2

∆ − c2Lq2

∆/c2NS)

(ω2∆ − c2

Lq2∆/c2

NS)2 + (ω∆q2

∆DL/D∆)2

+

[x− 1 + 2

(x− 1)2

x2

ρs0

ρ0

]q4∆c2

2D2/c2NSD∆

(ω2∆ − c2

2q2∆/c2

NS)2 + (ω∆q2

∆D2/D∆)2

+

{(x− 1)2

x2− 2

[(x− 1)3(x− 3)

x3− x− 1

x2y

]ρs0

ρ0

}q2∆(ω2

∆ − c22q

2∆/c2

NS)

(ω2∆ − c2

2q2∆/c2

NS)2 + (ω∆q2

∆D2/D∆)2,

(4–46)

where x = ρ0χc2NS, y = ζ/ρ0D∆, ω∆ ≡ (D∆/c2

NS)ω and q∆ ≡ (D∆/cNS)q with the

defect diffusion constant D∆ = Γ/χ and the longitudinal sound speed of normal solid cNS,

Eq. 3–99. We show in Fig. 4-3 the normalized density-density correlation functions of a

normal solid and a supersolid. To do so, we have used the first sound speed cNS = 550

m/s, the density ρ = 0.19048 g/cm3, the isothermal compressibility χ = 0.29615 × 10−8

cm s2/g for solid 4He [85, 86] at the molar volume 21 cm3/mole, the viscosity of 4He fluid

2 × 10−5 g/cm s, the typical wave number involved in light scattering q−1 = 100 nm, and

Γ = 8 × 10−11 cm3s/g. In Fig. 4-4 we show the splitting of the Rayleigh peak due to the

defect diffusion of normal solid into an additional Brillouin doublet of the second sound for

different values of supersolid fraction.

Therefore, we conclude that the light scattering on a solid 4He could provide us with

very rich information about the supersolid phase analogous to the superfluid phase. The

detection of the additional Brillouin doublet in the spectrum of scattered light will give

another signature for the existence of the supersolid. However, a detection of the diffusion

mode of defects must be preceded by that of the propagating modes. The defect-mediated

supersolid predicted by Andreev and Lifshitz assumes the existence of a sufficiently large

number of defects. But the small number of thermal defects present in a solid 4He [87] and

the large activation energies of vacancies and interstitials [32] are problematic in realizing

the proposed Brillouin doublet of the second sound modes.

81

Figure 4-1. Brillouin spectrum of liquid argon. Reprinted figure 3 with permission from P.A. Fleury and J. P. Boon, Phys. Rev. 186, 244 (1969). Copyright (1969) bythe American Physical Society(http://link.aps.org/doi/10.1103/PhysRev.186.244).

Figure 4-2. Brillouin spectra of 4He superfluid. Reprinted figure 6 with permission from J.A. Tarvin, F. Vidal, and T. J. Greytak, Phys. Rev. B 15, 4193 (1977).Copyright (1977) by the American Physical Society(http://link.aps.org/doi/10.1103/PhysRevB.15.4193).

82

−0.06 −0.04 −0.02 0 0.02 0.04 0.060

200

400

600

800

1000

1200

1400

(D∆ / cNS2 ) ω

Nor

mal

ized

Den

sity

−D

ensi

ty C

orre

latio

n F

unct

ion

normal solid

ρs / ρ = 0.1

Figure 4-3. Density-density correlation functions of isothermal and isotropic normal solids(dashed line) and supersolids (solid line).

83

−0.01 −0.008 −0.006 −0.004 −0.002 0 0.002 0.004 0.006 0.008 0.010

100

200

300

400

500

600

700

(D∆ / cNS2 ) ω

Nor

mal

ized

Den

sity

−D

ensi

ty C

orre

latio

n F

unct

ion

normal solidρ

s / ρ = 0.01

ρs / ρ = 0.02

Figure 4-4. Splitting of the Rayleigh peak (dashed line) due to the defect diffusion modeof a normal solid into the Brillouin doublet of the second sound modes.

84

CHAPTER 5DYNAMICS OF TOPOLOGICAL DEFECTS IN SUPERSOLIDS

In Chapter 1 we discussed the growing theoretical and experimental interest in

dislocations and grain boundaries in solids and vortices in superfluids, due to the

anomalous behaviors of solid 4He at low temperatures. In this Chapter we study the

dynamics of vortices and dislocations present in supersolids. Our primary objective in this

chapter is to derive an effective action for vortices and dislocations in supersolids, and

study their properties such as the inertial mass of a topological defect.

We start with the Lagrangian density derived in Chapter 3, Eq. 3–68, for supersolids

without thermal fluctuations:

LquadSS = −ρ∂tθ − λijwij − δρ∂tθ − 1

2ρ(∂iθ)

2 − ∂µ

∂wij

∣∣∣∣ρ

δρwij − 1

2

∂µ

∂ρ

∣∣∣∣wij

(δρ)2

+1

2ρnij (∂tui − ∂iθ) (∂tuj − ∂jθ)− 1

2

∂λij

∂wlk

∣∣∣∣ρ

wijwlk, (5–1)

where we have shifted φ = θ − µt and omitted, for simplicity, the subscript ‘0’ used to

indicate the constant equilibrium values. The dynamical variables in Eq. 5–1 are δρ, φ,

and ui. As we discussed in Chapter 3 the first two terms generally can be ignored because

they are total derivatives. However, when vortices and dislocations are considered, it is

not possible to simply drop these terms because of singularities due to defects. In fact as

we discuss in Sections 5.1 and 5.2, the first term is responsible for the Magnus force for

vortices [70] and the second term the Peach-Koehler force for dislocations [71, 72]. Before

we proceed, we “integrate out” the density fluctuations to obtain

LquadSS = −ρ∂tθ − λijwij +

1

2

∂ρ

∂µ

∣∣∣∣wij

(∂tθ)2 − 1

2ρsij∂iθ∂jθ +

1

2ρnij∂tui∂tuj

−1

2

∂λij

∂wlk

∣∣∣∣µ

wijwlk − 1

2

(ρnij +

∂ρ

∂wij

∣∣∣∣µ

)(∂tui∂jθ + wij∂tθ

), (5–2)

where we have dropped constants which do not affect on the dynamics, and used

Eqs. 3–75 and 3–79. In addition, for the interaction terms between superfluidity

and elasticity we distinguished explicitly ∂tui∂iθ from wij∂tθ, even though they can

85

be merged into one by integrating by parts (provided that ui and θ are analytic);

nevertheless, when topological defects are taken into account, the integration by parts

is no longer possible. Consequently, when one deals with partial derivatives associated

with vortices and dislocations, extra caution must be taken. It is worthwhile to note that

the density fluctuation “renormalizes” the coupling constant between the elasticity and the

superfluidity from −ρnij in the case of constant density (δρ = 0) to −ρnij − (∂ρ/∂wij)µ.

Therefore, when ρnij = −(∂ρ/∂wij)µ, the elastic part becomes completely decoupled from

the superfluid part.

We consider a two-dimensional isotropic supersolid for simplicity. With the

assumption of isotropy and the representations of Eqs. 3–89 through 3–91, the Euclidean

action (τ = it) of the Lagrangian Eq. 5–2 can be written as

SE =1

2

∫dτ

∫d2x

[2iρ∂τθ + 2λijuij + ρ2χ(∂τθ)

2 + ρs(∂iθ)2 + ρn(∂τui)

2

+(λ− ρ2γ2χ)u2ii + 2µu2

ij + i(ρn − ρ2γχ)(∂τui∂iθ + uii∂τθ)

], (5–3)

which is our starting point to derive the effective action for vortices and dislocations, and

study their dynamics in supersolids.

5.1 Vortex Dynamics

Let us first consider vortices which can be characterized by the single-valued line

integral of θ along a closed path enclosing only one vortex of strength (charge) of some

integer e in units of h/m ∮dθ =

h

me, (5–4)

where m is the mass of particle of supersolid. For two-dimensional supersolids vortices are

point-like particles, and the line integral is taken in the counterclockwise direction around

the axis z perpendicular to the two-dimensional supersolid system. We now can separate θ

into two parts such that

θ = θS + θV, (5–5)

86

where θS is the analytical part and θV is the singular part due to vortices. Then we have

Eq. 5–4 along a path enclosing α vortices

∮dθ =

∮dθV =

h

m

∑α

eα, (5–6)

which implies

∇×∇θV =h

m

∑α

eαzδ(2)(x− xα), (5–7)

where eα = ±1 is the charge of the α-th vortex located at xα(τ) = [xα(τ), yα(τ)]. In

addition, the velocity of the super-component can be written as the sum of a longitudinal

part ∂iθS and a transverse part ∂iθ

V. Since ∂2θV = 0, for θV we take the ansatz

θV =∑

α

θα

=~m

∑α

eα arctan

[y − yα(τ)

x− xα(τ)

], (5–8)

and, in the following, we derive an effective action for vortices in supersolids in terms of

vortex coordinates xα(τ). Let’s first insert Eq. 5–5 into Eq. 5–3. Then the action becomes

SE = S1 + S2, (5–9)

where

S1 = iρ

∫dτ

∫d2x∂τθ

V, (5–10)

S2 =1

2

∫dτ

∫d2x

[ρ2χ(∂τθ

S)2 + ρs(∂iθS)2 + ρ2χ(∂τθ

V)2 + ρs(∂iθV)2

+2ρ2χ(∂τθV)(∂τθ

S) + ρn(∂τui)2 + (λ− ρ2γ2χ)u2

ii + 2µu2ij

+2i(ρn − ρ2γχ)∂τui∂iθS + i(ρn − ρ2γχ)(∂τui∂iθ

V + uii∂τθV)

]. (5–11)

The linear term in uij is dropped in SE because there are no dislocations. As shown in

Ref. [70], from S1 one can derive a transverse force, known as the Magnus force, that

acts on a vortex. Since ∂τ =∑

α sαi (τ)∂/∂xα

i (τ) with sαi (τ) = dxα

i (τ)/dτ being the

vortex velocity, S1 can be rewritten in terms of the vortex velocity and an effective vector

87

potential,

S1 = −i∑

α

∫dτ

∫d2xAα

i [xα(τ)]sαi (τ), (5–12)

where we have defined

Aαi (xα) = −ρ

∫d2x

∂θα(x− xα)

∂xαi

. (5–13)

Therefore, the α-th vortex moves with a velocity sα(τ) in an effective magnetic field given

by ∇xa × Aα(xα) = −hρeαz/m. The Magnus force [70, 88] acting on the α-th vortex is

then given by

FαMagnus = i

hρeα

msα(τ)× z, (5–14)

which shows a motion of vortex perpendicular to the applied force.

Now we eliminate the displacement vector u and the analytic phase θS in S2 by using

the equations of motion to express S2 in terms of θV. As usual, the equations of motion

can be calculated by taking variations of SE with respect to θS and u. The decomposition

of u into the longitudinal part uL and the transverse part uT leads us to the equations of

motion:

ρn∂2τuT + µ∂2uT = −i

1

2(ρn − ρ2χγ)∂τ (∂θV), (5–15)

ρ2χ∂2τθ

S + ρs∂2θS + i(ρn − ρ2γχ)∂i∂τui = −ρ2χ∂τ (∂τθ

V), (5–16)

ρn∂2τuL + (λ− ρ2γ2χ)∂2uL + i(ρn − ρ2γχ)∂τ∂θS = −i

1

2(ρn − ρ2γχ)∂(∂τθ

V), (5–17)

where λ = λ + 2µ. These equations of motion are inhomogeneous differential equations

for u and θS with source terms from θV: ∂τθV for longitudinal wave modes and ∂iθ

V for

transverse wave modes. When ρn = ρ2γχ, there is no source term for the displacement

vector as expected because there is no coupling between superfluidity and elasticity. In

this case, Eqs. 5–15 through 5–17 become ordinary wave equations. The equation of

motion for the transverse part has the same form as Eq. 10 of Ref. [89] in which the

vortex dynamics in a superconductor was studied. In their work, they estimated the size

of the shear deformation caused by a vortex which is moving with a velocity much less

88

than the transverse sound speed. They concluded that the shear deformation generated

by a vortex was negligible. Following Ref. [89] we can also estimate the maximal shear

deformation due to a vortex for solid 4He:

umaxT ∼ ~

mHe

(ρn − ρ2χγ)s

ρnc2T

∼ 1.57× 10−8[m2/s](ρn − ρ2χγ)s

ρnc2T

. (5–18)

Taking the phenomenological coupling constant γ to be zero, and the vortex speed to be

s ∼ cT ∼ 300 m/s, the maximal shear deformation umaxT ∼ 5 × 10−11 m which is negligible

similar to the conclusion obtained in Ref. [89].

We now solve for uL, θS, and uT , in Fourier space, resulting in

uL(q, ω) = −q(ρn − ρ2χγ)ω2 − (ρs/ρ

2χ)q2

2∆A

(∂τθV)(q, ω), (5–19)

θS(q, ω) = −iωρnω

2 + (λ− ρ2χγ2/2 + ρn2/2ρ2χ− ρnγ)q2

∆A

(∂τθV)(q, ω), (5–20)

uT (q, ω) =(ρn − ρ2χγ)ω

ρnω2 + µq2(∂iθ

V)(q, ω), (5–21)

where ∆A is given by Eq. 3–103, and (∂iθV)(q, ω) and (∂τθ

V)(q, ω) are the temporal

Fourier transforms of

(∂iθV)(q, τ) = i

h

mεik

qk

q2

∑α

eαe−iq·xα(τ), (5–22)

(∂τθV)(q, τ) = −i

h

mεik

qk

q2

∑α

eαsαi (τ)e−iq·xα(τ), (5–23)

which are easily calculated from Eq. 5–8. By replacing Eqs. 5–19 and 5–21 into Eq. 5–11,

S2 reduces to the following form:

S2 =ρs~2

2m2

∑

α,β

eαeβ

∫dτ F0(x

α(τ)− xβ(τ))

+~2

2m2

∑

α,β

eαeβ

∫dτ

∫dτ ′sα

i (τ)sβi (τ ′)∂α

j ∂βj G2(x

α(τ)− xβ(τ ′), τ − τ ′; cL, c2)

− ~2

2m2

∑

α,β

eαeβ

∫dτ

∫dτ ′sα

i (τ)sβj (τ ′)∂α

i ∂βj G2(x

α(τ)− xβ(τ ′), τ − τ ′; cL, c2)

+~2(ρn − ρ2χγ)2

8m2ρn

∑

α,β

eαeβ

∫dτ

∫dτ ′ G1(x

α(τ)− xβ(τ ′), τ − τ ′; cT ), (5–24)

89

where ∂αi ≡ ∂/∂xα

i (τ) and ∂βi ≡ ∂/∂xβ

i (τ ′), and we have defined some useful integrals

F0(x) =

∫d2q

1

q2e−iq·x, (5–25)

G1(x, τ ; c) =

∫d2q

1

q2e−iq·x

∫dω

2π

ω2

ω2 + c2q2eiωτ , (5–26)

G2(x, τ ; c, c) =

∫d2q

1

q2e−iq·x

∫dω

2π

aω2 + bq2

Deiωτ , (5–27)

with c and c being some sound speeds, and

D = ρn(ω2 + c2q2)(ω2 + c2q2). (5–28)

In Eq. 5–27, we have D = ∆A, a = ρnρs + (ρn − ρ2χγ)2/4 and b = ρs(λ − ρ2χγ2) + (ρn −ρ2χγ)2ρs/4ρ

2χ for S2 written in Eq. 5–24. Therefore, we derive the action by evaluating

the integrals defined in Eqs. 5–25 through 5–26. First, the term with F0 in Eq. 5–24 is

local in time and has a logarithmic divergence [F0(x) ∼ − ln(x)]. However, we show below

that this logarithmic divergence is canceled out with contributions from the other two

terms in Eq. 5–24. Second, we now evaluate G1 by separating out the local term, resulting

in

G1(x, τ ; c) =

∫d2q

1

q2e−iq·x

∫dω

2π

(1− c2q2

ω2 + c2q2

)eiωτ

= δ(τ)F0(x)− c2F1(x, τ ; c), (5–29)

where

F1(x, τ ; c) =

∫d2qe−iq·x

∫dω

2π

1

ω2 + c2q2eiωτ

=π

c√

c2τ 2 + |x|2 . (5–30)

Third, to calculate G2 we divide it into two terms by expanding in partial fractions: one

with the sound speed c and the other with c

G2(x, τ ; c, c) =A

ρn

F2(x, τ ; c) +B

ρn

F2(x, τ ; c), (5–31)

90

where

A ≡ ac2 − b

c2 − c2, (5–32)

B ≡ ac2 − b

c2 − c2, (5–33)

F2(x, τ ; c) =

∫d2q

1

q2e−iq·x

∫dω

2π

1

ω2 + c2q2eiωτ . (5–34)

The second term in Eq. 5–24 contains spatial derivatives of G2. The first spatial

derivatives equal minus the Laplacian of G2 because ∂βk = −∂α

k . Next, in Ref. [90] it

is shown that the second spatial derivatives with vortex velocities can be converted into

temporal derivatives: from sαi (τ)sβ

j (τ ′)∂αi ∂β

j to ∂τ∂τ ′ . Hence, we can evaluate G2 as well as

the third term in Eq. 5–24 by calculating ∂2τF2 and ∂2F2. In fact, they can be written in

terms of F0 and F1:

∂2F2(x, τ ; c) = −F1(x, τ ; c), (5–35)

and

∂2τF2(x, τ ; c) = −F0(x)δ(τ) + c2F1(x, τ ; c). (5–36)

Using Eqs. 5–30, 5–35 and 5–36, we obtain

S2 =~2

m2

[ρs

2+

(ρn − ρ2χγ)2

8ρn

− a

2ρn

] ∑

α,β

eαeβ

∫dτ F0(x

α(τ)− xβ(τ))

+~2A

2m2ρn

∑

α,β

eαeβ

∫dτ

∫dτ ′

[sα

i (τ)sβi (τ ′) + c2

L

]F1(x

α(τ)− xβ(τ ′), τ − τ ′; cL)

+~2B

2m2ρn

∑

α,β

eαeβ

∫dτ

∫dτ ′

[sα

i (τ)sβi (τ ′) + c2

2

]F1(x

α(τ)− xβ(τ ′), τ − τ ′; c2)

− ~2c2T (ρn − ρ2χγ)2

8m2ρn

∑

α,β

eαeβ

∫dτ

∫dτ ′ F1(x

α(τ)− xβ(τ ′), τ − τ ′; cT ), (5–37)

where

A =[ρnρs + (ρn − ρ2χ2γ)2/4]c2

L − ρs(λ− ρ2χγ2)− (ρn − ρ2χ2γ)2ρs/4ρ2χ

c2L − c2

2

, (5–38)

91

and

B =[ρnρs + (ρn − ρ2χ2γ)2/4]c2

2 − ρs(λ− ρ2χγ2)− (ρn − ρ2χ2γ)2ρs/4ρ2χ

c22 − c2

L

. (5–39)

Note that the local term cancels out because a = ρsρn + (ρn − ρ2χγ)2/4, and we finally get

the action for vortices in supersolids

SE = S1 − π~2(ρn − ρ2χγ)2

8m2ρncT

∑

αβ

eαeβ

∫dτ

∫dτ ′

c2T√

c2T (τ − τ ′)2 + |xα(τ)− xβ(τ ′)|2

+π~2A

2m2ρncL

∑

αβ

eαeβ

∫dτ

∫dτ ′

sαi (τ)sβ

i (τ ′) + c2L√

c2L(τ − τ ′)2 + |xα(τ)− xβ(τ ′)|2

+π~2B

2m2ρnc2

∑

αβ

eαeβ

∫dτ

∫dτ ′

sαi (τ)sβ

i (τ ′) + c22√

c22(τ − τ ′)2 + |xα(τ)− xβ(τ ′)|2 . (5–40)

From the derived action we can infer that each wave mode coupling to vortices produces

a non-local “Coulomb” potential term. In particular, a wave mode perpendicular to the

vortex velocity generates an additional term which is proportional to the product of two

vortex velocities at different times. Without coupling, the elastic variable is not related

to the phase variable so that one can integrate out the elastic part in Eq. 5–11, and the

remaining part of action is of a superfluid with vortices. In this case (ρn = ρ2χγ), we get

A = 0 and B = ρnρs, and the derived Lagrangian for vortices reduces to

SE = S1 +π~2ρs

2m2c2

∑

αβ

eαeβ

∫dτ

∫dτ ′

sαi (τ)sβ

i (τ ′) + c22√

c22(τ − τ ′)2 + |xα(τ)− xβ(τ ′)|2 , (5–41)

which is the result obtained by Eckern and Schmid [90].

We calculate the vortex mass by taking the limit as c2(τ − τ ′)2 À |xαi (τ) − xβ

i (τ ′)|2

from the derived action, Eq. 5–40. Then we obtain a frequency-dependent mass of vortex

in a supersolid

Mvortex = − π~2

2m2

[ρ2χ +

1

4(ρn − ρ2χγ)2

(1

µ+

1

λ− ρ2χγ2

) ][γ + ln(ωε)] , (5–42)

where γ = 0.5772 . . . is Euler’s constant, and ε is a cut-off introduced to regulate the

divergence in temporal integrals. For the case in which γ = 0, the inertial mass of a vortex

92

reduces to

Mvortex ' − π~2

2m2

[ρ2χ +

ρn2

4

(1

µ+

1

λ

) ][γ + ln(ωε)] . (5–43)

Once again, without the coupling of superfluid to the displacement vector we recover the

mass of vortex in superfluid, obtained by Eckern and Schimd.

5.2 Dislocation Dynamics

Having completed the dynamics of vortices, let us now consider dislocations and study

their dynamics in supersolids by deriving an effective action. Analogous to vortices in

superfluid, dislocations produce singularities breaking the discrete translational symmetries

in lattices: the line integral along a closed path enclosing a dislocation yields

∮dui = bi, (5–44)

where b is the Burgers vector. In a complete circulation around a dislocation a Burgers

vector causes a mismatch to the circulation in an ideal lattice by an amount of b. There

are two types of dislocation lines: edge dislocations and screw dislocations. If l is the

tangent unit vector to the dislocation line, for edge dislocations l ⊥ b while for screw

dislocations l ‖ b. Since we work with two-dimensional supersolids, there are only edge

dislocations because screw dislocations produce a displacement perpendicular to the plane

of the system.

The derivation of the effective action for dislocations is analogous to the calculation

done for vortices in Section 5.1. First we take into account dislocations by introducing

explicitly the non-analytical part due to dislocations in the displacement vector:

u = uS + uD, (5–45)

where uS is analytic and uD is singular. Consequently, the contour integral for a system of

dislocations results in ∮dui =

∮duD

i =∑

α

bαi , (5–46)

93

where the summation is over all the enclosed dislocations. Thus we take the ansatz for uD

uDi =

∑α

bαi

2πarctan

[y − yα(τ)

x− xα(τ)

], (5–47)

where bα is the Burgers vector of the α-th edge dislocation located at xα(τ) =

[xα(τ), yα(τ)]. Then upon replacing the decomposition of the displacement vector into

Eq. 5–3, the action can be written as the sum of two parts:

S1 =

∫dτ

∫d2xλiju

Dij, (5–48)

and

S2 =1

2

∫dτ

∫d2x

[ρn(∂τu

Si )

2 + 2ρn(∂τuSi )(∂τu

Di ) + ρn(∂τu

Di )2 + uS

ijσSij + uS

ijσDij + uD

ijσSij

+uDijσ

Dij + χ(∂τθ)

2 + ρs(∂iθ)2 + 2iΓuS

ii∂τθ + iΓ∂τuDi ∂iθ + iΓuD

ii∂τθ

], (5–49)

where

uS,Dij =

∂iuS,Dj + ∂ju

S,Di

2, (5–50)

and

σS,Dij = Λδiju

S,Dkk + 2µuS,D

ij . (5–51)

In S2 we also introduced some simple notations: Γ ≡ ρn − ρ2γχ, χ ≡ ρ2χ and Λ ≡λ − ρ2γ2χ. Second, we remove the analytic variables uS and θ in the action by means of

the equations of motion. The equations of motion are obtained by taking the variations of

the action Eq. 5–3 with respect to θ and uSi :

χ∂2τθ + ρs∂

2θ + iΓ∂τuSii = − i

2Γ∂i(∂τu

Di )− i

2Γ∂τu

Dii , (5–52)

ρn∂2τu

Sk + ∂jσ

Skj + iΓ∂k∂τθ = −ρn∂τ (∂τu

Dk )− ∂jσ

Dkj. (5–53)

When the coupling between the superfluidity and the elasticity is neglected (Γ = 0), uS

and θ become completely decoupled from each other, and dislocations produce only elastic

94

deformations. We invert the equations of motion to obtain uS and θ, resulting in

uSi (q, ω) =

(−iρnωT−1

ij − Γ

2qjT

−1i0

)∂τu

Dj (q, ω) + iqkT

−1ij σD

jk(q, ω) +Γ

2ωT−1

i0 uDjj, (5–54)

and

θ(q, ω) =

(−iρnωT−1

0j − Γ

2qjT

−100

)∂τu

Dj (q, ω) + iqkT

−10j σD

jk(q, ω) +Γ

2ωT−1

00 uDjj (5–55)

where

T−1ij =

(δij − qiqj

q2

)1

ρnω2 + µq2+

qiqj

q2

χω2 + ρsq2

χ∆A

, (5–56)

T−10i = T−1

0i = iΓqiω

χ∆A

, (5–57)

T−100 =

ρnω2 +

(Λ + 2µ

)q2

χ∆A

. (5–58)

Now we use Eqs. 5–54 and 5–55 to write the action in terms of uD (along with σDij). Then

the last step in deriving the action for dislocations is to use the ansatz for uD, Eq. 5–47,

and express the action in terms of the coordinates of dislocations (Appendix F for details).

The result is

95

S2 =(2π)2

2

(Λ + 2µ +

Γ2

4χ

) ∑

α,β

bαi bβ

i

∫dτF0(x

α(τ)− xβ(τ))

+(2π)2

2µ

∑

α,β

bαi bβ

i

∫dτ

∫dτ ′

[sα

j (τ)sβj (τ ′) + c2

T

]F1(x

α(τ)− xβ(τ ′), τ − τ ′; cT )

− (2π)2

2µ

∑

α,β

bαi bβ

j

∫dτ

∫dτ ′

[sα

k (τ)sβk(τ ′) + 4c2

T

]∂α

i ∂βj F2(x

α(τ)− xβ(τ ′), τ − τ ′; cT )

− (2π)2µ∑

α,β

bαi bβ

j

∫dτ

∫dτ ′sα

i (τ)sβk(τ ′)∂α

j ∂βk F2(x

α(τ)− xβ(τ ′), τ − τ ′; cT )

+(2π)2

2

∑

α,β

bαi bβ

j

∫dτ

∫dτ ′sα

k (τ)sβk(τ ′)∂α

i ∂βj G

(1)2 (xα(τ)− xβ(τ ′), τ − τ ′; cL, c2)

− (2π)2∑

α,β

bαi bβ

j

∫dτ

∫dτ ′sα

j (τ)sβk(τ ′)∂α

i ∂βk G

(2)2 (xα(τ)− xβ(τ ′), τ − τ ′; cL, c2)

− (2π)2

2

∑

α,β

bαi bβ

i

∫dτ

∫dτ ′∂α

j ∂βj G

(3)2 (xα(τ)− xβ(τ ′), τ − τ ′; cL, c2)

+ 2(2π)2µ2∑

α,β

bαi bβ

j

∫dτ

∫dτ ′∂α

i ∂βj G

(4)2 (xα(τ)− xβ(τ ′), τ − τ ′; cL, c2), (5–59)

where G(1)2 , G

(2)2 , G

(3)2 and G

(4)2 can be written in terms of F2 by using Eqs. 5–31 through

5–33 with

a(1) = ρn

(Λ + 2µ

)+

ρnΓ

4χ, (5–60)

b(1) =Λ + 2µ

χ

(ρnρs +

Γ2

4

), (5–61)

a(2) = ρnΛ +ρnΓ

4χ, (5–62)

b(2) = ρnρsΛ

χ+

Γ2

4

(Λ− 2µ

), (5–63)

a(3) = Λ2 + ΛΓ2

χ+

(Γ2 + ρsρn

) Γ2

4χ2, (5–64)

b(3) =ρs

χ

[Λ2 +

Γ2

4χ

(Λ + 2µ

) ], (5–65)

a(4) = 1, (5–66)

96

b(4) =ρs

χ. (5–67)

In Eq. 5–59 the spatial derivatives of the function F2 is involved. Since ∂αi = [xα

i (τ) −xβ

i (τ ′)]/X∂X with X = |xα(τ)− xβ(τ ′)|, we have

∂i∂jF2(x, τ ; c) = −π

cδij

1

c|τ |+√

c2|τ |2 + |x|2 +π

c

xixj

|x|4(c|τ | −

√c2|τ |2 + |x|2)2

√c2|τ |2 + |x|2 . (5–68)

As in the case of vortices, we obtained non-local “Coulomb” potentials for the propagating

modes existing in supersolids (the second and the penultimate terms in Eq. 5–59). It is

worthwhile to notice that the derived action for dislocations is anisotropic even though

the solid is isotropic. This is because of the vectorial characteristics of the singularity

(Burgers vector) due to dislocations which breaks the discrete symmetry of the lattice

and leaves the lattice anisotropic. In the case of vortices the effective action Eq. 5–40

remains isotropic because the singularity caused by a vortex is a scalar. In addition,

with dislocations a term local in time appears in the action. More interestingly, it does

not depend on the superfluid density. Thus this local term seems to be intrinsic to the

dynamics of dislocations in normal solids; however, superfluidity makes a contribution

through the coupling with dislocations (the term with Γ). If we neglect the interaction

between the superfluidity and the elasticity (Γ = 0), the coefficient of the local term

becomes ρnc2L which could suggest that its origin is due to the longitudinal elastic sound

modes.

On the other hand, we can obtain the Peach-Koehler force from S1. Since the stress

tensor λ is constant, it can be taken out of the integral, and the integral of uDij becomes a

summation over each dislocation of a “surface” integral over the two sides of the cut along

which the singular displacement vector of the α-th dislocation undergoes a discontinuity

of bα. The two surfaces of the cut are denoted by Σ− on which uD = 0 and Σ+ on which

uD = bα, where dΣ−i and dΣ+

i are given by −[z × dxα(τ)]i and [z × dxα(τ)]i, respectively

97

(Fig. 5-1). Then we have

S1 =∑

α

λij

∫dτ

(uα

i |Σ+ − uαi |Σ−

) ∫[z× dxα(τ)]i

=∑

α

λij

∫dτbα

j

∫[z× dxα(τ)]i , (5–69)

where uα is the singular part of the α-th dislocation’s displacement vector. The negative

of coefficient of dxα(τ) in the integrand of Eq. 5–69 is the force acting on the α-th

dislocation. By the cyclic property we obtain

FαPK = − (λ · ba)× z, (5–70)

which is the Peach-Koehler force perpendicular to the vector λ · bα in the lattice plane

[71, 72].

Similar to vortices we now can define an inertial mass for a slowly moving dislocation.

In the limit c2(τ − τ ′)2 À |xαi (τ)− xβ

i (τ ′)|2, we obtain

Mdislij = 2π3 [γ + ln(ωε)]

{2bibj

ρs(Λ + 2µ)2

[ρsρn(Λ + µ)2 +

Γ2

4Λ2

]

−δijbkbk

[2ρn +

ρn(Λ2 − µ2)

(Λ + 2µ)2+

Γ2

2ρs

Λ2 + 2µ2

(Λ + 2µ)2

]}. (5–71)

The effective mass of a dislocation is a tensor because of the reason discussed earlier. If γ

is neglected, we get

Mdislij = 2π3ρn [γ + ln(ωε)]

{2bibj

ρs(λ + 2µ)2

[ρs(λ + µ)2 +

ρn

4λ2

]

−δijbkbk

[2 +

λ2 − µ2

(λ + 2µ)2+

ρn(λ2 + 2µ2)

2ρs(λ + 2µ)2

]}. (5–72)

An effective action for dislocations in normal solids can be obtained from Eq. 5–59 by

neglecting the coupling between superfluidity and elasticity (Γ = 0). For a normal solid,

98

we get the mass of a dislocation

Mdislij = 2π3ρn [γ + ln(ωε)]

{2bibj

(Λ + µ

Λ + 2µ

)2

− δijbkbk

[2 +

Λ2 − µ2

(Λ + 2µ)2

]}. (5–73)

In summary, in this chapter we have considered two basic topological defects (vortices

and edge dislocations) which could exist in a supersolid phase. We have derived the

effective actions in terms of the coordinates of topological defects. The inertial mass

associated with the kinetic energy of such defects coupling with the elastic deformation

and superfluidity were obtained.

99

z

z ×^

^dxα

^dxα

^dxα

Σ-

-Σ+

z ×^

Figure 5-1. Cut for an edge dislocation. On surface Σ− uD = 0 whereas on Σ+ uD = b.The unit normals of Σ± are given by ±z × dxα.

100

CHAPTER 6CONCLUSION

We proposed a viscoelastic solid model as an alternative explanation for the recent

torsional oscillator observations on solid 4He. We have first studied the dynamical

response of isotropic viscoelastic solids with cylindrical symmetry to an oscillatory

shear stress using the no-slip boundary condition. We found the elastic resonance effect

in the effective moment of inertia and the effective damping coefficients. At low driving

frequencies, the effective moment of inertia decreases as the shear modulus increases.

This leads us to a possible connection between the anomalous increase in shear modulus

obtained by Day and Beamish [34] and the NCRI observed by Kim and Chan [19, 20].

However, the quantitative estimate of the period shift due to the 10 % increase in the

shear modulus was found to be only about one hundredth of the the observed period shift

in TO experiments. We also found that the frequency-dependent complex shear modulus

of a viscoelastic solid can explain the results of the TO experiments. In our model, a

characteristic time, which is related to the viscosity of solids increases rapidly as the

temperature is lowered, and causes both the drop in the resonant period and the peak in

the inverse of Q-factor. Our viscoelastic model predicts ∆Q−1max/(∆P/P0) = 1 as other

theoretical works [49, 50]; however, the experimental values of this ratio is less than the

predicted value varying from 0.01 to 0.65. The consequence of this discrepancy between

the theories and the experiments is that any theoretical model does not fit both the period

shift and the change in the inverse of Q-factor. In this work, we found that we could

identify the change in the inverse of Q-factor, but only the 10 % of the resonant period

shift. This fact could suggest that the unexplained part of the period shift might be due to

supersolidity.

In the second part of work we investigated the hydrodynamic properties of a model

supersolid. We introduced the variational principle to derive a Lagrangian density as well

as the non-dissipative hydrodynamics for supersolids. One of our main results is that

101

the coupling constant between the elastic variable and the superfluid, which is set by the

Galilean covariance, the conservation laws and the broken symmetries, is the density of the

normal component. Using the hydrodynamic equations of motion we calculated the second

and fourth sound speeds of defects for a model supersolid.

Next, we have calculated the density-density correlation function for an isotropic

supersolid in local thermal equilibrium using linear response theory. First, we found

that when dissipation is neglected, each pair of propagating modes whose sound speed

is c produces a δ-function pair located at ω = ±cq in the density-density correlation

function. Second, we have extended the study to include dissipation. Due to dissipation

the neglected defect diffusion mode of a normal solid appears as a central Rayleigh peak

in the density-density correlation function. Analogous to the superfluid transition of

liquid 4He, we found that the Rayleigh peak due to the defect diffusion mode of a normal

solid becomes a Brillouin doublet of the pair of longitudinal second sound modes in the

supersolid transition. We proposed a Brillouin light scattering experiment to observe this

splitting as an alternative way to detect the supersolid phase.

Finally, we have studied the dynamics of vortices and dislocations in supersolids

using the derived Lagrangian for supersolids. An effective action for vortices and another

for dislocations were obtained in terms of the coordinates of vortices and dislocations.

In the case of vortices, we found that each sound mode existing in a supersolid couples

with vortices, and generates a non-local “Coulomb” potential. Contrary to a superfluid,

transverse propagating modes exist in a supersolid and these transverse elastic modes

contribute to the action for vortices as well. For dislocations each mode in supersolids

produces both Coulomb and non-Coulomb potentials which are non-local in time. Another

contrast to vortices is that a local term with a logarithmic divergence arises. As a result,

we calculated the frequency-dependent inertial mass of vortices and dislocations. The

vectorial nature of singularity of dislocations destroys the isotropy of solids and the

derived mass becomes a second rank tensor. In both cases the effective mass has a

102

logarithmic dependence on the frequency. On the other hand, we showed that the terms

linear in the displacement vector u and the superfluid variable θ of the action, which are

usually neglected in studying the dynamics without topological defects because of their

total derivative form, are responsible for the Magnus force acting on vortices and the

Peach-Koehler force on dislocations.

This thesis lays the groundwork for several future studies. First, the viscoelastic

model for solid 4He provided us with a relaxation time which might be related to

dislocations in solids. We hope that it would be possible to derive analytically the

relaxation time of viscoelasticity from the dynamic equations of dislocations. As suggested

by Day and Beamish, the dislocation motion is controlled by 3He impurities in solid

4He, and effectively changes the shear modulus and, possibly, the relaxation time of

solid 4He. Consequently, we would understand better the connection between the shear

modulus experiment and the TO experiment. In this regard, the derived action of a model

(super)solid with dislocation will be useful because it describes the dislocation dynamics

in (super)solids. Second, we can extend the viscoelastic model for solid 4He in a TO in

several directions. We can investigate the response of a supersolid in the TO experiment

by using the derived hydrodynamic equation of supersolids. We expect that when the

supersolid hydrodynamics is combined with the viscoelastic model, it would be possible

to fit all the TO results more precisely. Alternatively, we hope that the model could be

improved by considering the inhomogeneity of the system and/or a different boundary

condition. We think that the local variation of the relaxation time could smear out one

of the TO responses of the homogeneous system, and the slip boundary condition would

provide us with a larger period shift. Finally, we showed that the variational method is

systematically effective in deriving the non-dissipative hydrodynamics and the Lagrangian

density of a continuum medium which is characterized with conservation laws and broken

symmetries. The derived Lagrangian density was useful in studying the dynamics of

topological defects. We plan to apply the variational principle in other systems; possible

103

systems include Wigner crystals formed by excitons in electron-hole bilayers, striped

superfluids, and so on.

104

APPENDIX ACALCULATION OF BACK ACTION TERMS

In this appendix we show the detailed calculation to get the back action terms for

different geometries of torsion cells. Taking uφ(t) ∼ exp(−iωt), Eq. 2–12 reduces to

(∂2

r +1

r∂r − 1

r2+ q2

E + ∂2z

)uφ = 0, (A–1)

where qE ≡√

ρω2/µ(ω) with µ(ω) = µ(1 − iωτ). In the following we solve Eq. A–1 for

three different boundary conditions: infinite cylinder, finite cylinder, and infinite annulus.

For the coordinate system used in this appendix, refer to Fig. 2-1.

A.1 Infinite Cylinder

First, we consider an infinite cylinder of radius R. The boundary condition for uφ is

Rφ0e−iωt for r = R. Since there is no z dependence for infinite cylinder, Eq. A–1 becomes

∂2ru +

1

r∂ru +

[q2E −

1

r2

]u = 0. (A–2)

The differential equation Eq. A–2 can be reduced to the Bessel equation of order one.

Applying the boundary condition and the finiteness at the center we get the displacement

field

uφ = Rφ0J1(qEr)

J1(qER)e−iωt, (A–3)

with qER = ωτE. The obtained displacement field uφ yields a non-vanishing stress

σφr(r, t) = µ(ω)

(∂uφ

∂r− uφ

r

)= −qEµ(ω)Rφ0

J2(qEr)

J1(qER)e−iωt. (A–4)

The corresponding total torque for a height h is

M inf cyl(t) = −ω2φ0IRBH(qER)e−iωt, (A–5)

where the moment of inertia of rigid body IRB = πρR4h/2. Using Eqs. 2–17 and 2–18,

from Eq. A–5, we finally get the effective moment of inertia

I inf cyleff (ω) = IRB<[H(qER)], (A–6)

105

and the effective damping coefficient

γinf cyleff (ω) = ωIRB=[H(qER)]. (A–7)

Also, we get the back action term using Eq. 2–6

ginf cyl(ω) = ω2IRBH(qER). (A–8)

A.2 Finite Cylinder

For the second case, we consider a finite cylinder with radius R and height h. The

boundary condition for uφ is

uφ =

Rφ0e−iωt for r = R

rφ0e−iωt for z = ±h/2.

(A–9)

In this case we need to solve Eq. A–1:

(∂2

r +1

r∂r − 1

r2+ q2

E + ∂2z

)uφ = 0. (A–10)

We decompose uφ into two parts: uφ = φ0r + V (r, z) with V = 0 at the boundary. Then

Eq. A–10 reduces to

∂2rV +

1

r∂rV +

[q2E −

1

r2

]V + ∂2

zV = −q2Eφ0r. (A–11)

The solution of Eq. A–11 is

V (r, z) = q2Eφ0

∫dx′ r′G(x,x′), (A–12)

where the Green’s function G(x,x′) satisfies the inhomogeneous differential equation

[∂2x −

1

r2+ q2

E

]G(x,x′) = −4π

rδ(r − r′)δ(φ− φ′)δ(z − z′), (A–13)

106

with G = 0 at the boundary. Using

δ(z − z′) =1

π

∫ ∞

0

dk cos[k(z − z′)], (A–14)

δ(φ− φ′) =1

2π

∞∑m=−∞

eim(φ−φ′), (A–15)

we can expand G(x,x′) as

G(x,x′) =1

2π2

∞∑m=−∞

∫ ∞

0

dk eim(φ−φ′) cos[k(z − z′)] gm(k, r, r′). (A–16)

Replacing this into Eq. A–13, we get the modified Bessel equation of order m′ for

gm(k, r, r′):∂2gm

∂r2+

1

r

∂gm

∂r−

(k′2 +

m′2

r2

)gm = −4π

rδ(r − r′), (A–17)

where k′2 ≡ k2 − q2E and m′2 ≡ m2 + 1. The general solution, using the finiteness condition

at the center, is

gm(k, r, r′) = Im′(k′r<) [AIm′(k′r>) + BKm′(k′r>)] , (A–18)

where r< = min{r, r′} and r> = max{r, r′}. Then the use of the boundary condition at

r = r′,dgm

dr

∣∣∣∣r′+ε

− dgm

dr

∣∣∣∣r′−ε

= −4π

r′, (A–19)

yields the Green’s function

G(x,x′) = − 2

π

∞∑m=−∞

∫ ∞

0

dk eim(φ−φ′) cos[k(z − z′)]×

×Im′(k′r<)

Im′(k′R)

[Km′(k′R)Im′(k′r>)− Im′(k′R)Km′(k′r>)

]. (A–20)

Finally, using Eq. A–12 we get the displacement field

uφ(r, z, t) = φ0re−iωt +

4q2Eφ0

π

∞∑m=1

(−1)m+1

2m− 1cos

[(2m− 1)π

hz

] [r

β2m

− RI1(βmr)

β2mI1(βmR)

]e−iωt,

(A–21)

107

where

βm =

√(2m− 1)2π2

h2− q2

E. (A–22)

First we see that we recover the results of the infinite cylinder case taking the limit that

h → ∞. In this limit, βm becomes iqE, and the first term in the second squared bracket

cancels the first term in Eq. A–21, and the second term in the second squared bracket

becomes the displacement field for infinite cylinder Eq. A–3.

There are two non-vanishing components of the stress tensor which are

σφr(r, z, t) = µ(ω)

(∂uφ

∂r− uφ

r

)

= −4µ(ω)q2Eφ0R

πe−iωt

∞∑m=1

(−1)m+1

2m− 1

[cos

(2m− 1)π

hz

]I2(βmr)

βmI1(βmR),

(A–23)

σφz(r, z, t) = µ(ω)∂uφ

∂z

= −4µ(ω)q2Eφ0

he−iωt

∞∑m=1

(−1)m+1 sin(2m− 1)π

hz

[r

β2m

− RI1(βmr)

β2mI1(βmR)

].

(A–24)

The total torque is

M(t) =

∫ 2π

0

dφ

∫ d

−d

dz R2σφr|r=R + 2

∫ 2π

0

dφ

∫ R

0

r2σφz|z=−h/2

= −8ω2φ0

π2IRBe−iωt

∞∑m=1

[1

(2m− 1)2−

(πR

hαm

)2]H(iαm)

−8ω2R2φ0

h2IRBe−iωt

∞∑m=1

1

α2m

, (A–25)

where

αm ≡ Rβm =

√(2m− 1)2π2R2

h2− q2

ER2. (A–26)

Then, the effective moment of inertia for a viscoelastic cylinder is

Icyleff (ω)

IRB

=8

π2

∞∑m=1

<{[

1

(2m− 1)2− π2R2

h2α2m

]H(iαm)

}+

8R2

h2

∞∑m=1

<{

1

α2m

}, (A–27)

108

and the effective damping coefficient is

γfin cyleff (ω)

ωIRB

=8

π2

∞∑m=1

={[

1

(2m− 1)2− π2R2

h2α2m

]H(iαm)

}+

8R2

h2

∞∑m=1

={

1

α2m

}. (A–28)

The back action term is

gcyl(ω) =8ω2

π2IRB

∞∑m=1

[1

(2m− 1)2−

(πR

hαm

)2]H(iαm) +

8ω2R2

h2IRB

∞∑m=1

1

α2m

. (A–29)

A.3 Infinite Annulus

We lastly consider an infinite annulus of inner radius Ri and outer radius R. In this

case the boundary condition for uφ becomes

uφ =

Riφ0e−iωt for r = Ri

Rφ0e−iωt for r = R.

(A–30)

The differential equation for uφ is the same as that of an infinite cylinder Eq. A–2 whose

general solutions are J1(qEr) and N1(qEr). Since the finiteness at the origin is no longer

necessary, the displacement field is given by

uφ = φ0AJ1(qEr) + φ0BN1(qEr), (A–31)

where

A =RiN1(qER)−RN1(qERi)

J1(qERi)N1(qER)− J1(qER)N1(qERi), (A–32)

B =RiJ1(qER)−RJ1(qERi)

N1(qERi)J1(qER)−N1(qER)J1(qERi). (A–33)

The non-vanishing stress and the total torque for this case are

σφr = µ(ω)

(∂ruφ − uφ

r

)= −qEφ0µ(ω)e−iωt

[AJ2(qEr) + BN2(qEr)

], (A–34)

109

M(t) = −h

∫ 2π

0

dφ σ|r=RiR2

i + h

∫ 2π

0

dφ σ|r=RR2

=2πφ0ρhR2

i ω2

qE

e−iωt

[AJ2(qERi) + BN2(qERi)

]

−2πφ0ρhR2ω2

qE

e−iωt

[AJ2(qER) + BN2(qER)

]. (A–35)

The effective moment of inertia is

I inf anneff (ω) = <

{2πρhR2

qE

[AJ2(qER) + BN2(qER)

]}

−<{

2πρhR2i

qE

[AJ2(qERi) + BN2(qERi)

]}. (A–36)

The effective dampting coefficient is

γinf anneff (ω) = =

{2πρhR2ω

qE

[AJ2(qER) + BN2(qER)

]}

−={

2πρhR2i ω

qE

[AJ2(qERi) + BN2(qERi)

]}. (A–37)

The back action term is

ginf ann(ω) =2πρhR2

i ω2

qE

[AJ2(qERi) + BN2(qERi)

]− 2πρhR2ω2

qE

[AJ2(qER) + BN2(qER)

].

(A–38)

110

APPENDIX BVARIATIONAL PRINCIPLE IN SUPERSOLIDS WITH THE ROTATIONAL

VELOCITY OF SUPER COMPONENTS

In Chapter 3 we derived the hydrodynamics of a supersolid using the variational

principle, and showed that the velocity of the super-component is irrotational. In this

Appendix we show that it is possible to derive systematically the hydrodynamics of

a supersolid with the transverse part of vs which generated by vortices present in the

supersolid. The transverse part of vs can be obtained by imposing another Lin’s constraint

[61]:

∂tχ +1

ρ

[ρsvsi + (ρ− ρs)vni

]∂iχ = 0. (B–1)

Then the Lagrangian of the isotropic supersolid with all the constraints, Eqs. 3–22, 3–24,

3–25 and B–1 becomes

LSS =1

2ρsvs

2 +1

2(ρ− ρs)vn

2 − USS(ρ, ρs, s, Rij) + φ

{∂tρ + ∂i

[ρsvsi + (ρ− ρs)vni

]}

+α

[∂ts + ∂i(svni)

]+ γ

{∂t(ρχ) + ∂i

[ρsχvsi + (ρ− ρs)χvni

]}

+βi

[∂t(sRi) + ∂j(sRivnj)

], (B–2)

where φ, α, β, and γ are Lagrangian multipliers, and we have used the second Lin’s

condition combined with the mass conservation equation.

The equations of motion are calculated by taking the variations with respect to the

dynamical variables:

• δρ1

2vn

2 − ∂USS

∂ρ− Dnφ

Dt− χ

Dnγ

Dt= 0 (B–3)

• δρs1

2vs

2 − 1

2vn

2 − ∂USS

∂ρs

− (vsi − vni)(∂iφ + χ∂iγ) = 0 (B–4)

• δsDnα

Dt+ Ri

Dnβi

Dt+

∂USS

∂s= 0 (B–5)

111

• δvsi

ρsvsi − ρs∂iφ− ρsχ∂iγ = 0 (B–6)

• δvni

(ρ− ρs)vni − (ρ− ρs)∂iφ− s∂iα− sRj∂iβj − (ρ− ρs)χ∂iγ = 0 (B–7)

• δRi

sDnβi

Dt− ∂j

(∂USS

∂Rji

)= 0 (B–8)

• δχρ∂tγ + ρsvsi∂iγ + (ρ− ρs)vni∂iγ = 0 (B–9)

From Eq. B–6, we obtain the following Clebsch potential representations for the

velocity of the super-component

vsi = ∂iφ + χ∂iγ. (B–10)

Therefore, we get

∇× vs = ∇χ×∇γ 6= 0. (B–11)

On the other hand, the use of the Clebsch representation of vs leads us Eq. 3–34.

Taking the gradient of Eq. B–3, we get the Euler equation for vs

Dsvsi

Dt= −∂i

(∂USS

∂ρ

)− ∂i

(∂USS

∂ρs

)− ρ− ρs

ρ

[(vs − vn)× (∇× vs)

]

i

= −∂iµ− ρ− ρs

ρ

[(vs − vn)× (∇× vs)

]

i

. (B–12)

112

The Euler equation for vn can be derived following the same calculation done in previous

sections:

(ρ− ρs)Dnvni

Dt= −(ρ− ρs)∂i

(∂USS

∂ρ

)− (ρ− ρs)∂i

(∂USS

∂ρs

)− ∂j

(∂USS

∂Rjk

)∇Rik

−s∂i

(∂USS

∂s

)− 1

2(ρ− ρs)∂i(vnj − vsj)

2 −[∂tρs + ∂j(ρsvnj)

]vsi

−{

∂t(ρ− ρs) + ∂j

[(ρ− ρs)vnj

]}vni

+ρs

ρ(ρ− ρs)

[(vs − vn)× (∇× vs)

]

i

.

= −(ρ− ρs)∂iµ− ∂jλjkRik − s∂iT − 1

2(ρ− ρs)∂i(vnj − vsj)

2

−{

∂t(ρ− ρs) + ∂j

[(ρ− ρs)vnj

]}(vni − vsi),

+ρs

ρ(ρ− ρs)

[(vs − vn)× (∇× vs)

]

i

. (B–13)

However, the momentum conservation equation for this case in which ∇ × vs = 0, is the

same as that of the case of the longitudinal vs, Eq. 3–41.

113

APPENDIX CSTATIC CORRELATION FUNCTIONS OF ISOTROPIC SUPERSOLIDS

Given Eq. 3–19, the thermodynamic relation of the static free energy density FSS =

ESS − Ts is calculated

dFSS = −sdT − λikdRik + µdρ. (C–1)

Without thermal fluctuations we expand the static free energy density FSS up to the

second order in fluctuations in the density δρ and the displacement vector u:

FSS =1

2

∂µ

∂ρ

∣∣∣∣wij

(δρ)2 +∂µ

∂wij

∣∣∣∣ρ

δρwij +1

2

∂λij

∂wlk

∣∣∣∣ρ

wijwlk. (C–2)

Using Eqs. 3–89 through 3–91 the free energy can be written as, in Fourier space,

FSS =1

2µq2u2

T +1

2

(δρ(q) uL(q)

)B

δρ(−q)

uL(−q)

, (C–3)

where

B =

1ρ20χ

−iqγ

iqγ q2λ

. (C–4)

Then the static correlation functions can be easily read out from Eq. C–3. We obtain

χρρ(q) = β 〈δρ(q)δρ(−q)〉 =ρ2

0χλ

λ− ρ20γ

2χ, (C–5)

χuLρ(q) = β 〈uL(q)δρ(−q)〉 =iρ2

0χγ

q(λ− ρ20γ

2χ). (C–6)

114

APPENDIX DKUBO FUNCTIONS AND CORRELATION FUNCTIONS

In general, the Laplace-Fourier transformations of hydrodynamic equations of a

physical system will have a form:

θα(q, ω) = Aαγ(q, ω)θγ(q, t = 0), (D–1)

where θα are some hydrodynamic variables and Aαγ is an arbitrary matrix which can be

obtained from the hydrodynamic equations. The index α indicates the hydrodynamic

variable θα. On the other hand, the linear response theory tells us how θα(q, ω) is related

to θγ(q, t = 0) through the response function χαβ [74],

θα(q, ω) =1

iω[χαβ(q, ω)− χαβ(q)] χ−1

βγ (q)θγ(q, t = 0). (D–2)

Note that χγβ(q) are the static susceptibilities which can be calculated from statistical

mechanics. Then, Eq. D–1 and D–2 imply that

1

iω[χαβ(q, ω)− χαβ(q)] = Aαγ(q, ω)χγβ(q), (D–3)

where we have used χ−1βγ χγδ = δβδ.

Now, we define a Kubo function Kαβ [78] as

i∂tKαβ(x,x′, t) =2

βχ′′αβ(x,x′, t), (D–4)

where χ′′(x,x′, t) is defined as

χ(x,x′, t) = 2iη(t)χ′′(x,x′, ), (D–5)

115

where η(t) is the step function. The Laplace transform of Kαβ is

Kαβ(x,x′, z) =

∫ ∞

0

dteiztKαβ(x,x′, t)

=1

β

∫ ∞

−∞

dω

πi

χ′′αβ(x,x′, ω)

ω (ω − z)

=1

izβ[χαβ(x,x′, z)− χαβ(x,x′, t = 0)] .

(D–6)

Therefore, using Eq. D–3, the Kubo function can be obtained

Kαβ(q′, ω) =1

βAαγ(q, ω)χγβ(q), (D–7)

and the susceptibility χ′′αβ is related to this Kubo function through Eq. D–4.

116

APPENDIX ECALCULATION OF THE DENSITY-DENSITY CORRELATION FUNCTION

Using Eq. D–7 the density-density Kubo function is obtained from Eq. 4–31

Kρρ(q, z) =χρρ(q)

kBTC−1

ρρ (q, z) +χuLρ(q)

kBTC−1

uLρ(q, z)

=χρρ(q)

kBT

iz3 + bρρz2 + dρρq

2z + eρρq2

Z+

χuLρ(q)

kBT

dρuLq2z + eρuL

q2

Z, (E–1)

where the static susceptibilities are given in Appendix C, and

Z ≡ (z2 − c2Lq2 + izq2DL)(z2 − c2

2q2 + izq2D2), (E–2)

bρρ = −Γγq2 − ρs0(ρn0 − ρs0)

ρn0

Λq2 − ζ − 2ρs0σ

ρn0

q2, (E–3)

dρρ = −iλ

ρn0

+ iγ, (E–4)

eρρ =ρs0Λλ

ρn0

q2 − ρs0Λγq2 +ρs0σγ

ρn0

q2, (E–5)

dρuL= (λ− ρ0γ)q, (E–6)

eρuL= iρs0

(Λ− σ

ρn0

)λq3 + i

ρs0

ρn0

[2σρ0 − ζ − ρ0Λ(ρ0 − 2ρs0)

]γq3. (E–7)

We can rewrite the density-density Kubo function by doing partial fraction expansion.

Then each term in Eq. E–1 can be separated into the first sound part and the second

sound part as follows

aijz3 + bijz

2 + dijq2z + q2eij

(z2 − c2Lq2 + izDLq2)(z2 − c2

2q2 + izD2q2)

=Aijz + Bij

z2 − c2Lq2 + izDLq2

+Dijz + Eij

z2 − c22q

2 + izD2q2,

(E–8)

where Aij, Bij, Dij and Eij can be written as a function of aij, bij, dij and eij, Eqs. E–3

through E–7:

Aij =aij [c4

L − c22c

2L + q2DL(DLc2

2 −D2c2L)]

(c2L − c2

2)2 + q2(DL −D2)(DLc2

2 −D2c2L)

+ibij(c

22DL −D2c

2L) + dij(c

2L − c2

2) + ieij(DL −D2)

(c2L − c2

2)2 + q2(DL −D2)(DLc2

2 −D2c2L)

, (E–9)

117

Bij =iaijc

2Lq2(DLc2

2 −D2c2L) + bijc

2L(c2

L − c22)

(c2L − c2

2)2 + q2(DL −D2)(DLc2

2 −D2c2L)

+idijc

2Lq2(DL −D2) + eij [c2

L − c22 + q2DL(D2 −DL)]

(c2L − c2

2)2 + q2(DL −D2)(DLc2

2 −D2c2L)

, (E–10)

Dij =aij [c4

2 − c2Lc2

2 + q2D2(D2c2L − Γ1c

22)]

(c2L − c2

2)2 + q2(DL −D2)(DLc2

2 −D2c2L)

++ibij(c

2LD2 − Γ1c

22) + dij(c

22 − c2

L) + ieij(D2 − Γ1)

(c2L − c2

2)2 + q2(DL −D2)(DLc2

2 −D2c2L)

, (E–11)

Eij =iaijc

22q

2(D2c2L − Γ1c

22) + bijc

22(c

22 − c2

L)

(c2L − c2

2)2 + q2(DL −D2)(DLc2

2 −D2c2L)

+idijc

22q

2(D2 − Γ1) + eij [c22 − c2

L + q2D2(Γ1 −D2)]

(c2L − c2

2)2 + q2(DL −D2)(DLc2

2 −D2c2L)

. (E–12)

The functions in the density-density correlation function, Eq. 2–7, are given by

I1(q) = χρρ(q)Aρρ + χuLρ(q)AρuL, (E–13)

I2(q) = χρρ(q)Dρρ + χuLρ(q)DρuL, (E–14)

I3(q) = χρρ(q)Bρρ + χuLρ(q)BρuL− iq2DLI1(q), (E–15)

I4(q) = χρρ(q)Eρρ + χuLρ(q)EρuL− iq2D2I2(q). (E–16)

118

APPENDIX FDERIVATION OF AN EFFECTIVE ACTION FOR EDGE DISLOCATIONS

In this appendix we show the derivation of S2, Eq. 5–59, in detail starting with

Eq. 5–49 and the ansatz used for uD, Eq. 5–47. The use of the equations of motion

Eqs. 5–54 and 5–55 with Eqs. 5–56 through 5–58 leads us to the action in terms of the

singular displacement vector

S2 = S(1)2 + S

(2)2 + S

(3)2 + S

(4)2 , (F–1)

where

S(1)2 =

µ

2

∫dω

2π

∫d2q

(2π)2

q2

ω2 + c2T q2

(δij − qiqj

q2

)∂τu

Dj (q, ω)∂τu

Di (−q,−ω)

+1

2

∫dω

2π

∫d2q

(2π)2

qiqj

∆A

{ρn(Λ + 2µ)(χω2 + ρsq

2)

+Γ2

4

[ρnω

2 + (Λ + 2µ)q2]}

∂τuDj (q, ω)∂τu

Di (−q,−ω), (F–2)

S(2)2 =

1

2

∫dω

2π

∫d2q

(2π)2

[uD

ij(q, ω)σDij(−q,−ω)− qiqkT

−1il σD

lk(q, ω)σDij(−q,−ω)

+Γ2

4ω2T−1

00 uDjj(q, ω)uD

ii(−q,−ω) + iΓqkωT−10j σD

jk(q, ω)uDii(−q,−ω)

],(F–3)

S(3)2 =

1

2

∫dω

2π

∫d2q

(2π)2

Γ2

2qjω

ρnω2 − (Λ + 2µ)q2

∆A

∂τuDj (q, ω)uD

ii(−q,−ω), (F–4)

S(4)2 =

∫dω

2π

∫d2q

(2π)2qjω

(δil − qiql

q2

)1

ω2 + c2T q2

∂τuDl (q, ω)σD

ij(−q,−ω)

+

∫dω

2π

∫d2q

(2π)2

ωqiqjql

q2

ρnχω2 + (ρnρs + Γ2/2)q2

∆A

∂τuDl (q, ω)σD

ij(−q,−ω).(F–5)

119

Taking the ansatz, Eq. 5–47, one can easily calculate the spatial Fourier transforms:

∂juDi (q, τ) = iεjk

qk

q2

∑α

bαi e−iq·xa(τ), (F–6)

∂τuDi (q, τ) = −iεjk

qk

q2

∑α

bαi sα

j (τ)e−iq·xa(τ), (F–7)

σDij(q, τ) = iδijεlkΛ

qk

q2

∑α

bαl e−iq·xa(τ) + iµ

qk

q2

∑α

(bαj εik + bα

i εjk

)e−iq·xa(τ). (F–8)

Using these equations we get:

S(1)2 = −(2π)2µ

2

∑

α,β

bαi bβ

i

∫dτF0(x

α(τ)− xβ(τ))

+(2π)2

2

(µ− a1

ρn

) ∑

α,β

bαi bβ

j

∫dτ∂α

i ∂βj F3(x

α(τ)− xβ(τ))

+(2π)2µ

2

∑

α,β

bαi bβ

i

∫dτ

∫dτ ′

[sα

j (τ)sβj (τ ′) + c2

T

]F1(x

α(τ)− xβ(τ ′), τ − τ ′; cT )

−(2π)2µ

2

∑

α,β

bαi bβ

j

∫dτ

∫dτ ′

[sα

k (τ)sβk(τ ′) + c2

T

]∂α

i ∂βj F2(x

α(τ)− xβ(τ ′), τ − τ ′; cT )

+(2π)2

2

∑

α,β

bαi bβ

j

∫dτ

∫dτ ′sα

k (τ)sβk(τ ′)∂α

i ∂βj G

(1)2 (xα(τ)− xβ(τ ′), τ − τ ′; cL, c2)

−(2π)2

2

∑

α,β

bαi bβ

j

∫dτ

∫dτ ′∂α

i ∂βj G

(1)2 (xα(τ)− xβ(τ ′), τ − τ ′; cL, c2), (F–9)

where G(1)2 is defined by Eq. 5–31 with

a(1) = ρn

(Λ + 2µ

)+

ρnΓ

4χ, (F–10)

b(1) =Λ + 2µ

χ

(ρnρs +

Γ2

4

), (F–11)

120

S(2)2 =

(2π)2

2

(Λ + 2µ +

Γ2

4χ

) ∑

α,β

bαi bβ

i

∫dτF0(x

α(τ)− xβ(τ))

− (2π)2

2

(Λ + 2µ +

Γ2

4χ

) ∑

α,β

bαi bβ

j

∫dτ∂α

i ∂βj F3(x

α(τ)− xβ(τ))

− (2π)2µ

2c2T

∑

α,β

bαi bβ

j

∫dτ

∫dτ ′∂α

i ∂βj F2(x

α(τ)− xβ(τ ′), τ − τ ′; cT )

− (2π)2

2Λ2

∑

α,β

bαi bβ

j

∫dτ

∫dτ ′

(δij∂

αk ∂β

k − ∂αi ∂β

j

)G

(4)2 (xα(τ)− xβ(τ ′), τ − τ ′; cL, c2)

− (2π)2

2Γ2Λ

∑

α,β

bαi bβ

j

∫dτ

∫dτ ′

(δij∂

αk ∂β

k − ∂αi ∂β

j

)G

(5)2 (xα(τ)− xβ(τ ′), τ − τ ′; cL, c2)

+(2π)2

2

Γ2

4

∑

α,β

bαi bβ

i

∫dτ

∫dτ ′∂α

j ∂βj G

(6)2 (xα(τ)− xβ(τ ′), τ − τ ′; cL, c2)

− (2π)2

2

Γ2

4

∑

α,β

bαi bβ

j

∫dτ

∫dτ ′∂α

i ∂βj G

(6)2 (xα(τ)− xβ(τ ′), τ − τ ′; cL, c2), (F–12)

where G(4)2 , G

(5)2 , and G

(6)2 are defined by Eq. 5–31 with

a(4) = 1, (F–13)

b(4) =ρs

χ, (F–14)

a(5) =1

χ, (F–15)

b(5) = 0, (F–16)

a(6) =ρn

χ, (F–17)

b(6) =Λ + 2µ

χ, (F–18)

S(3)2 = π2 Γ2a2

ρn

∑

α,β

bαi bβ

j

∫dτ∂α

i ∂βj F3(x

α(τ)− xβ(τ))

−π2Γ2∑

α,β

bαi bβ

j

∫dτ

∫dτ ′sα

j (τ)sβk(τ ′)∂α

i ∂βk G

(7)2 (xα(τ)− xβ(τ ′), τ − τ ′; cL, c2)

+π2Γ2∑

α,β

bαi bβ

j

∫dτ

∫dτ ′∂α

i ∂βj G

(7)2 (xα(τ)− xβ(τ ′), τ − τ ′; cL, c2), (F–19)

121

where G(7)2 is defined by Eq. 5–31 with

a(7) =ρn

χ, (F–20)

b(7) = −Λ + 2µ

χ, (F–21)

S(4)2 = (2π)2

(Λ + µ

) ∑

α,β

bαi bβ

j

∫dτ∂α

i ∂βj F3(x

α(τ)− xβ(τ))

−(2π)2Λ∑

α,β

bαi bβ

j

∫dτ

∫dτ ′sα

j (τ)sβk(τ ′)∂α

i ∂βk G

(8)2 (xα(τ)− xβ(τ ′), τ − τ ′; cL, c2)

+(2π)2Λ∑

α,β

bαi bβ

j

∫dτ

∫dτ ′∂α

i ∂βj G

(8)2 (xα(τ)− xβ(τ ′), τ − τ ′; cL, c2)

−(2π)2µ∑

α,β

bαi bβ

j

∫dτ

∫dτ ′sα

i (τ)sβk(τ ′)∂α

j ∂βk F2(x

α(τ)− xβ(τ ′), τ − τ ′; cT )

−(2π)2µc2T

∑

α,β

bαi bβ

j

∫dτ

∫dτ ′∂α

i ∂βj F2(x

α(τ)− xβ(τ ′), τ − τ ′; cT ) (F–22)

where G(8)2 is defined by Eq. 5–31 with

a(8) = ρn, (F–23)

b(8) = − 1

χ

(ρnρs +

Γ2

2

), (F–24)

In Eqs. F–9 through F–22 G2 is defined as

G2(x, τ ; c, c) =A

ρn

F2(x, τ ; c) +B

ρn

F2(x, τ ; c), (F–25)

where

A ≡ ac2 − b

c2 − c2, (F–26)

B ≡ ac2 − b

c2 − c2, (F–27)

with a = b − a(c2 + c2) and b = −ac2c2. In deriving Eqs. F–9 through F–22 one can find

useful the fact that sαi (τ)∂α

i can be changed to −∂τ . Now we can combine S(1)2 , S

(2)2 , S

(3)2 ,

and S(4)2 together by using Eqs. 5–29, 5–35 and 5–36. We find that the local term with F0

122

still remains unlike to the case of vortices, but the local term with F3 cancels out. Then

we finally obtain the result that is given by Eq. 5–59.

123

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127

BIOGRAPHICAL SKETCH

Chi-Deuk Yoo was born in Seoul, Korea and immigrated to Argentina during his

high school years. He finished, overcoming cultural shock, high school in Buenos Aires,

Argentina, and enrolled in the Physics Department of the University of Buenos Aires,

where he received a Licenciado en Ciencias Fısicas in 2002. In the same year he was

admitted to the Physics Department of the University of Florida. During the early years

at the University of Florida he was interested in high-energy theory, but in 2004 he joined

Prof. Alan T. Dorsey’s group to study theoretical condensed matter physics.

128