Download The role of gauge invariance in the theory of superconductivity*

The role of gauge invariance in the
theory of superconductivity*
Dietrich Einzel
Walther-Meißner-Institut für Tieftemperaturforschung
Bayerische Akademie der Wissenschaften
D-85748 Garching
g
Outline
• Electrodynamics
•
•
•
•
Quantum mechanics
London‘s theory
Nambu‐BCS theory (w/o Greens functions!)
Summary and conclusion
* Nobel Lecture 2008, Y. Nambu, Rev. Mod. Phys. 81, 1015 – 1018 (2009)
Seminar on Advances in Solid State Physics, WMI, June 8, 2010
1
Two fundamental theorems
Noether theorem:
Goldstone theorem:
(Emmy Noether, 1918)
(Jeffrey Goldstone, 1961)
„Every continuous symmetry
of a system is to be associated
with a conserved quantity“
„The spontaneous breaking of
a continuous symmetry is to
be associated with a massless
and spinless particle, the socalled
ll d Nambu-Goldstone
N b G ld t
b
boson“
“
symmetry
t operation
ti
conservation
ti law
l
translation in time
energy
translation in space
momentum
b k symmetry
broken
t
G ld t
Goldstone
b
boson
liquids
Galilean
g
phonon
p
longitudinal
longit.+transv. phonon
magnon
phonon
rotation in space
angular momentum
solids
Galilean
spin rotation
phase
charge
gauge
2
Electrodynamics: potentials and fields
scalar potential
t potential
t ti l
vector
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
Copenhagen
h
i t
interpretation:
t ti
probability
b bilit density
d
it
Schrödinger equation in the presence of
Gauge transformation
and
,
fluxoid quantum/2π
corresponds to local U(1) trafo:
with
4
London‘s* theory in a nutshell
Macroscopic (pseudo-) bosonic condensate wave function
i. e. p
postulate of macroscopic
p p
phase coherence associated with a
macroscopic number of
(i) bosons
(k = 1, Bose condensate)
(ii) fermion pairs (k = 2, pair condensate)
superfluid
density
Schrödinger
equation
conservation
law
continuity
Josephson
gauge-invariant
current density
* F & H London,1935, 1950
5
Nambu-BCS* theory: route to superconductivity
energy variable
particles
holes
(i) pair attraction
(exchange boson)
((ii)) pair
p formation
in k-space
(iii) broken gauge
[U(1)] symmetry
* Bardeen, Cooper & Schrieffer, 1957
6
Nambu-BCS theory: particle-hole structure
(i ) pair
(iv)
i potential
t ti l
(v) energy becomes a matrix in particle-hole (Nambu) space for T<Tc
Yoishiro Nambu, 1962
particles
mixture
diagonalization
0
Bogoliubov,
Valatin, 1957
mixture
holes
off-diagonal long range order (ODLRO)
7
Nambu-BCS theory: thermal excitations
(vi) gap formation: energy
dispersion of the thermal
excitations (Bogoliubov(Bogoliubov
Valatin quasiparticles,
„bogolons“)
Ek
b
bogolons
l
Δk
0
0
kF
k
8
Nambu-BCS theory: thermal excitations
Thermal excitations: Bogoliubov-Valatin quasiparticles
momentum distribution
isotropic
nodal
Fermi surface
Bogoliubov
q
quasiparticles
p
9
Momentum distribution in Nambu space
particles
pairing
Nambu matrix
pairing
holes
BCS coherence factors
diagonal density
particles
holes
off-diagonal
g
densityy
Gorkov pairing amplitude
10
Momentum distribution functions revisited
Which of
these two
functions
is relevant
for superconductivity
Fermi-Dirac
?
thermal excitations
11
Nonequilibrium description: electromagnetic response
scalar potential
vector potential
external
perturbation
potentials
nonequilibrium
phase space
distribution
linear
response
12
Nonequilibrium description: normal state
density fluctuations: particle-hole excitations, local equilibrium
vertex (e,, evk)
„minimal coupling“
13
A Nambu space, discovered at a German University
N b matrices:
Nambu
ti
generall structure
t t
Nambu space
Entrance for
particles |k,σ>
holes
|-k,-σ>
only
14
Nonequilibrium description: Nambu structure
electromagnetic potentials in Nambu space
diagonalization
shifted quasiparticle energy
shifted momentum distribution
!
15
Nonequilibrium description: Nambu structure
integral properties
Yosida function
Nambu-BCS current density
superfluid density tensor
16
Nonequilibrium description: Yosida function
BCS Yo
B
osida ffunctio
on
1
Y(T)
l
low
t
temperature
t
regime
GL
regime
0
0.1
T/Tc
1
17
Nonequilibrium description: supercurrent density
violation of
the number
conservation
law
gauge
transformation
gauge-invariant
current density
18
Nonequilibrium description: supercurrent density
how to restore the number conservation law
gauge-invariant (strictly tansverse) current density
„backflow term“ from gauge mode
19
Nonequilibrium description: electromagnetic response
quantum
q
ant m dynamics:
d namics
von Neumann
equation
linearization,
collisionless
limit Ik = 0
streaming in phase space
Fourier space
external and molecular forces
Betbeder-Matibet & Nozieres, 1969; Wölfle, 1976; Einzel & Klam, 2006
20
Nonequilibrium description: integral equations
diagonal
g
energy
gy
Consequences:
Coulomb-I.
Fermi liquid-I.
: external perturbations cause electromagnetic response
: dielectric screening,
screening plasma oscillations
: collective density (sound) oscillations
off-diagonal energy
Consequences:
pairing-I.
: order parameter collective modes (amplitude, phase),
gauge-invariance
21
Nonequilibrium description: dynamics of the gap
amplitude
phase
off-diagonal
g
energy
gy decomposition
p
order p
parameter p
phase
fluctuations
Gauge mode, Anderson-Bogoliubov mode, Nambu-Goldstone mode
order
d parameter
t amplitude
lit d
fluctuations
2Δ mode, coupling O(pha)
22
Nonequilibrium description: the gauge mode
solution for δΔk: generali d Josephson
lized
J
h
relation
l ti
gauge mode
gauge mode frequency
sound
d velocity
l it
23
Nonequilibrium description: condensate response
The Tsuneto function:
complicated expression
long
wavelength
g
limit
condensate densityy response
stationary
limit
condensate current response
24
Nonequilibrium description: solution for δnk
solution for δnk
macroscopic particle density
density conservation/relaxation
macroscopic 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
no homogeneous density response!
stationary current 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 ψ
gauge-invariant formulation possible
Local equilibrium BCS response theory:
spontaneously
t
l broken
b k gauge U(1) symmetry
t
Nambu space description
correct microscopic form of superfluid density tensor ns
lacks g
gauge
g invariance and therefore p
particle number conservation
Nonequilibrium BCS response theory:
order parameter phase fluctuations: gauge mode
generall condensate
d
t response (Tsuneto)
(T
t ) function
f
ti
determines 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
gauge mode frequency [vF2/3]1/2 unaffected by unconventionality of pairing
gauge mode frequencies different from [vF2/3]1/2 in non-centrosymmetric superconductors
28
Appendix: quantum mechanics, Madelung description
quantum mechanical wave function
(Erwin Madelung,
Madelung 1926)
probability density
conservation law for np
probability current
density
(gauge invariant !)
(gauge-invariant
Hamilton-Jacobi
(Josephson) equation
Euler equation for
dissipationless
„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 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 of
macroscopic phase coherence associated with a macroscopic number of
(i) bosons
(k = 1, Bose condensate)
(ii) fermion pairs (k = 2,
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
p
of
Replacement
* F & H London,1935, 1950
byy the electrochemical p
potential
A3
Appendix: basic results of the London theory
condensate
d
t currentt
persistent currents
Screening and magnetic
field penetration depth
Fluxoid
Fl
id quantization
ti ti
(„2e or not 2e“)
A4
Appendix: The London functional
London energy density: gradients, fields, magnetodynamics
Madelung
transformation
( )
(MT)
WKB
London penetration depth
gaugeinvariant!
A5
Appendix: The Ginzburg-Landau functional
Ginzburg-Landau energy density
gradients, fields,
magnetodynamics
thermodynamics
Madelung transformation
SC p
phase transition:
condensation of
superfluid density a2
A6
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)
A7
Appendix: classification of pair potentials
Condensate: p
pair p
potential ((total spin
p s)
node
structure
Conventional:
singlet
(s=0)
triplet
(s=1)
Unconventional:
fp(s) shares lattice
symmetry
broken gauge
symmetry
( ) breaks lattice
fp(s)
symmetry
additional broken
symmetries
ti
A8
Appendix: unconventional pairing
Organic
SC‘s (1980)
Heavy-fermion
y
SC‘s (1979)
Superfluid
3He (1971)
Cuprate- (high-Tc)
SC‘s (1986)
Ruddlesden-Popper
SC‘s Sr2RuO4 (1994)
NCS superconductors* (2004)
* D.E. + Klam/Manske: PRL 102, 027004 (2009)
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 energy
dispersion of
Bogoliubov quasiparticles
1
n(ξξp)
n(|ξp|)
Momentum distribution of Bogoliubov quasiparticles
Δ/kBT T = 0
=0
1
ν(Εp)
2
3
0
‐4
0
ξp/kBT
A11
Appendix: BCS quasiparticle response functions
Yosida kernel
normal fluid density
Yosida function
spin susceptibility
specific heat
A12
Appendix: BCS quasiparticle response functions
vertex
quantity
vertex
normal fluid density
spin susceptibility
entropy
specific
p
heat
Generalized Yosida functions
A13
Appendix: BCS theory and temperature dependencies
Δ2(T)/Δ2(0)
1
1
energy gap
0
0.1
1
T/Tc
1
Y(T)
( )
0
0.1
2.5
normal fluid density,
spin susc
spin susc.
σ(T)/σN(T)
entropy
T/Tc
1
)/ N((T))
CV((T)/C
specific heat
1
0
0.1
T/Tc
1
0
0.1
T/Tc
1
A14
Appendix: BCS theory: magnetic penetration depth
1 4λL(0)
1.4λ
d-wave
λL(T)
1.2λL(0)
λL(0)
s-wave
0
T/T
T/Tc
08
0.8
A15
Appendix: density & current revisited
homogeneous density response
no homogeneous density response!
stationary current 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
generalized
condensate
density
stationary
i it
imit
long
wavelength
l
th
limit
generalized
Yosida
functions
„condensate“ density
„superfluid“ density
A18