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The role of gauge invariance in theThe role of gauge invariance in thetheory of superconductivity*
Dietrich EinzelWalther-Meißner-Institut für Tieftemperaturforschung
Bayerische Akademie der WissenschaftenD-85748 Garching g
Outline• Electrodynamics• Electrodynamics
• Quantum mechanics
• London‘s theory
• Nambu‐BCS theory (w/o Greens functions!)
• Summary and conclusion
1
* Nobel Lecture 2008, Y. Nambu, Rev. Mod. Phys. 81, 1015 – 1018 (2009) Seminar on Advances in Solid State Physics, WMI, June 8, 2010
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Two fundamental theorems
Noether theorem: Goldstone theorem:(Emmy Noether, 1918) (Jeffrey Goldstone, 1961)
„The spontaneous breaking of„Every continuous symmetry
of a system is to be associatedwith a conserved quantity“
a continuous symmetry is tobe associated with a masslessand spinless particle, the so-ll d N b G ld t b “called Nambu-Goldstone boson“
t ti ti l b k t G ld t bsymmetry operation conservation law
translation in time energy
broken symmetry Goldstone boson
liquidsGalilean longitudinal phonon
translation in space momentum
rotation in space angular momentum
g psolidsGalilean longit.+transv. phononspin rotation magnon
phase charge gauge phonon
2
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Electrodynamics: potentials and fields
t t ti lscalar potential vector potential
magnetic field
electric field
gauge invariance
3
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Quantum mechanics: gauge invariance
Consider a non-relativistic (Bose-) particle of charge q=ke and mass m=km0
Quantum-mechanical description: wave function
C h i t t ti b bilit d it
Schrödinger equation in the presence of and
Copenhagen interpretation: probability density
Gauge transformation ,
corresponds to local U(1) trafo:
fluxoid quantum/2π
4
with
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London‘s* theory in a nutshell
Macroscopic (pseudo-) bosonic condensate wave functioni. e. postulate of macroscopic phase coherence associated with a p p pmacroscopic number of
(i) bosons (k = 1, Bose condensate)(ii) fermion pairs (k = 2, pair condensate)(ii) fermion pairs (k 2, pair condensate)
superfluiddensity
Schrödingerequation
Josephson
conservationlaw
continuity Josephsongauge-invariantcurrent density
continuity
* F & H London,1935, 1950 5
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Nambu-BCS* theory: route to superconductivity
energy variableparticles
energy variableholes
(i) pair attraction(exchange boson)
(ii) pair formation( ) pin k-space
(iii) broken gauge[U(1)] symmetry
6* Bardeen, Cooper & Schrieffer, 1957
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Nambu-BCS theory: particle-hole structure
(i ) i t ti l(iv) pair potential
(v) energy becomes a matrix in particle-hole (Nambu) space for T<TcYoishiro Nambu, 1962
particles mixture
diagonalization
Bogoliubov, 0
Valatin, 1957holesmixture
off-diagonal long range order (ODLRO)7
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Nambu-BCS theory: thermal excitations
Ek(vi) gap formation: energy
dispersion of the thermal excitations (Bogoliubov
b l
excitations (Bogoliubov-Valatin quasiparticles, „bogolons“)
bogolons
ΔkΔk
k k00
kF k0
8
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Nambu-BCS theory: thermal excitations
Thermal excitations: Bogoliubov-Valatin quasiparticles
momentum distribution
isotropicnodal
Fermi surface
Bogoliubovquasiparticlesq p
9
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Momentum distribution in Nambu space
particles pairing
Nambu matrix
holespairing
BCS coherence factors
diagonal density
BCS coherence factors
diagonal density
particles holes
off-diagonal densityg y
Gorkov pairing amplitude 10
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Momentum distribution functions revisited
Which ofWhich ofthese twofunctionsis relevant for super-
conductivity Fermi-Dirac
?thermal excitations
11
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Nonequilibrium description: electromagnetic response
scalar potential vector potential
externalperturbationperturbationpotentials
nonequilibriumphase spacedistribution
linear response
12
response
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Nonequilibrium description: normal state
density fluctuations: particle-hole excitations, local equilibrium
vertex (e, evk)( , k)
13„minimal coupling“
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A Nambu space, discovered at a German University
Nambu spaceN b t i l t t Nambu spaceNambu matrices: general structure
Entrance forEntrance for
particles |k,σ>holes |-k,-σ>
only
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Nonequilibrium description: Nambu structure
electromagnetic potentials in Nambu space
shifted quasiparticle energy
diagonalization
shifted momentum distribution
!15
!
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Nonequilibrium description: Nambu structure
integral properties Yosida function
Nambu-BCS current density
superfluid density tensor
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Nonequilibrium description: Yosida function
11on
func
tio
Y(T)
osid
af Y(T)
BC
S Yo
GLl t t
0
B GLregime
low temperatureregime
0.1 10
T/Tc17
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Nonequilibrium description: supercurrent density
violation ofthe numberconservationlaw
gaugetransformation
gauge-invariant
18
current density
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Nonequilibrium description: supercurrent density
how to restore the number conservation lawhow to restore the number conservation law
gauge-invariant (strictly tansverse) current density
19„backflow term“ from gauge mode
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Nonequilibrium description: electromagnetic response
q ant m d namicsquantum dynamics:von Neumann equation
linearization,collisionless
Fourier space
limit Ik = 0
streaming in phase space external and molecular forces
20Betbeder-Matibet & Nozieres, 1969; Wölfle, 1976; Einzel & Klam, 2006
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Nonequilibrium description: integral equations
diagonal energyg gy
Coulomb-I. Fermi liquid-I.Consequences:
: external perturbations cause electromagnetic response
: dielectric screening plasma oscillations: dielectric screening, plasma oscillations
: collective density (sound) oscillations
off-diagonal energy
pairing-I.Consequences:
: order parameter collective modes (amplitude, phase),
21
gauge-invariance
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Nonequilibrium description: dynamics of the gap
amplitude phaseoff-diagonal energy decompositiong gy p
order parameter phasep pfluctuations
d t lit d
Gauge mode, Anderson-Bogoliubov mode, Nambu-Goldstone mode
order parameter amplitudefluctuations
2Δ mode, coupling O(pha)22
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Nonequilibrium description: the gauge mode
solution for δΔk: genera-li d J h l tilized Josephson relation
gauge mode
gauge mode frequency
d l it
23
sound velocity
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Nonequilibrium description: condensate response
The Tsuneto function:
complicated expression
longwavelengthglimit
condensate density responsey
stationarylimit
24condensate current response
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Nonequilibrium description: solution for δnk
solution for δnk
macroscopic particle density
density conservation/relaxation
macroscopic current densitymacroscopic current density
25
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Nonequilibrium description: continuity equation
condensate terms on r.h.s. revisited
gauge mode
gauge-invariance charge/particle number conservation law
26
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Nonequilibrium description: density & current
density response long wavelength limit
Josephson
stationary current response
no homogeneous density response!
backflow“„backflow
stationary current purely transverse!27
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Summary and conclusion
Gauge invariance in the theory of superconductivity
London‘s theory (Madelung version):
postulate of phase-coherent macroscopic wave function ψgauge-invariant formulation possible
Local equilibrium BCS response theory:
t l b k U(1) tspontaneously broken gauge U(1) symmetryNambu space descriptioncorrect microscopic form of superfluid density tensor ns
lacks gauge invariance and therefore particle number conservationg g p
Nonequilibrium BCS response theory: order parameter phase fluctuations: gauge mode
l d t (T t ) f tigeneral condensate response (Tsuneto) functiondetermines χs, ns, dynamic conductivity, Raman response, … occurrence of „backflow“ terms in current response, Raman response, etc. gauge invariance and hence particle number conservation can be restored
28
gauge invariance and hence particle number conservation can be restoredgauge mode frequency [vF
2/3]1/2 unaffected by unconventionality of pairinggauge mode frequencies different from [vF
2/3]1/2 in non-centrosymmetric superconductors
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Appendix: quantum mechanics, Madelung description
quantum mechanical wave function(Erwin Madelung 1926)(Erwin Madelung, 1926)
probability density
conservation law for np
probability currentdensity(gauge invariant !)(gauge-invariant !)
Hamilton-JacobiHamilton-Jacobi(Josephson) equation
Euler equation forEuler equation fordissipationless„Madelung fluid“
A1
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Appendix: Comments on the Madelung description
Schrödinger equations for Ψ and Ψ∗ equivalent to
( i) probability density conservation law (magnitude of Ψ)
(ii) Hamilton Jacobi-equation (phase of Ψ)
in the quasiclassical limit
Identification of gauge-invariant probability current densityIdentification of gauge-invariant probability current density
Acceleration equation for vp: Euler equation for the Madelung (probability) fluid
A2
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Appendix: London* theory in a nutshell
Reinterpretation of
as macroscopic (pseudo-) bosonic condensate wave function, postulate ofmacroscopic phase coherence associated with a macroscopic number ofmacroscopic phase coherence associated with a macroscopic number of
(i) bosons (k = 1, Bose condensate)
(ii) fermion pairs (k = 2 pair condensate)(ii) fermion pairs (k = 2, pair condensate)
Reinterpretation of as the macroscopic condensate density
Reinterpretation of as the supercurrent or condensate current density
Reinterpretation of as the superfluid condensate velocity
Replacement of by the electrochemical potential p y p
* F & H London,1935, 1950 A3
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Appendix: basic results of the London theory
d t tcondensate current
persistent currents
Screening and magneticfield penetration depthfield penetration depth
Fl id ti tiFluxoid quantization(„2e or not 2e“)
A4
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Appendix: The London functional
London energy density: gradients, fields, magnetodynamics
Madelungtransformation(MT)( )
WKB
London penetration depth gauge-invariant!
A5
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Appendix: The Ginzburg-Landau functional
Ginzburg-Landau energy density
gradients, fields,magnetodynamics
thermodynamics
Madelung transformation
SC phase transition:pcondensation of
superfluid density a2A6
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Appendix: The Higgs mechanism in particle physics
Klein-Gordon equation: relativistic spinless Bose particles
London functional (gauge-invariant)μ2
Ginzurg-Landau functional
mass condensation(Higgs mechanism)(Higgs mechanism)
A7
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Appendix: classification of pair potentials
Condensate: pair potential (total spin s)p p ( p )
node singlet tripletnodestructure
singlet(s=0)
triplet(s=1)
Conventional: Unconventional:
( )fp(s) shares latticesymmetrybroken gauge
fp(s) breaks latticesymmetryadditional broken
t isymmetry symmetries
A8
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Appendix: unconventional pairing
Heavy-fermion OrganicySC‘s (1979) SC‘s (1980)
Superfluid 3He (1971)
Cuprate- (high-Tc)SC‘s (1986)
Ruddlesden-PopperSC‘s Sr2RuO4 (1994)
NCS supercon-ductors* (2004)
* D.E. + Klam/Manske: PRL 102, 027004 (2009)
Book on NCS superconductors,Book on NCS superconductors, (M. Sigrist, Ed., Springer, Heidelberg), Chapter: Kinetic Theory of NCS Superconductors
A9
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Appendix: isotropic vs. nodal gaps
Examples for fk(s)
conv. BCS, 3He-B(pseudo-) isotropic( )
3He-A, UBe13: axial
UPt3: E1g
UPt3: E2u
cuprates: B1g
A10
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Appendix: statistics of Bogoliubov quasiparticles
energy dispersion of 1
n(ξp)
energy dispersion of Bogoliubov quasiparticles
ξp
n(|ξ |)Momentum distribution of Bogoliubov quasiparticles
Δ/kBT = 0
n(|ξp|)
Δ/kBT = 01 ν(Εp)
0
23
‐4 0 ξp/kBTA11
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Appendix: BCS quasiparticle response functions
Yosida kernel
normal fluid density
Yosida function
spin susceptibility
specific heatspecific heat
A12
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Appendix: BCS quasiparticle response functions
vertexvertex
quantity vertex
normal fluid densitynormal fluid density
spin susceptibility
entropy
specific heatp
Generalized Yosida functionsA13
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Appendix: BCS theory and temperature dependencies
1 1σ(T)/σN(T)Δ2(T)/Δ2(0)
entropy1 1
energy gap000.10.1 11
CV(T)/CN(T)
T/TcT/Tc
Y(T)
00
normal fluid density,spin susc
specific heat1 2.5
V( )/ N( )( )
spin susc.1
000.1 0.11 1T/Tc T/Tc
00
A14
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Appendix: BCS theory: magnetic penetration depth
1 4λ (0)1.4λL(0)
d-wave s-wave
λL(T)λL(T)1.2λL(0)
T/T0 0 8λL(0)
T/Tc0 0.8A15
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Appendix: density & current revisited
homogeneous density response
no homogeneous density response!
stationary current response
no homogeneous density response!
stationary current purely transverse!A16
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Appendix: r.h.s of continuity equation
particle number conservation and gauge invariance
condensate response kernel: Tsuneto function
A17
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Appendix: condensate response
generalizedcondensatedensity
stationaryi itimit
longl thwavelength
limit
generalizedgeneralizedYosidafunctions
„superfluid“ density„condensate“ density
A18