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: |Ψ 0 i = |k|≤k F Y k ˆ c † k↑ ˆ c † -k↓ |0i (1) note the states created by ˆ c † k↑ and ˆ c † -k↓ , |k ↑i and |k ↑i are degenerate k↑ = -k↓ = k single-electron energy (2) guaranteed by time reversal symmetry (time reversal operator ˆ K): |k ↑i ˆ K ←→ |- k ↓i (3) 1
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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
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