Lecture Notes

Aspects of Symmetry in Unconventional Superconductors

Manfred Sigrist, ETH Zurich

Unconventional Superconductors

Many novel superconductors show properties different from standard superconductors (overview);

Aim of this lecture:

• discuss structure of Cooper pairs from a symmetry point of view - key symmetries: time

reversal and inversion symmetry

• learn the techniques of the phenomenological approach: generalized Ginzburg-Landau

theories - broken symmetries and order parameters

• discuss phenomena due to symmetry breaking: example broken time reversal symmetry

• analyze consequences of lack of key symmetries

Some literature:

• V.P. Mineev and K.V. Samokhin, Introduction to Unconventional Superconductivity, Gor-

don and Breach Science Publisher (1999).

• M. Sigrist, Introduction to Unconventional Superconductivity, AIP Conf. Proc. 789, 165

(2005).

• M. Sigrist, Introduction to unconventional superconductivity in non-centrosymmetric met-

als, AIP Conf. Proc. 1162, 55 (2009).

1. General form of Cooper pairing and BCS theory

BCS theory of superconductivity describes an instability of a normal metal state,

normal metal ground state:

|Ψ0〉 =

|k|≤kF∏k

c†k↑c†−k↓|0〉 (1)

note the states created by c†k↑ and c†−k↓, |k ↑〉 and |k ↑〉 are degenerate

εk↑ = ε−k↓ = εk single-electron energy (2)

guaranteed by time reversal symmetry (time reversal operator K):

|k ↑〉 K←→ | − k ↓〉 (3)

1

BCS ground state:

|ΨBCS〉 =∏k

[uk + vkc

†k↑c†−k↓

]|0〉 (4)

coherent state of electron pairs of opposite momenta

bk = 〈ΨBCS |c−k↓ck↑|ΨBCS〉 = u∗kvk (5)

non-vanishing for |k| very close to kF : BCS state affects mainly Fermi surface. electron number

not fixed (grand canonical viewpoint - coherent state like BEC)

1.1. Cooper problem - generalized

Cooper instability through interaction between two electrons added to normal state |Ψ0〉 of free

electrons (εk = ~2k2/2m)

2 electron states |k1s1〉 and |k2s2〉, assume k1 + k2 and |k1|, |k2| > kF (restricted by Pauli

exclusion principle)

Schrodinger equation for 2 interacting electrons− ~2

2m

(∇2r1 + ∇2

r2

)+ V (r1 − r2)

ψ(r1, s1; r2, s2) = E ψ(r1, s1; r2, s2) (6)

V (r1 − r2): 2 -particle interaction;

change to center of mass and relative coordinates: R = 12(r1 + r2) and r = r1 − r2

ψ(r1, s1; r2, s2) = φs1s2(r)eiq·R = φs1s2(r) since q = k1 + k2 = 0 (7)

Symmetry aspect:

Pauli exclusion principle → antisymmetric wave function: φs1s2(r) = −φs2s1(−r)

φs1s2(r) = φ(r)χs1s2 =

φ(r) = φ(−r), χs1s2 = −χs2s1 even parity, spin singlet

φ(r) = −φ(−r), χs1s2 = χs2s1 odd parity, spin triplet(8)

with φ(r) orbital and χs1s2 spin part of wave function1

⇒ −~2

m∇2φ(r) + V (r)φ(r) = E φ(r) (10)

turn to Fourier (momentum) space:

gk =

∫d3r e−ik·rφ(r) , Vq =

∫d3r e−iq·rV (r) (11)

1Spin part of wave function:

spin singlet S = 0 : χs1s2 = 1√2(| ↑↓〉 − | ↓↑〉) χs1s2 = −χs2s1

spin triplet S = 1 : χs1s2 = | ↑↑〉, 1√2(| ↑↓〉+ | ↓↑〉), | ↓↓〉 χs1s2 = χs2s1

(9)

2

⇒ ~2k2

mgk +

1

Ω

∑k′

Vk−k′gk′ = E gk (12)

with Ω = L3 volume k = 2πL (nx, ny, nz) (nx,y,z: integers)

Symmetry aspect: assume full spherical rotation symmetry, symmetry group O(3)

expansion in spherical harmonics |lm〉

Vk−k′ =∑∞

l=0 Vl(k, k′)∑+l

m=−l Ylm(k)Y ∗lm(k′)

gk =∑∞

l=0

∑+lm=−l glm(k)Ylm(k)

(13)

with k = |k| and k = k/k; 2

set |lm〉| − l ≤ m ≤ +l is (2l + 1)-dimensional basis of the irreducible representation of O(3)

labelled by l

define ξk = εk − εF = ~2k22m − εF , ∆E = E − εF and rewrite Vl(k, k

′)→ Vl(ξk, ξk′) and glm(k)→glm(ξk)

1

Ω

∑k

→∫

d3k

(2π)3→∫dξ N(ξ)

dΩk4π

(15)

with density of states

N(ξ) =1

Ω

∑k

δ(ξ − ξk) (16)

Schrodinger equation decouples in different channels l (angular momentum):

(2ξ −∆E)glm +

∫dξ′ N(ξ′)Vl(ξ, ξ

′)glm(ξ′) = 0 (17)

for practical reasons: N(ξ) ≈ N(0) and

Vl(ξ, ξ′) =

νl −εc ≤ ξ, ξ′ ≤ εc

0 otherwise(18)

with εc εF ;

solving the equation, searching for bound state of 2 electrons, ie. ∆E < 0:

glm(ξ) =−N(0)νl2ξ −∆E

∫ εc

0dξ′ glm(ξ′) =

−N(0)νl2ξ −∆E

Ilm for0 ≤ ξ ≤ εc (19)

for νl < 0:

Ilm = −IlmN(0)νl

∫ εc

0

dξ′

2ξ′ −∆E= −Ilm

N(0)νl2

ln

(∆E − 2εc

∆E

)(20)

⇒ ∆E = −2εce2/N(0)νl (21)

2Spherical harmonics: orthogonality relation∫dΩk

4πY ∗lm(k)Yl′m′(k) = δll′δmm′ (14)

3

if ∆E εc (note: εc functions as a somewhat arbitrary cutoff for the integral);

bound state with energy E < 2εF ⇒ instability: lowest bound state for strongest ”attractive”

channel l (νl < νl′ for l 6= l′).

Symmetry aspect:

bound state parity distinguished by l: (−1)l ⇒

l = 0, 2, 4, . . . even parity, spin singlet

l = 1, 3, 5, . . . odd parity, spin triplet.

Examples:

electron-phonon interaction:

Vk−k′ =

ν0(ξ, ξ′) < 0 −εD ≤ ξ, ξ′ ≤ εD

0 otherwise(22)

interaction without angular dependence (contact interaction): pairing channel l = 0, S = 0

”s-wave” (complete symmetric in orbital and spin space

simple anisotropic ”repulsive” interaction: Vk−k′ = V (ξ, ξ′)(k − k′)2

V (ξ, ξ′) =

ν > 0 −εc ≤ ξ, ξ′ ≤ εc

0 otherwise(23)

but

ν(k − k′)2 = 2ν[1− k · k′] = 8πν︸︷︷︸=ν0>0

Y00(k)Y ∗00(k′)−8π

3ν︸ ︷︷ ︸

=ν1<0

+1∑m=−1

Y1m(k)Y ∗1m(k′) (24)

no bound state in l = 0, S = 0 (repulsive) channel; bound state in (attractive) l = 1, S1 channel:

odd parity spin triplet ”p-wave”.

1.2. Generalized BCS theory

we introduce a general form of a BCS Hamiltonian:

HBCS =∑k,s

ξkc†kscks +

1

2

∑k,k′

∑s1,s2,s3,s4

Vk,k′;s1s2s3s4c†ks1c†−ks2c−k′s3ck′s4 (25)

we restrict to the BCS scattering channel

Vk,k′;s1s2s3s4 = 〈−k, s1;k, s2|V | − k′, s3;k′, s4〉 . (26)

and the Pauli exclusion principle requires

Vk,k′;s1s2s3s4 = −V−k,k′;s2s1s3s4 = −Vk,−k′;s1s2s4s3 = V−k,−k′;s2s1s4s3 . (27)

instability discussed by decoupling through generalized mean field like bk in Eq.5:

c−k′sck′s′ = bk,ss′ + (c−k′sck′s′ − bk,ss′) with bk,ss′ = 〈c−k′sck′s′〉 (28)

4

inserting leads to mean field Hamiltonian

Hmf =1

2

∑k,s

ξk(c†kscks − c−ksc†−ks)−

1

2

∑k,s1,s2

[∆k,s1s2c

†ks1c†−ks2 + ∆∗k,s1s2cks1c−ks2

]+K , (29)

where

K = −1

2

∑k,k′

∑s1,s2,s3,s4

Vk,k′;s1s2s3s4〈c†ks1c†−ks2〉〈c−k′s3ck′s4〉+

1

2

∑k,s

ξk. (30)

and the self-consistency equations

∆k,ss′ = −∑k′,s3s4

Vk,k′;ss′s3s4bk,s3s4 and ∆∗k,ss′ = −∑k′s1s2

Vk′,k;s1s2s′sb∗k,s2s1 . (31)

∆k,ss′ or bk,ss′ characterize BCS state

Bogolyubov quasiparticle spectrum: matrix formulation

Hmf =∑k

C†kEkCk +K , (32)

with

Ck =

ck↑ck↓c†−k↑c†−k↓

, Ek =1

2

ξkσ0 ∆k

∆†k −ξkσ0

and ∆k =

∆k,↑↑ ∆k,↑↓

∆k,↓↑ ∆k,↓↓

.

(33)

to be diagonalized into

H =∑k

A†kEkAk +K (34)

where

Ak =

ak↑ak↓a†−k↑a†−k↓

and Ek =

Ekσ0 0

0 −Ekσ0

(35)

Bogolyubov transformation with unitary matrix

Uk =

uk vk

v∗−k u∗−k

⇒ Ck = UkAk and Ek = U †kEkUk (36)

and UkU†k = U †kUk = 1. with

uk =(Ek + ξk)σ0√2Ek(Ek + ξk)

and vk =−∆k√

2Ek(Ek + ξk)(37)

and the quasiparticle energy

Ek =√ξ2k + |∆k|2 with |∆k|2 =

1

2tr(

∆†k∆k

). (38)

Quasiparticle spectrum with excitation gap (electron-hole hybridization at the Fermi energy):

5

hole−like

k

kF

2∆

hole−like

electron−like

electron−like

E

Self-consistence equation (gap equation):

∆k,s1s2 = −∑k′,s3s4

Vk,k′;s1s2s3s4∆k′,s4s3

2Ektanh

(Ek

2kBT

)(39)

and BCS coherent ground state:

|ΨBCS〉 =∏k,s,s′

uk,ss′ + vk,ss′ c

†ksc†−ks′

|0〉 (40)

gap matrix parametrization:

structure of the mean field: bk,ss′ = φ(k)χss′ ⇒

φ(k) =

+φ(−k) even parity, spin singlet

−φ(−k) odd parity, spin triplet(41)

Pauli exclusion principle: ∆k = −∆T−k for both even and odd parity

even parity - spin singlet:

∆k =

(∆k,↑↑ ∆k,↑↓∆k,↓↑ ∆k,↓↓

)=

(0 ψ(k)

−ψ(k) 0

)= iσyψ(k) . (42)

with even scalar gap function, ψ(k) = ψ(−k) ,

odd parity - spin triplet:

∆k =

(∆k,↑↑ ∆k,↑↓∆k,↓↑ ∆k,↓↓

)=

(−dx(k) + idy(k) dz(k)

dz(k) dx(k) + idy(k)

)= i (d(k) · σ) σy , (43)

with odd vector gap function, d(k) = −d(−k)

Note: spin configuration d ⊥ S because

dx(| ↓↓〉 − | ↑↑〉)− idy((| ↓↓〉+ | ↑↑〉) + dz(| ↑↓〉+ | ↓↑〉) (44)

excitation gap:

6

we use |∆k|2 = 12 tr

∆k∆†k

:

∆k∆†k = |ψ(k)|2σ0 ⇒ |∆k|2 = |ψ(k)|2 even parity - spin singlet

∆k∆†k = |d(k)|2σ0 + i(d(k)× d(k)∗) · σ ⇒ |∆k|2 = |d(k)|2 odd parity spin triplet(45)

note for ”unitary states”: d(k)× d(k)∗ = 0.

New parametrization:

rewrite interaction to separate even and odd parity:

Vk,k′;s1s2s3s4 = J0k,k′ σ

0s1s4 σ

0s2s3 + Jk,k′σs1s4 · σs2s3 , (46)

leads to gap equations for even parity,

ψ(k) = −∑k′

(J0k,k′ − 3Jk,k′)︸ ︷︷ ︸

= vsk,k′

ψ(k′)

2Ek′tanh

(Ek′

2kBT

)(47)

for odd parity,

d(k) = −∑k′

(J0k,k′ + Jk,k′)︸ ︷︷ ︸

= vtk,k′

d(k′)

2Ek′tanh

(Ek′

2kBT

)(48)

where

vs,tk,k′

=′∑l

νs,tl (ξk, ξk′)+l∑

m=−lYlm(k)Y ∗lm(k

′) (49)

note sum over l restricts to given parity (−1)l and νs,tl (ξ, ξ′) with the usual restrictions

linearized gap equation: T → Tc−, ∆k → 0 and Ek = |ξk|,case: even parity

ψ(k) = −∑k′

νsk,k′ψ(k′)

2ξk′tanh

(ξk′

2kBT

)

= −N(0)〈νsk,k′ψ(k′)〉k′,FS∫ εc

0dξ

1

ξtanh

(ξ

2kBT

)︸ ︷︷ ︸

= ln(1.14εc/kBT )

(50)

eigenvalue equation (λ: dimensionless eigenvalue defining Tc)

− λψ(k) = −N(0)〈νsk,k′ψ(k′)〉k′,FS with kBTc = 1.14εce−1/λ (51)

analogous for odd parity

− λd(k) = −N(0)〈vtk,k′d(k′)〉k′,FS (52)

where 〈· · · 〉k,FS angular average on Fermi surface

7

largest eigenvalue determines highest Tc → superconducting instability

Symmetry operations:

symmetries of the normal state:

• orbital rotation O(3): g|k, s〉 = |Rgk, s〉 with Rg; rotation matrix of element g ∈ O(3)

• spin rotation SU(2): g|k, s〉 =∑

s′ D(g)ss′ |k, s′〉; D(g) = exp[iS · θg] with g ∈ SU(2)

• time reversal K: K|k, s〉 =∑

s′(−iσy)ss′ | − k, s′〉; K = −iσyC with C complex conju-

gation (K ∈ K = E, K.

• inversion I: I|k, s〉 = | − k, s〉 with I ∈ I = E, I

• gauge U(1): Φ|k, s〉 = eiφ/2|k, s〉 with Φ ∈ U(1)

Symmetry operations on gap function:

• Fermion exchange: ∆k = −∆T−k

• orbital rotation: g∆k = ∆Rgk

• spin rotation: g∆k = D(g)†∆kD(g)

• time reversal: K∆k = σy∆∗kσy

• inversion: I∆k = ∆−k

• gauge: Φ∆k = eiφ∆k

transferred to the gap functions ψ(k) and d(k):

even parity odd parity

Fermion exchange ψ(k) = ψ(−k) d(k) = −d(−k)

Orbital rotation gψ(k) = ψ(Rgk) gd(k) = d(Rgk)

Spin rotation gψ(k) = ψ(k) gd(k) = Rgd(k)

Time-reversal Kψ(k) = ψ∗(−k) Kd(k) = −d∗(−k)

Inversion Iψ(k) = ψ(−k) Id(k) = d(−k)

U(1)-gauge Φψ(k) = eiφψ(k) Φd(k) = eiφd(k)

Conventional pairing state: most symmetric pairing state l = 0, S = 0 ⇒ ψ(k) = ψ0

Unconventional pairing state: l 6= 0, S = 0, 1

8

Examples of unconventional pairing states:

cuprate high-temperature superconductors: quasi-two-dimensional ψ(k) = ∆0(k2x − k2

y) with l =

2, S = 0 ”d-wave”;

excitation gap with line nodes: |∆k| = |∆0||k2x − k2

y| .

3He B-phase : d(k) = ∆0k with l = 1, S = 1 ”p-wave”;

excitation gap without nodes (isotropic): |∆k| = |∆0||k|.

3He A-phase : d(k) = ∆0kz(kx ± iky) ”p-wave” (2-fold degenerate);

excitation gap point nodes: |∆k| = |∆0||kx ± iky|.

gap nodes influence low-temperature thermodynamic properties, e.g. specific heat

C(T ) ∝

T−3/2e−∆/kBT nodeless

T 3 point nodes

T 2 line nodes

(53)

power laws versus thermally activated behavior, also observable in other quantities relying on

a thermal average over low-energy states (London penetration depth, NMR-T−11 , ultrasound

absorption, . . . )

2. Generalized Ginzburg-Landau theory of superconductivity

Ginzburg-Landau theory for 2nd-order phase transitions based on concept of spontaneous sym-

metry breaking

key quantity: order parameter which grows continuously from zero crossing the transition

temperature into the order phase

2.1. Conventional Ginzburg-Landau theory

order parameter: gap function ∆k or pair mean field b,ss′

free energy expansion in order parameter η = η(r, T ) for T ≈ Tc:

F [η,A;T ] =

∫Ωd3r

[a(T )|η|2 + b(T )|η|4 +K(T )|Πη|2 +

1

8π(∇×A)2

], (54)

with Π = ~i∇+ 2e

c A and a(T ) ≈ a′(T −Tc), b(T ) ≈ b(Tc) = b > 0 and K(T ) = K(Tc) = K > 0;

A is vector potential and ∇×A−B is magnetic field.

variational minimization of F with respect to η and A ⇒ Ginzburg-Landau equations

aη + 2bη|η|2 −KΠ∗ ·Πη = 0

2e

cK η∗Πη + ηΠ∗η∗︸ ︷︷ ︸

=j/c

− 1

4π∇× (∇×A)︸ ︷︷ ︸

=B

= 0 (55)

9

1. equation: uniform case for phase transition

0 = a(T )η + 2bη|η|2 ⇒ |η|2 =

0 T > Tc

−a(T )

2bT ≤ Tc

(56)

2. equation: London equation

∇× (∇×B) = −4π

c

8e2

cK|η|2B ⇒ ∇2B =

1

λ2L

B , (57)

with

λ−2L =

32πe2

c2K|η|2 =

4πe2nsmc2

(58)

ns density of superfluid electrons

describes the phenomenology of the superconducting phase: phase transition and Meissner-

Ochsenfeld effect (screening of magnetic fields)

construction of free energy functional

F [η,A] is a scalar under all symmetries of the normal state: G = O(3)× SU(2)×K×I ×U(1)

order parameter: η is a scalar under O(3) and SU(2) because of pairing in fully symmetric

channel l = 0, S = 0

time reversal : Kη = η∗

U(1) gauge : Φη = ηeiφ(59)

gradient and vector potential: Π invariant under SU(2)

orbital rotation: gΠ = RgΠ

time reversal : KΠ = −Π∗ =~i∇− 2e

cA

inversion: IΠ = −Π

U(1) gauge : ΦΠ = Π +e

c∇φ

(60)

scalar combinations: η∗η, (η∗η)2 and (Πη)∗ · (Πη)∗ as well as (∇×A)2

broken symmetry at phase transition: U(1) with G′ = O(3)× SU(2)×K × I2.2. Ginzburg-Landau theory for unconventional pairing

linearized gap equation:

−λψ(k) = −N(0)〈vsk,k′ψ(k′)〉k′,FS

−λd(k) = −N(0)〈vtk,k′d(k′)〉k′,FS(61)

generally degenerate solutions for eigenvalues λ: largest λ ⇒ highest Tc

10

degenerate solution form the basis of an irreducible representation of the normal state symmetry

group, e.g. for O(3) states classified according to angular momentum l with degeneracy 2l + 1

(dimension of representation)

assume even parity spin singlet Cooper pairs with l 6= 0: scalar gap function ψ(k) =∑

m ηm(r)ψm(k)

where ψm(k) are basis function of the irreducible representation Dl and ηm is the order pa-

rameter

generalized scalar free energy involves invariant terms of ηm, η∗m and Π

example O(3)× I → D4h: discrete rotation symmetry

D4h tetragonal point group (16 elements = 8 rotations + 8 rotations × inversion )

moreover we assume spin-orbit coupling: orbital and spin part rotate simultaneously

irrelevant for even parity spin singlet Cooper pairing

odd-parity spin-triplet states:

rotation: gd(k) = Rgd(Rgk)

inversion: Id(k) = d(−k) = −d(k)

(62)

basis functions for the irreducible representations of D4h:

Γ E 2C4 C2 2C ′2 2C ′′2 I 2S4 σh 2σv 2σd basis function used names

A1g 1 1 1 1 1 1 1 1 1 1 ψ = 1 s-waveA2g 1 1 1 -1 -1 1 1 1 -1 -1 ψ = kxky(k

2x − k2

y) g-wave

B1g 1 -1 1 1 -1 1 -1 1 1 -1 ψ = k2x − k2

y dx2−y2-wave

B2g 1 -1 1 -1 1 1 -1 1 -1 1 ψ = kxky dxy-waveEg 2 0 -2 0 0 2 0 -2 0 0 ψ = kxkz, kykz d-wave

A1u 1 1 1 1 1 -1 -1 -1 -1 -1 d = xkx + yky p-waveA2u 1 1 1 -1 -1 -1 -1 -1 1 1 d = xky − ykx p-waveB1u 1 -1 1 1 -1 -1 1 - 1 - 1 1 d = xkx − yky p-waveB2u 1 -1 1 -1 1 -1 1 -1 1 - 1 d = xky + ykx p-waveEu 2 0 -2 0 0 -2 0 2 0 0 d = zkx, zky p-wave

there are 4 non-degenerate and 1 2-fold degenerate order parameter for each even and odd parity

case

for the one-dimensional representations the free energy functional looks identical to the case of

conventional order parameters

two-dimensional representation:

ψ(k) = ηxkxkz + ηykykz or d(k) = ηxzkx + ηy zky (63)

with the scalar free energy of the order parameter η = (ηx, ηy) and vector potential A:

11

F [η,A;T ] =

∫d3r

[a(T )|η|2 + b1|η|4 +

b22η∗2x η2

y + η2xη∗2y + b3|ηx|2|ηy|2

+K1|Πxηx|2 + |Πyηy|2+K2|Πxηy|2 + |Πyηx|2

+K3(Πxηx)∗(Πyηy) + c.c.+K4(Πxηy)∗(Πyηx) + c.c.

+K5|Πzηx|2 + |Πzηy|2+1

8π(∇×A)2

](64)

where a(T ) = a′(T − Tc), b1 > 0, 4b1 − |b2|+ b3 > 0 and K1,...,5 > 0.

uniform superconducting phase: 3 phases possible for symmetry reasons (table of basis functions)

Phase ψ(k) d(k) broken symmetry

A kz(kx ± iky) z(kx ± iky) U(1),K broken time reversal symmetry

B kz(kx ± ky) z(kx ± ky) U(1), D4h → D2h broken rotation symmetry

C kzkx, kzky zkx, zky U(1), D4h → D2h broken rotation symmetry

b / b2 1

b / b13

4b + b + b = 01 2 3

4b − b + b = 01 2 3

A

C

B

Which phase is most stable from a microscopic view point ? Consider T = 0 condensation

energy (weak coupling),

12

Econd = 〈H′〉∆ − 〈H′〉∆=0 =1

2

∑k,s

(ξk − Ek) +1

2

∑k,s1,s2

∆∗k,s1s2∆k,s2s1

2Ek

= 2N(0)

∫ εc

0dξ (ξ − 〈

√ξ2 + |∆k|2〉k,FS) +

⟨|∆k|2

∫ εc

0dξ

1√ξ2 + |∆k|2

⟩k,FS

≈ −N(0)

2〈|∆k|2〉k,FS

(65)

Gap structure important for stability: simple discussion assuming spherical Fermi surface: de-

termine 〈|∆k|2〉k,FS

Phase 〈|ψ(k)|2〉k,FS 〈|d(k)2〉k,FSA 2/15 2/3

B 1/15 1/3

C 1/15 1/3

result: for even and odd parity the A-phase is most stable as it has least nodes.

Broken symmetries and physical properties

Normal state Symmetry including spin-orbit coupling: G = D4h ×K × U(1)

Broken U(1)-gauge symmetry yields London equation (Meissner-Ochsenfeld effect) and flux

quantization

Impact of further broken symmetries:

”Nematic phase” through broken crystal rotation symmetry as in phase B and C: G′ = D2h×K

η =

(1, 1), (1,−1) B-phase

(1, 0), (0, 1) C-phase(66)

coupling to lattice strain εµν : invariant terms in free energy

Fε−η =

∫d3r

[γ1(εxx + εyy) + γ′1εzz

|η|2 + γ2(εxx − εyy)(|ηx|2 − |ηy|2) + γ3εxy(η

∗xηy + ηxη

∗y)]

(67)

with γi real coefficients. This free energy has to be supplemented by the elastic energy:

Fel =∫d3r

∑µ1,...,µ4

12Cµ1···µ4εµ1µ2εµ3µ4 .

• B-phase couples to the strain εxy ⇒ uniaxial distorting along [110] or [110]

• C-phase couples to the strain εxx − εyy ⇒ uniaxial distorting along [100] or [010]

• A-phase does not coupling to anisotropic strain ⇒ not nematic

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Single domain phase through cooling under application of uniaxial stress to sample.

”Magnetic phase” through broken time reversal symmetric as in A-phase: G′ = D4h

define Cooper pair angular moment:

L = i~〈ψ(k)∗(k ×∇k)ψ(k)〉k,FS (68)

and analog for odd parity state

with ψ(k) = ηxkzkx + ηykzky

L = i~zη∗xηy〈k2

zk2x〉k,FS − η∗yηx〈k2

zk2y〉k,FS

∝ i~(η∗xηy − η∗yηx)z (69)

Categories of time reversal symmetry breaking phases (G.E. Volovik and L.P. Gor’kov):

”Ferromagnetic” phase: L 6= 0 example: A-phase (see above)

”Antiferromagnetic” phase: L = 0 example: s+ idx2−y2-wave state ψ(k) = ηs + iηd(k2x − k2

y)

L = i~

〈(η∗s − iηd(k2

x − k2y))2iηdkykz〉k,FS

〈(η∗s − iηd(k2x − k2

y))2iηdkzkx〉k,FS

−〈(η∗s − iηd(k2x − k2

y))2iηdkxky〉k,FS

= 0 (70)

from a group theoretical point of view: components of L are basis functions of irreducible

representations of point group

example D4h: Lx, Ly → Eg and Lz → A2g

• order parameter of A-phase: η = ηx, ηy → Eg: Eg ⊗ Eg = A1g ⊕ A2g ⊕ B1g ⊕ B2g ; the

decomposition of this Kronecker product contains A2g which is connected with Lz; thus

the Lz-component can be finite.

• order parameter of s + id-wave phase: η = ηs, ηd → A1g ⊕ B1g: (A1g ⊕ B1g) ⊗ (A1g ⊕B1g) = 2A1g ⊕ 2B1g ; the decomposition of this Kronecker product does not contain any

representation connected with L; thus L cannot be constructed for the order parameter

and vanishes.

topological view point:

- ferromagnetic ←→ chiral ⇒ phase has chiral subgap edge states (spontaneous edge

currents)

- antiferromagnetic ←→ not chiral ⇒ edge subgap states exist and give rise to sponta-

neous currents, but not connected with topological bulk properties.

Conserved charge (G.E. Volovik):

assume full rotation symmetry around z-axis (cylindrical instead of tetragonal symmetry)

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U(1) gauge Φψ(k) = eiφψ(k)

rotation around z-axis Θψ(k) = eiθψ(k)

⇒ ΦΘ−1 = E for φ = θ (71)

thus there is a conserved charge: Q = Lz − N/2: changes of angular momentum and charge are

coupled

removing a Cooper pair (N → N − 2) changes the angular momentum of the system by ±~z:spatial fluctuations in Cooper pair density induces local angular momentum density or orbital

magnetic flux

⇒ ”anomalous electro-magnetism” ⇒ spontaneous edge currents and current patterns

around defects ⇒ polar Kerr effect (theory still incomplete)

Spontaneous edge currents: Ginzburg-Landau description of the time reversal symmetry

breaking phase

consider planar edge with normal vector n = (100) (x ≥ 0 superconductor and x < 0 vacuum)

boundary conditions for the order parameter (scattering at the edge is pair breaking), simplified

as matching condition for mirror operation on order parameter at planar edge (x < 0 virtual)

(ηx, ηy) ←→ (−ηx, ηy) ⇒

ηx(x) = −ηx(−x)

ηy(x) = ηy(−x)(72)

Ginzburg-Landau equation give simplified solution:

ηx(x) = η0 tanh(x/ξ) and ηy(x) = iη0 with η20 =

a′(T − Tc)4b1 − b2 + b3

(73)

supercurrent density: j = −c∂F/∂A,

jx = 8πe[K1η

∗xΠxηx +K2η

∗yΠxηy +K3η

∗xΠyηy +K4η

∗yΠyηx + c.c.

]jy = 8πe

[K1η

∗yΠyηy +K2η

∗xΠyηx +K3η

∗yΠxηx +K4η

∗xΠxηy + c.c.

]jz = 8πeK5η∗xΠzηx + η∗yΠzηy + c.c. .

(74)

with Ax = Az = 0 we find jx(x) = 0 ⇒ no current flows through the edge

jy(x) = 16πeK3ηy~i

∂ηx∂x

+c

4πλ2Ay =

16πe~ξ

η20

cosh2(x/ξ)︸ ︷︷ ︸= j(0)

y (x)

+c

4πλ2Ay . (75)

which enters the London equation

∂2Ay∂x2

− 1

λ2Ay =

4π

cj(0)y (x) (76)

where j(0)y spontaneous current parallel to the edge on a width ξ =

√−K1/a(T ); there is

a Meissner screening current which compensates j(0)y such that the magnetic flux induced is

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located close to the surface only and penetrates over the London penetration depth λ,

1

λ2=

32π2e2

c2(K1 +K2)|η0|2 (77)

observation of spontaneous magnetic fields by zero-field µSR, measures internal magnetic field

ξ λ ξ λ

jyBz

0 0x x

0

spread in sample, enhancement of magnetism in superconducting phase: Sr2RuO4, PrOs4Sb12,

(U,Th)Be13, SrPtAs, Re6Zr, . . .

no direct observation of edge currents so far

Possible realizations:

Sr2RuO4: d(k) = ∆0z(kx ± iky)

URu2Si2: ψ(k) = ∆0kz(kx ± iky)

3. Role of key symmetry

two key symmetries, time reversal and inversion, to form zero-momentum Cooper pairs of two

partners of identical energy (Anderson, 1959, 1984)

search Cooper pair partner for |k ↑〉

time reversal: K|k ↑〉 = | − k ↓〉 ⇒ |k ↑〉, | − k ↓〉 form even-parity spin-singlet pair

inversion: I|k ↑〉 = | − k ↓〉 ⇒ |k ↑〉, | − k ↓〉 form odd-parity spin-triplet pair(78)

What happens if one of the two key symmetries is absent?

Implementation in Hamiltonian:

H −→ H+H′ = H+∑k

∑s,s′

gk · c†ksσss′ cks′ (79)

the term H′ conserves

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(a) inversion symmetry, if gk = g−k

(b) time reversal symmetry, if gk = −g−k(80)

examples:

case (a): gk = µBH Zeeman field, leads to spin splitting of the Fermi surface (majority /

minority spin Fermi sea)

case (b): gk = αz × k Rashba spin-orbit coupling, leads to spin splitting of Fermi surface with

k dependent quantization axis

3.1. Ferromagnetic superconductor or superconductor in magnetic Zeeman field

uniform spin polarization leads to paramagnetic limiting (Pauli or Clogston-Chandrasekhar

limit): breaking of spin singlet Cooper pairs

this is mostly not observable, because the upper critical field Hc2 of orbital depairing is usually

much lower than the limiting field Hp3

Hc2(T = 0) =Φ0

2πξ20

and Hp(T = 0) =Hc(0)√

4πχp(82)

where Hc(0) is the thermodynamic critical field at T = 0 and χp is the Pauli spin susceptibility.

coupling terms to the free energy expansion due to magnetic field H for order parameters

ψ(k) =∑

j ηjψi(k) and d(k) =∑

µ,j ηµjµkj

2nd-order coupling to H for suppression (paramagnetic limit)

F(2)H = β

∑µ,ν

∑j

HµHν

|ηj |2δµν + η∗µjηνj

∝H2〈|ψ(k)|2〉k,FS + 〈|H · d(k)|2〉k,FS (83)

with β > 0 (this terms gives correction to spin susceptibility, Yosida)

1st-order coupling to H for the structure of state

F(1)H = iβ′

∑λ,µ,ν

ελµνHλη∗µjηνj ∝ iH · 〈d(k)∗ × d(k)〉k,FS (84)

3Paramagnetic limit: comparison of spin polarization and condensation energy:

Hc(0)

8π=χp

2H2 ⇒ Hp =

Hc√4πχp

(81)

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with ελµν completely antisymmetric tensor

Assuming H ‖ z we find that spin singlet order parameters generally and spin-triplet order

parameters with d ‖ H are suppressed, while spin triplet components with d ⊥ H is stable

yielding H · (d∗ × d) 6= 0. This is a non-unitary state with

∆k∆†k = |d(k)|2σ0 + i(d(k)× d(k)∗) · σ (85)

which is not equal to σ0: |∆↑↑| 6= |∆↓↓| on the two split Fermi surfaces. Note, the A1-phase of

Helium in a magnetic field is non-unitary with |∆↑↑| 6= 0 and |∆↓↓| = 0.

3.2. Non-centrosymmetric superconductors

non-centrosymmetric compounds have a crystal lattice lacking an inversion center, this yields

spin-orbit coupling e.g. like Rashba spin-orbit coupling

inversion symmetry is important for spin-triplet Cooper pairs:

coupling terms for the spin-orbit coupling term represented by gk =∑

µ,j gµjµkj

2nd-order coupling to gk for suppression spin-triplet pairing

F (2)g = β

∑µ,ν

∑j,j′

|gµj |2|ηνj′ |2 − (gµjηµj)

∗(gνj′ηνj′)∝ 〈|gk × d(k)|2〉k,FS (86)

vskip 0.2 cm 1st-order coupling to gk yields parity-mixing

F (1)g = β′

∑µ,j

gµj(η∗µjηs + ηµjη

∗s) ∝ 〈gk · d(k)∗ψ(k)〉k,FS + c.c. (87)

where for simplicity we take conventional s-wave pairing for the spin singlet component (different

spin singlet states are also possible)

this suggests that d(k) and gk have the same symmetry properties and the gap matrix is given

by

∆k = (ψ(k) + d(k) · σ)iσy ⇒ ∆k∆†k = (|ψ|2 + |d|2)σ0 + ψ∗d+ ψd∗ · σ (88)

which means the mixed-parity state is non-unitary with a different gap on the two spin-split

Fermi surfaces.

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