superﬂuid-insulator transition - harvard...

M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Nature 415, 39 (2002).

Ultracold 87Rbatoms - bosons

Superfluid-insulator transition

g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

c

g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

c

“Boltzmann” theory of Nambu-Goldstone-Higgs excitations and

vortices

Boltzmann theory of

particles/holes

g

T

gc

0

InsulatorSuperfluid

Quantumcritical

TKT

c

CFT3 at T>0

No quasiparticles:Boltzmann theory does not apply

Condensed Matter > Strongly Correlated Electrons

Title: Conformal field theories at non-zerotemperature: operator product expansions, MonteCarlo, and holographyAuthors: Emanuel Katz, Subir Sachdev, Erik S. Sorensen, William Witczak-Krempa(Submitted on 12 Sep 2014 (v1), last revised 2 Dec 2014 (this version, v2))

Abstract: We compute the non-zero temperature conductivity of conserved flavor currents inconformal field theories (CFTs) in 2+1 spacetime dimensions. At frequencies much greater thanthe temperature, , the dependence can be computed from the operator productexpansion (OPE) between the currents and operators which acquire a non-zero expectation valueat T > 0. Such results are found to be in excellent agreement with quantum Monte Carlo studiesof the O(2) Wilson-Fisher CFT. Results for the conductivity and other observables are alsoobtained in vector 1/N expansions. We match these large results to the correspondingcorrelators of holographic representations of the CFT: the holographic approach then allows us toextrapolate to small . Other holographic studies implicitly only used the OPEbetween the currents and the energy-momentum tensor, and this yields the correct leading large

behavior for a large class of CFTs. However, for the Wilson-Fisher CFT a relevant "thermal"operator must also be considered, and then consistency with the Monte Carlo results is obtainedwithout a previously needed ad hoc rescaling of the T value. We also establish sum rules obeyedby the conductivity of a wide class of CFTs.

Comments: 25+16 pages; 5+2 figures. Single column. v2: Added new appendix about numericalmethods, expanded discussion, typos corrected

Subjects:Strongly Correlated Electrons (cond-mat.str-el); General Relativity and QuantumCosmology (gr-qc); High Energy Physics - Lattice (hep-lat); High Energy Physics -Theory (hep-th)

Journal reference: Phys. Rev. B 90, 245109 (2014)DOI: 10.1103/PhysRevB.90.245109Cite as: arXiv:1409.3841 [cond-mat.str-el] (or arXiv:1409.3841v2 [cond-mat.str-el] for this version)

Submission historyFrom: William Witczak-Krempa [view email] [v1] Fri, 12 Sep 2014 20:00:03 GMT (867kb,D)[v2] Tue, 2 Dec 2014 02:03:36 GMT (833kb,D)Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)

Link back to: arXiv, form interface, contact.

ℏω >> TkB ω

ω

ℏω/( T)kB

ω

Recent extension away from the critical point

Quantum critical fluid in graphene

ARO MURI meeting, BerkeleyJanuary 11, 2016

Subir Sachdev

HARVARDTalk online: sachdev.physics.harvard.edu

Andrew Lucas

Philip KimJesse Crossno

Kin Chung Fong

kx

ky

Graphene

Electron Fermi surface

Graphene

Hole Fermi surface

Electron Fermi surface

Graphene

-1 -0.5 0 0.5 1

100

200

300

400

500

600

-1 -0.5 0 0.5 1

100

200

300

400

500

600

∼ 1√n(1 + λ ln Λ√

n)

n1012/m2

T (K)

Dirac liquid

ElectronFermi liquid

HoleFermi liquid

Quantum critical

Graphene

D. E. Sheehy and J. Schmalian, PRL 99, 226803 (2007)M. Müller, L. Fritz, and S. Sachdev, PRB 78, 115406 (2008)

M. Müller and S. Sachdev, PRB 78, 115419 (2008)

-1 -0.5 0 0.5 1

100

200

300

400

500

600

-1 -0.5 0 0.5 1

100

200

300

400

500

600

∼ 1√n(1 + λ ln Λ√

n)

n1012/m2

T (K)

Dirac liquid


HoleFermi liquid

Quantum critical

Graphene

M. Müller, L. Fritz, and S. Sachdev, PRB 78, 115406 (2008)M. Müller and S. Sachdev, PRB 78, 115419 (2008)

Predictedstrange metal

g

μ

0

Commensurate Mott insulator

gc

Superfluid

Superfluid

For cuprates, this insulator is realized e.g. at hole density = 1/8.

FIG. 1: Zero temperature (T = 0), zero field (B = 0) phase diagram in the vicinity of the

quantum critical point described by the CFT, represented by the filled circle. The coupling g

represents a parameter which tunes between a superfluid and a Mott insulator which is at a

density commensurate with the underlying lattice. The chemical potential µ introduces variations

in the density and ρ is difference in the density of pairs of holes in the superfluid from that in the

Mott insulator. The thin dashed lines are contours of constant ρ. In the application to the cuprate

superconductors, the Mott insulator with ρ = 0 could be, e.g., an insulating state at hole density

δI = 1/8 in a generalized phase diagram; then ρ = (δ − δI)/(2a2), where a is the lattice spacing.

The thick dotted line represents a possible trajectory of a particular compound as its hole density

is decreased; note that the ground state is always a superconductor along this trajectory, even at

δ = 1/8 (although there will be a dip in Tc near δ = 1/8 as is also clear from Fig. 2). Note that the

parent Mott insulator with zero hole density is not shown above. This paper will describe electrical

and thermal transport in the above phase diagram perturbed by an applied magnetic field B and

a small density of impurities.

The discussion so far applies, strictly speaking, only to systems which are exactly at the

commensurate density for which a gapped Mott insulator can form. The cuprates, and other

experimental systems, are not generically at these special densities, and so it is crucial to

develop a theory that is applicable at generic densities. Such a theory will emerge as a

special case of our more general results below. We allow the density to take values ρ by

applying a chemical potential µ, as shown in Figs. 1 and 2.

We emphasize that ρ measures the deviation in particle number density from the density

of the commensurate insulator [19, 20]; so ρ can be positive or negative, and we will see

3

c

Fermisurface

kx

ky

• Fermi surface separates empty

and occupied states in mo-

mentum space.

• Area enclosed by Fermi sur-

face = Q. Momenta of low

energy excitations fixed by

density of all electrons.

• Long-lived electron-like quasi-

particle excitations near the

Fermi surface: lifetime of quasi-

particles 1/T 2.

Ordinary metals: the Fermi liquid

• (Thermal conductivity)

T (Electrical conductivity)

=

2k2B3e2

L0

Dirac Fluid in Graphene 26

Wiedemann-Franz Law

I Wiedemann-Franz law in a Fermi liquid:

T 2k2

B

3e2 2.45 108 W ·

K2.

I in hydrodynamics one finds

T=

Lhydro

(1 + (Q/Q0)2)2 , Lhydro 1.

hence the Lorenz ratio, L, departs from the Sommer- feld value, L o

L - efT (4)

The important scattering processes in thermal and electrical conduction are: (i) elastic scattering by solute atoms, impurities and lattice defects, (ii) scattering of the electrons by phonons, and (iii) electron-electron interactions. In the elastic scattering region, i.e. at very low temperature, IE = IT and hence L = L 0. At higher temperatures, electron-electron scattering and electron-phonon scattering dominate and the collisions are inelastic. Then IE#l T and hence L deviates from L o.

Deviations from the Sommerfeld value of the Lorenz number are due to various reasons. In metals,

1 0 2 A g /e

"73 t - O C~

"~I01

0) >

rr"

10 ~

Cu o ~ ,", /

Brass O V , "

Sn o z~ /v Fe GerP~n Pt Q~ z~/~z~

silv U

Wiedemann-Franz ;< 0 o

Figure l Relative

I I I .I 101 10 2

Re la t i ve e lec t r ica l c o n d u c t i v i t y

thermal conductivities, A, measured by Wiedemann and Franz (AAg assumed to be = 100) and relative electrical conductivities, ~, measured by ( 9 Riess, (A) Becquerel, and (V) Lorenz. C~Ag assumed to be - 100. After Wiedemann and Franz [1].

at low temperatures the deviations are due to the inelastic nature of electron-phonon interactions. In some cases, a higher Lorenz number is due to the presence of impurities. The phonon contribution to thermal conductivity sometimes increases the Lorenz number, and this contribution, when phonon Umklapp scattering is present, is inversely propor- tional to the temperature. The deviations in Lorenz number can also be due to the changes in band structure. In magnetic materials, the presence of mag- nons also can change the Lorenz number at low temperatures. In the presence of a magnetic field, the Lorenz number varies directly with magnetic field. Changes in Lorenz number are sometimes due to structural phase transitions. In recent years, the Lorenz number has also been investigated at higher temperatures and has been found to deviate from the Sommerfeld value [14-20] and it is sometimes attributed to the incomplete degeneracy (Fermi smearing) [21] of electron gas. The Lorenz number has also been found to vary with pressure [-22, 23].

In alloys, the thermal conductivity and hence the Lorenz number have contributions from the electronic and lattice parts at low temperatures. The apparent Lorenz ratio (L/Lo) for many alloys has a peak at low temperatures. At higher temperatures the apparent Lorenz ratio is constant for each sample and approaches Lo as the percentage of alloying, x, increases. In certain alloys at high temperatures, the ordering causes a peak in L/L o.

The Lorenz number of degenerate semiconductors also shows a similar deviation to that observed in metals and alloys. Up to a certain temperature, inelastic scattering determines the Lorenz number value, and below this the scattering is elastic which is due to impurities. Supression of the electronic contribution to thermal conductivity and hence the separation of the lattice and electronic parts of conductivity can be done by application of a transverse magnetic field and hence the Lorenz number can be evaluated. The deviation of the Lorenz number in some degenerate semiconductors is attributed to phonon drag. In some

1 0 16

~'v 3"0 t ~ 2.5 O 2.0

1017 1018 I ~ ' I

Bi

Carr ier concen t ra t i on (cm 3) 1 019 1 0 2~ 1 0 21 1 0 22 1 0 23

I 1 ~ t I t ; I

SiGe 2 Cs AI Au Mg

SiBe3 /. / Nb~b.,~o/Cu F i i e 9

Sb As Ag Ga

1 024

30 c N 2.5 c-

O " 2.0

SiGe z ~oe \ SiG%

i \ SiGe 1

I I r I I t f

1 0 2 1 0 3 1 0 4

Se H~ Y W1 Cs2 Fe Ir Co z Li2 AUz Au3 AI2 All

\." \ \ \ ! ! , ,L T, o y t w , p, t Feq z , ~ / Cr '' \ X~., K Pt , \ Ni2 u A ~ "

Yb ~o 1 Er / / 2 As Sb ~ Rh g t31~ Bi 2 Bi I Cq

I [ I I [ I i I t I I I I I I

1 0 6 1 0 e 1 0 7 1 0 8 1 0 9 Electr ical conduc t i v i t y (~,~-1 cm-1)

! I

101o

Figure 2 Experimental Lorenz number of elemental metals in the low-temperature residual resistance regime, see Table I. Also shown are our own data points on a doped, degenerate semiconductor (Table III). Data are plotted versus electrical conductivity and also versus carrier concentration, taken from Ashcroft and Mermin [24] except for the semiconductors.

4262 G. S. Kumar, G. Prasad, and R.O. Pohl, J. Mat. Sci. 28, 4261 (1993)

Key properties of a strange metal

• No quasiparticle excitations

• Shortest possible “collision time”, or

more precisely, fastest possible local

equilibration time ~kBT

• Continuously variable density, Q(conformal field theories are usually

at fixed density, Q = 0)

-1 -0.5 0 0.5 1

100

200

300

400

500

600

-1 -0.5 0 0.5 1

100

200

300

400

500

600

∼ 1√n(1 + λ ln Λ√

n)

n1012/m2

T (K)

Dirac liquid


HoleFermi liquid

Quantum critical

Graphene

D. E. Sheehy and J. Schmalian, PRL 99, 226803 (2007)M. Müller, L. Fritz, and S. Sachdev, PRB 78, 115406 (2008)

M. Müller and S. Sachdev, PRB 78, 115419 (2008)


Wiedemann-Franz Law Violations in Experiment

0

4

8

12

16

20L

/ L0

10

20

30

40

50

60

70

80

90

100T

ba

th (

K)

−10−15 15−5 0 5 10

n (109 cm-2)

disorder-limited

phonon-limited

[Crossno et al, submitted]

Strange metal in graphene

J. Crossno et al. arXiv:1509.04713; Science, to appear

L =

(Thermal conductivity)


; L0 2k2B3e2



0

4

8

12

16

20L

/ L0

10

20

30

40

50

60

70

80

90

100T

ba

th (

K)

−10−15 15−5 0 5 10

n (109 cm-2)

disorder-limited

phonon-limited




L =



; L0 2k2B3e2

Wiedemann-Franz obeyed



0

4

8

12

16

20L

/ L0

10

20

30

40

50

60

70

80

90

100T

ba

th (

K)

−10−15 15−5 0 5 10

n (109 cm-2)

disorder-limited

phonon-limited




L =



; L0 2k2B3e2

Wiedemann-Franz violated !

Quasiparticle transport in metals:

• Focus on infinite number of (near) conserva-

tion laws (momenta of quasiparticles on the

Fermi surface) and compute how they are

slowly violated by the lattice or impurities

Transport in strange metals

• There are no quasiparticles, and so the Fermi

surface is not a central actor in transport

(although a Fermi surface can be precisely

defined in some cases).

• Focus on relaxation of total momentum (in-

cluding contributions of the Fermi surface (if

present) and all critical bosons) by the lat-

tice or impurities

Beyond Perturbation Theory 23

Summary

hydrodynamics

memory matrix holography

universal constraints on transport

appropriate microscopics for cuprates

few conserved quantities

perturbative limit

long time dynamics; “renormalized IR fluid”

emerges

matrix large N theory; non-perturbative computations

[Lucas JHEP]

[Lucas 1506][Donos, Gauntlett 1506]

[Lucas, Sachdev PRB]

[Forster ’70s]

[Hartnoll, others]

figure from [Lucas, Sachdev, Physical Review B91 195122 (2015)]

Transport in Strange Metals

S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, 144502 (2007)

Dynamics of chargedblack hole horizons

Prediction for transport in the graphene strange metal

S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, 144502 (2007)M. Müller and S. Sachdev, PRB 78, 115419 (2008)

Recall that in a Fermi liquid, the Lorenz ratio L = /(T), where is the thermal conductivity, and is the conductivity, is given byL = 2k2B/(3e

2).For a strange metal with a “relativistic” Hamiltonian, hydrody-

namic, holographic, and memory function methods yield

= Q

1 +

e2v2FQ2imp

HQ

, =

v2FHimp

T

1 +

e2v2FQ2imp

HQ

1

L =v2FHimp

T 2Q

1 +

e2v2FQ2imp

HQ

2

,

where H is the enthalpy density, imp is the momentum relaxation time(from impurities), while = Q, an intrinsic, finite, “quantum criti-cal” conductivity. Note that the limits Q ! 0 and imp ! 1 do notcommute.

0

4

8

12

16

20

L / L

0

10

20

30

40

50

60

70

80

90

100

Tb

ath

(K

)

−10−15 15−5 0 5 10

n (109 cm-2)

2

−10 −5 0 5 10

0

0.6

1.2

Vg (V)

R (

kΩ)

250 K150 K

100 K

50 K4 K

A

10

-1012 -1011 -1010 1010 1011 1012

100

n (cm-2)

Ele

c. C

on

du

ctiv

ity

(4 e

2/h

)

Tb

ath (K)

e-h+

B

∆Vg (V) Tbath (K)-0.5 0.50 0 50 100 150

Th

erm

al C

on

du

ctiv

ity

(n

W/K

)

0

2

4

6

8

0

2

4

6

0

2

4

6

8

0

01

1

10 mm

20 K

-0.5 V

0 V

40 K

75 K

C D

E

Tel-ph

Tdis

κe

σTL0

FIG. 1. Temperature and density dependent electrical and thermal conductivity. (A) Resistance versus gate voltageat various temperatures. (B) Electrical conductivity (blue) as a function of the charge density set by the back gate for di↵erentbath temperatures. The residual carrier density at the neutrality point (green) is estimated by the intersection of the minimumconductivity with a linear fit to log() away from neutrality (dashed grey lines). Curves have been o↵set vertically such thatthe minimum density (green) aligns with the temperature axis to the right. Solid black lines correspond to 4e2/h. At lowtemperature, the minimum density is limited by disorder (charge puddles). However, above Tdis 40 K, a crossover markedin the half-tone background, thermal excitations begin to dominate and the sample enters the non-degenerate regime nearthe neutrality point. (C-D) Thermal conductivity (red points) as a function of (C) gate voltage and (D) bath temperaturecompared to the Wiedemann-Franz law, TL0 (blue lines). At low temperature and/or high doping (|µ| kBT ), we find theWF law to hold. This is a non-trivial check on the quality of our measurement. In the non-degenerate regime (|µ| < kBT )the thermal conductivity is enhanced and the WF law is violated. Above Telph 80 K, electron-phonon coupling becomesappreciable and begins to dominate thermal transport at all measured gate voltages. All data from this figure is taken fromsample S2 (inset 1E).

Realization of the Dirac fluid in graphene requires thatthe thermal energy be larger than the local chemical po-tential µ(r), defined at position r: kBT & |µ(r)|. Impu-rities cause spatial variations in the local chemical po-tential, and even when the sample is globally neutral, itis locally doped to form electron-hole puddles with finiteµ(r) [25–28]. Formation of the DF is further complicatedby phonon scattering at high temperature which can re-lax momentum by creating additional inelastic scatteringchannels. This high temperature limit occurs when theelectron-phonon scattering rate becomes comparable tothe electron-electron scattering rate. These two temper-atures set the experimental window in which the DF andthe breakdown of the WF law can be observed.

To minimize disorder, the monolayer graphene samplesused in this report are encapsulated in hexagonal boronnitride (hBN) [29]. All devices used in this study aretwo-terminal to keep a well-defined temperature profile

[30] with contacts fabricated using the one-dimensionaledge technique [31] in order to minimize contact resis-tance. We employ a back gate voltage Vg applied tothe silicon substrate to tune the charge carrier densityn = ne nh, where ne and nh are the electron and holedensity, respectively (see supplementary materials (SM)).All measurements are performed in a cryostat controllingthe temperature Tbath. Fig. 1A shows the resistance Rversus Vg measured at various fixed temperatures for arepresentative device (see SM for all samples). From this,we estimate the electrical conductivity (Fig. 1B) usingthe known sample dimensions. At the CNP, the residualcharge carrier density nmin can be estimated by extrap-olating a linear fit of log() as a function of log(n) outto the minimum conductivity [32]. At the lowest tem-peratures we find nmin saturates to 8109 cm2. Wenote that the extraction of nmin by this method overesti-mates the charge puddle energy, consistent with previous

4

0 100 2000

5

10

15

20

25

Temperature (K)

L /

L 0

B

1 10 100 1000109

1010

1011

Temperature (K)

nm

in (c

m-2

)

DisorderLimited

ThermallyLimited

S3S2S1

A

−6 −4 −2 0 2 4 60

4

8

12

16

20

n (1010 cm−2)

L/L 0

C

40 60 80 1000

2

4

6

8

10

T (K)

H

(eV

/µm

2)

CHe

h

-V+Ve

h

∆Vg = 0

FIG. 3. Disorder in the Dirac fluid. (A) Minimum car-rier density as a function of temperature for all three sam-ples. At low temperature each sample is limited by disorder.At high temperature all samples become limited by thermalexcitations. Dashed lines are a guide to the eye. (B) TheLorentz ratio of all three samples as a function of bath tem-perature. The largest WF violation is seen in the cleanestsample. (C) The gate dependence of the Lorentz ratio is wellfit to hydrodynamic theory of Ref. [5, 6]. Fits of all threesamples are shown at 60 K. All samples return to the Fermiliquid value (black dashed line) at high density. Inset showsthe fitted enthalpy density as a function of temperature andthe theoretical value in clean graphene (black dashed line).Schematic inset illustrates the di↵erence between heat andcharge current in the neutral Dirac plasma.

more pronounced peak but also a narrower density de-pendence, as predicted [5, 6].

More quantitative analysis of L(n) in our experimentcan be done by employing a quasi-relativistic hydrody-namic theory of the DF incorporating the e↵ects of weakimpurity scattering [5, 6, 39].

L =LDF

(1 + (n/n0)2)2 (2)

where

LDF =HvFlmT 2min

and n20 =

Hmin

e2vFlm. (3)

Here vF is the Fermi velocity in graphene, min is the elec-trical conductivity at the CNP, H is the fluid enthalpydensity, and lm is the momentum relaxation length from

impurities. Two parameters in Eqn. 2 are undeterminedfor any given sample: lm and H. For simplicity, we as-sume we are well within the DF limit where lm and Hare approximately independent of n. We fit the experi-mentally measured L(n) to Eqn. (2) for all temperaturesand densities in the Dirac fluid regime to obtain lm andH for each sample. Fig 3C shows three representative fitsto Eqn. (2) taken at 60 K. lm is estimated to be 1.5, 0.6,and 0.034 µm for samples S1, S2, and S3, respectively.For the system to be well described by hydrodynamics,lm should be long compared to the electron-electron scat-tering length of 0.1 µm expected for the Dirac fluid at60 K [18]. This is consistent with the pronounced sig-natures of hydrodynamics in S1 and S2, but not in S3,where only a glimpse of the DF appears in this moredisordered sample. Our analysis also allows us to es-timate the thermodynamic quantity H(T ) for the DF.The Fig. 3C inset shows the fitted enthalpy density asa function of temperature compared to that expected inclean graphene (dashed line) [18], excluding renormal-ization of the Fermi velocity. In the cleanest sample Hvaries from 1.1-2.3 eV/µm2 for Tdis < T < Telph. Thisenthalpy density corresponds to 20 meV or 4kBTper charge carrier — about a factor of 2 larger than themodel calculation without disorder [18].

In a hydrodynamic system, the ratio of shear viscosity to entropy density s is an indicator of the strength ofthe interactions between constituent particles. It is sug-gested that the DF can behave as a nearly perfect fluid[18]: /s approaches a “universal” lower bound conjec-ture by Kovtun-Son-Starinets, (/s)/(~/kB) 1/4 fora strongly interacting system [40]. Though we cannotdirectly measure , we comment on the implications ofour measurement for its value. Within relativistic hy-drodynamics, we can estimate the shear viscosity of theelectron-hole plasma in graphene from the enthalpy den-sity as Hee [40], where ee is the electron-electronscattering time. Increasing the strength of interactionsdecreases ee, which in turn decreases and /s. Employ-ing the expected Heisenberg limited inter-particle scat-tering time, ee ~/kBT [5, 6], we find a shear viscosityof 1020 kg/s in two-dimensional units, correspondingto 1010 Pa · s. The value of ee used here is consistentwith recent optical experiments on graphene [14, 16, 17].Using the theoretical entropy density for clean graphene(SM), we estimate (/s)/(~/kB) 3. This is comparableto 0.7 found in liquid helium at the Lambda-point [41],0.3 measured in cold atoms [3], and 0.4 for quark-gluon plasmas [4].

To fully incorporate the e↵ects of disorder, a hydrody-namic theory treating inhomogeneity non-perturbativelymay be needed [42]. The enthalpy densities reported hereare larger than the theoretical estimation obtained fordisorder free graphene; consistent with the picture thatchemical potential fluctuations prevent the sample fromreaching the Dirac point. While we find thermal conduc-

Lorentz ratio L = /(T)

=

v2FHimp

T 2Q

1

(1 + e2v2FQ2imp/(HQ))2


Relativistic hydrodynamics

S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, 144502 (2007)

Hydrodynamics of Quantum Electron Fluids 6

Relativistic Hydrodynamics

I hydrodynamics when l lee, t tee

I long time dynamics governed by conservation laws:

@

Tµ = J

F ext

µ

, @µ

Jµ = 0.

dynamics of relaxation to equilibrium

I expand Tµ , Jµ in perturbative parameter lee@µ

:

T

µ = P

µ + ( + P )uµ

u

2PµP

@

(

u

)

Pµ

2

d

@

u

+ · · · ,

J

µ = Qu

µ qPµ

@

µ µ

T

@

T u

F

ext

+ · · · ,

Pµ

µ + u

µ

u

,

Q

i = J

i µT

ti

I quantum physics ! values of P , q, etc...

[Hartnoll, Kovtun, Muller, Sachdev, Physical Review B76 144502 (2007)]

New (and only) transport co-ecient, Q:

“quantum critical” conductivity at Q = 0.

Qi = T ti µJ i

Translational symmetry breaking

Momentum relaxation by an external source h coupling to the operator O

H = H0 Z

ddxh(x)O(x).

M

= lim

!!0

Zddq |h(q)|2q2x

Im

GR

OO(q,!)H0

!+ higher orders in h

Leads to an additional term in equations of motion:

@µTµi

= . . . T it

imp+ . . .

S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, 144502 (2007)A. Lucas and S. Sachdev, PRB 91, 195122 (2015)

Translational symmetry breaking

Momentum relaxation by an external source h coupling to the operator O

H = H0 Z

ddxh(x)O(x).

M

= lim

!!0

Zddq |h(q)|2q2x

Im

GR

OO(q,!)H0

!+ higher orders in h

Leads to an additional term in equations of motion:

@µTµi

= . . . T it

imp+ . . .

“Memory function” methods yield an explicit expression for imp:

Mimp

= lim

!!0

Zddq |h(q)|2q2x

Im

GR

OO(q,!)H0

!+ . . .

S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, 144502 (2007)A. Lucas and S. Sachdev, PRB 91, 195122 (2015)

xi

AdS/CFT correspondence at zero temperature

AdSd+2

CFTd+1

R

d,1

Minkowski

Einstein gravity

ds

2 =

L

r

2 dr

2 dt

2 + d~x

2

SE =

Zd

d+2x

pg

1

22

R+

d(d+ 1)

L

2

r

r

xi

AdS/CFT correspondence at non-zero temperature

R

d,1

Minkowski

Einstein gravity

AdS-Schwarzschild

CFTd+1 atHawking

temperature Tof horizon

SE =

Zd

d+2x

pg

1

22

R+

d(d+ 1)

L

2

R

Entropy density of CFTd+1, S T d

Bekenstein-Hawking entropy density, SBH T d

Charged black branes

Einstein-Maxwell theory SEM =

Zd

d+2x

pg

1

22

R+

d(d+ 1)

L

2 R

2

g

2F

F

2

+

++

++

+Electric flux

hQi6= 0

rAdS-Reissner-Nordstrom Quantummatter on

the boundary with

a variable charge

densityQ of a global

U(1) symmetry.

A. Chamblin, R. Emparan, C. V. Johnson, and R. C. Myers, 99

Realizes a strange metal: a state with an unbroken global U(1)

symmetry with a continuously variable charge density, Q, at

T = 0 which does not have any quasiparticle excitations.

Charged black branes

+

++

++

+Electric flux

hQi6= 0

rQuantummatter on

the boundary with

a variable charge

densityQ of a global

U(1) symmetry.

ds

2 =1

r

2

dt

2

r

2d(z1)/(d)+ r

2/(d)dr

2 + dx

2i

More general theories have “hyperscaling violating metric”

at T=0

C. Charmousis, B. Gouteraux, B. S. Kim, E. Kiritsis and R. Meyer, JHEP 1011, 151 (2010).N. Iizuka, N. Kundu, P. Narayan and S. P. Trivedi, JHEP 1201, 94 (2012).L. Huijse, S. Sachdev, B. Swingle, Phys. Rev. B 85, 035121 (2012)

Inhomogeneous charged black branes

+

++

+Electric flux

hQi6= 0

r

+

Add a source term,Zd

dxh(x)O(x)

+

Weakly disordered charged black branes yield results identical to those obtained from memory functions and holography

• G.T. Horowitz, J.E. Santos, and D. Tong, JHEP 1207, 168 (2012), JHEP 1211, 102 (2012).• D. Vegh, arXiv:1301.0537. • M. Blake, D. Tong, and D. Vegh, PRL 112, 071602 (2013).•M. Blake and D. Tong, PRD 88, 106004 (2013). •A. Lucas, S. Sachdev, and K. Schalm, PRD89, 066018 (2014). • A. Lucas, JHEP 1503, 071 (2015). • R. A. Davison and B. Gouteraux,arXiv:1505.05092; arXiv:1507.07137. • M. Blake, arXiv:1505.06992; arXiv:1507.04870.

Prediction for transport in the graphene strange metal

S. A. Hartnoll, P. K. Kovtun, M. Müller, and S. Sachdev, PRB 76, 144502 (2007)M. Müller and S. Sachdev, PRB 78, 115419 (2008)

Recall that in a Fermi liquid, the Lorenz ratio L = /(T), where is the thermal conductivity, and is the conductivity, is given byL = 2k2B/(3e

2).For a strange metal with a “relativistic” Hamiltonian, hydrody-

namic, holographic, and memory function methods yield

= Q

1 +

e2v2FQ2imp

HQ

, =

v2FHimp

T

1 +

e2v2FQ2imp

HQ

1

L =v2FHimp

T 2Q

1 +

e2v2FQ2imp

HQ

2

,

where H is the enthalpy density, imp is the momentum relaxation time(from impurities), while = Q, an intrinsic, finite, “quantum criti-cal” conductivity. Note that the limits Q ! 0 and imp ! 1 do notcommute.

hole FL elec. FLDirac fluid

400 200 0 200 4000

0.5

1

1.5

2

n (µm2)

(k

1

)


400 200 0 200 4000

2

4

6

8

10

n (µm2)

(n

W/K

)

Figure 1: testingFigure 1: A comparison of our hydrodynamic theory of transport with the experimental results of[33] in clean samples of graphene at T = 75 K. We study the electrical and thermal conductancesat various charge densities n near the charge neutrality point. Experimental data is shownas circular red data markers, and numerical results of our theory, averaged over 30 disorderrealizations, are shown as the solid blue line. Our theory assumes the equations of state describedin (27) with the parameters C

0

11, C

2

9, C

4

200,

0

110,

0

1.7, and (28) withu

0

0.13. The yellow shaded region shows where Fermi liquid behavior is observed and theWiedemann-Franz law is restored, and our hydrodynamic theory is not valid in or near thisregime. We also show the predictions of (2) as dashed purple lines, and have chosen the 3parameter fit to be optimized for (n).

where e is the electron charge, s is the entropy density, n is the charge density (in units of length2),H is the enthalpy density, is a momentum relaxation time, and q is a quantum critical e↵ect, whoseexistence is a new e↵ect in the hydrodynamic gradient expansion of a relativistic fluid. Note that up toq, (n) is simply described by Drude physics. The Lorenz ratio then takes the general form

L(n) =L

DF

(1 + (n/n

0

)2)2, (3)

where

LDF

=v

2

F

H

T

2

q, (4a)

n

2

0

=Hq

e

2

v

2

F

. (4b)

L(n) can be parametrically larger than LWF

(as ! 1 and n n

0

), and much smaller (n n

0

).Both of these predictions were observed in the recent experiment, and fits of the measured L to (3) werequantitatively consistent, until large enough n where Fermi liquid behavior was restored. However, theexperiment also found that the conductivity did not grow rapidly away from n = 0 as predicted in (2),despite a large peak in (n) near n = 0, as we show in Figure 1. Furthermore, the theory of [25] does notmake clear predictions for the temperature dependence of , which determines (T ).

In this paper, we argue that there are two related reasons for the breakdown of (2). One is that thedominant source of disorder in graphene – fluctuations in the local charge density, commonly referred to ascharge puddles [43, 44, 45, 46] – are not perturbatively weak, and therefore a non-perturbative treatmentof their e↵ects is necessary.3 The second is that the parameter , even when it is sharply defined, is

3See [47, 48] for a theory of electrical conductivity in charge puddle dominated graphene at low temperatures.

4

A. Lucas, J. Crossno, K.C. Fong, P. Kim, and S. Sachdev, arXiv:1510.01738, PRB to appear

Q Q

Comparison to theory with a single momentum relaxation time imp.

Best fit of density dependence to thermal conductivity does not capture

the density dependence of electrical conductivity

x

n

n(x) < 0

n(x) > 0

s(x) > 0lee

Figure 3: A cartoon of a nearly quantum critical fluid where our hydrodynamic description oftransport is sensible. The local chemical potential µ(x) always obeys |µ| k

B

T , and so theentropy density s/k

B

is much larger than the charge density |n|; both electrons and holes areeverywhere excited, and the energy density does not fluctuate as much relative to the mean.Near charge neutrality the local charge density flips sign repeatedly. The correlation length ofdisorder is much larger than l

ee

, the electron-electron interaction length.

1.2 Outline

The outline of this paper is as follows. We briefly review the definitions of transport coecients in Section2. In Section 3 we develop a theory of hydrodynamic transport in the electron fluid, assuming that it isLorentz invariant. We discuss the peculiar case of the Dirac fluid in graphene in Section 4, and argue thatdeviations from Lorentz invariance are small. We describe the results of our numerical simulations of thistheory in Section 5, and directly compare our simulations with recent experimental data from graphene[33]. The experimentally relevant e↵ects of phonons are qualitatively described in Section 6. We concludethe paper with a discussion of future experimental directions. Appendices contain technical details of ourtheory.

In this paper we use index notation for vectors and tensors. Latin indices ij · · · run over spatialcoordinates x and y; Greek indices µ · · · run over time t as well. We will denote the time-like coordinateof A

µ as A

t. Indices are raised and lowered with the Minkowski metric

µ diag(1, 1, 1). The Einsteinsummation convention is always employed.

Transport Coecients2

Let us begin by defining the thermoelectric response coecients of interest in this paper. Suppose thatwe drive our fluid by a spatially uniform, externally applied, electric field E

i

(formally, an electrochemicalpotential gradient), and a temperature gradient @

i

T . We will refer to @

j

T as T

j

, with

j

= T

1

@

j

T ,for technical reasons later. As is standard in linear response theory, we decompose these perturbationsinto various frequencies, and focus on the response at a single frequency !. Time translation invarianceimplies that the (uniformly) spatially averaged charge current hJ

i

i and the spatially averaged heat currenthQ

i

i are also periodic in time of frequency !, and are related to E

i

and

i

by the thermoelectric transportcoecients: hJ

i

ihQ

i

i

ei!t =

ij

(!) T↵

ij

(!)T ↵

ij

(!) T

ij

(!)

E

j

j

ei!t

. (5)

6

Non-perturbative treatment of disorder

Note

n Q

Numerically solve the hydrodynamic equations in the presence of a

x-dependent chemical potential. The thermoelectric transport properties

will then depend upon the value of the shear viscosity, .



400 200 0 200 4000

0.5

1

1.5

2

n (µm2)

(k

1

)


400 200 0 200 4000

2

4

6

8

10

n (µm2)

(n

W/K

)

Figure 1: testingFigure 1: A comparison of our hydrodynamic theory of transport with the experimental results of[33] in clean samples of graphene at T = 75 K. We study the electrical and thermal conductancesat various charge densities n near the charge neutrality point. Experimental data is shownas circular red data markers, and numerical results of our theory, averaged over 30 disorderrealizations, are shown as the solid blue line. Our theory assumes the equations of state describedin (27) with the parameters C

0

11, C

2

9, C

4

200,

0

110,

0

1.7, and (28) withu

0

0.13. The yellow shaded region shows where Fermi liquid behavior is observed and theWiedemann-Franz law is restored, and our hydrodynamic theory is not valid in or near thisregime. We also show the predictions of (2) as dashed purple lines, and have chosen the 3parameter fit to be optimized for (n).

where e is the electron charge, s is the entropy density, n is the charge density (in units of length2),H is the enthalpy density, is a momentum relaxation time, and q is a quantum critical e↵ect, whoseexistence is a new e↵ect in the hydrodynamic gradient expansion of a relativistic fluid. Note that up toq, (n) is simply described by Drude physics. The Lorenz ratio then takes the general form

L(n) =L

DF

(1 + (n/n

0

)2)2, (3)

where

LDF

=v

2

F

H

T

2

q, (4a)

n

2

0

=Hq

e

2

v

2

F

. (4b)

L(n) can be parametrically larger than LWF

(as ! 1 and n n

0

), and much smaller (n n

0

).Both of these predictions were observed in the recent experiment, and fits of the measured L to (3) werequantitatively consistent, until large enough n where Fermi liquid behavior was restored. However, theexperiment also found that the conductivity did not grow rapidly away from n = 0 as predicted in (2),despite a large peak in (n) near n = 0, as we show in Figure 1. Furthermore, the theory of [25] does notmake clear predictions for the temperature dependence of , which determines (T ).

In this paper, we argue that there are two related reasons for the breakdown of (2). One is that thedominant source of disorder in graphene – fluctuations in the local charge density, commonly referred to ascharge puddles [43, 44, 45, 46] – are not perturbatively weak, and therefore a non-perturbative treatmentof their e↵ects is necessary.3 The second is that the parameter , even when it is sharply defined, is

3See [47, 48] for a theory of electrical conductivity in charge puddle dominated graphene at low temperatures.

4

Q Q

Solution of the hydrodynamic equations in the presence

of a space-dependent chemical potential.

Best fit of density dependence to thermal conductivity now gives a better fit to

the density dependence of the electrical conductivity (for /s 10). The Tdependencies of other parameters also agree well with expectation.


Quantum matter without quasiparticles

• No quasiparticle excitations

• Shortest possible “collision time”, or more precisely, fastest

possible local equilibration time ~kBT

• Continuously variable density, Q(conformal field theories are usually at fixed density, Q = 0)

• Theory built from hydrodynamics/holography

/memory-functions/strong-coupled-field-theory

• Exciting experimental realization in graphene.

superﬂuid-insulator transition - harvard...

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