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

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

Electrodynamics: potentials and fields

t t ti lscalar potential vector potential

magnetic field

electric field

gauge invariance

3

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

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

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

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

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

Nambu-BCS theory: thermal excitations

Thermal excitations: Bogoliubov-Valatin quasiparticles

momentum distribution

isotropicnodal

Fermi surface

Bogoliubovquasiparticlesq p

9

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

Momentum distribution functions revisited

Which ofWhich ofthese twofunctionsis relevant for super-

conductivity Fermi-Dirac

?thermal excitations

11

Nonequilibrium description: electromagnetic response

scalar potential vector potential

externalperturbationperturbationpotentials

nonequilibriumphase spacedistribution

linear response

12

response

Nonequilibrium description: normal state

density fluctuations: particle-hole excitations, local equilibrium

vertex (e, evk)( , k)

13„minimal coupling“

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

14

Nonequilibrium description: Nambu structure

electromagnetic potentials in Nambu space

shifted quasiparticle energy

diagonalization

shifted momentum distribution

!15

!

Nonequilibrium description: Nambu structure

integral properties Yosida function

Nambu-BCS current density

superfluid density tensor

16

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

Nonequilibrium description: supercurrent density

violation ofthe numberconservationlaw

gaugetransformation

gauge-invariant

18

current density

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

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

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

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

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

Nonequilibrium description: condensate response

The Tsuneto function:

complicated expression

longwavelengthglimit

condensate density responsey

stationarylimit

24condensate current response

Nonequilibrium description: solution for δnk

solution for δnk

macroscopic particle density

density conservation/relaxation

macroscopic current densitymacroscopic current density

25

Nonequilibrium description: continuity equation

condensate terms on r.h.s. revisited

gauge mode

gauge-invariance charge/particle number conservation law

26

Nonequilibrium description: density & current

density response long wavelength limit

Josephson

stationary current response

no homogeneous density response!

backflow“„backflow

stationary current purely transverse!27

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

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

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

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

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

Appendix: The London functional

London energy density: gradients, fields, magnetodynamics

Madelungtransformation(MT)( )

WKB

London penetration depth gauge-invariant!

A5

Appendix: The Ginzburg-Landau functional

Ginzburg-Landau energy density

gradients, fields,magnetodynamics

thermodynamics

Madelung transformation

SC phase transition:pcondensation of

superfluid density a2A6

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

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

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

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

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

Appendix: BCS quasiparticle response functions

Yosida kernel

normal fluid density

Yosida function

spin susceptibility

specific heatspecific heat

A12

Appendix: BCS quasiparticle response functions

vertexvertex

quantity vertex

normal fluid densitynormal fluid density

spin susceptibility

entropy

specific heatp

Generalized Yosida functionsA13

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

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

Appendix: density & current revisited

homogeneous density response

no homogeneous density response!

stationary current response

no homogeneous density response!

stationary current purely transverse!A16

Appendix: r.h.s of continuity equation

particle number conservation and gauge invariance

condensate response kernel: Tsuneto function

A17

Appendix: condensate response

generalizedcondensatedensity

stationaryi itimit

longl thwavelength

limit

generalizedgeneralizedYosidafunctions

„superfluid“ density„condensate“ density

A18