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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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