Lecture Notes. Aspects of Symmetry in Unconventional Superconductors

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1 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: theories - broken symmetries and order parameters generalized Ginzburg-Landau 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, Gordon and Breach Science Publisher (1999). M. Sigrist, Introduction to Unconventional Superconductivity, AIP Conf. Proc. 789, 165 (005). M. Sigrist, Introduction to unconventional superconductivity in non-centrosymmetric metals, AIP Conf. Proc. 116, 55 (009). 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 = k k F note the states created by ĉ k and ĉ k, k and k are degenerate k ĉ k ĉ k 0 (1) ɛ k = ɛ k = ɛ k single-electron energy () guaranteed by time reversal symmetry (time reversal operator ˆK): k ˆK k (3) 1

2 BCS ground state: Ψ BCS = k [ ] u k + v k ĉ k ĉ k 0 (4) coherent state of electron pairs of opposite momenta b k = Ψ BCS ĉ k ĉ k Ψ BCS = u k v k (5) non-vanishing for k very close to k F : 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 Ψ 0 of free electrons (ɛ k = k /m) electron states k 1 s 1 and k s, assume k 1 + k and k 1, k > k F (restricted by Pauli exclusion principle) Schrödinger equation for interacting electrons } { ( m r1 + ) r + V (r1 r ) ψ(r 1, s 1 ; r, s ) = E ψ(r 1, s 1 ; r, s ) (6) V (r 1 r ): -particle interaction; change to center of mass and relative coordinates: R = 1 (r 1 + r ) and r = r 1 r ψ(r 1, s 1 ; r, s ) = φ s1 s (r)e iq R = φ s1 s (r) since q = k 1 + k = 0 (7) Symmetry aspect: Pauli exclusion principle antisymmetric wave function: φ s1 s (r) = φ s s 1 ( r) φ(r) = φ( r), χ s1 s = χ s s 1 even parity, spin singlet φ s1 s (r) = φ(r)χ s1 s = φ(r) = φ( r), χ s1 s = χ s s 1 odd parity, spin triplet (8) with φ(r) orbital and χ s1 s spin part of wave function 1 turn to Fourier (momentum) space: g k = d 3 r e ik r φ(r), V q = m φ(r) + V (r)φ(r) = E φ(r) (10) d 3 r e iq r V (r) (11) 1 Spin part of wave function: spin singlet S = 0 : χ s1 s = 1 ( ) χ s1 s = χ s s 1 spin triplet S = 1 : χ s1 s =, 1 ( + ), χ s1 s = χ s s 1 (9)

3 with Ω = L 3 volume k = π L (n x, n y, n z ) (n x,y,z : integers) k m g k + 1 V Ω k k g k = E g k (1) k Symmetry aspect: assume full spherical rotation symmetry, symmetry group O(3) expansion in spherical harmonics lm with k = k and ˆk = k/k; V k k = l=0 V l(k, k ) +l m= l Y lm(ˆk)y lm (ˆk ) g k = +l l=0 m= l g lm(k)y lm (ˆk) set { lm l m +l} is (l + 1)-dimensional basis of the irreducible representation of O(3) labelled by l define ξ k = ɛ k ɛ F = k m g lm (ξ k ) with density of states (13) ɛ F, E = E ɛ F and rewrite V l (k, k ) V l (ξ k, ξ k ) and g lm (k) 1 Ω k d 3 k (π) 3 dξ N(ξ) dω k 4π (15) N(ξ) = 1 δ(ξ ξ k ) (16) Ω Schrödinger equation decouples in different channels l (angular momentum): (ξ E)g lm + dξ N(ξ )V l (ξ, ξ )g lm (ξ ) = 0 (17) k for practical reasons: N(ξ) N(0) and ν l ɛ c ξ, ξ ɛ c V l (ξ, ξ ) = 0 otherwise (18) with ɛ c ɛ F ; solving the equation, searching for bound state of electrons, ie. E < 0: for ν l < 0: g lm (ξ) = N(0)ν l ξ E ɛc 0 0 dξ g lm (ξ ) = N(0)ν l ξ E I lm for0 ξ ɛ c (19) ɛc dξ ( ) I lm = I lm N(0)ν l ξ E = I N(0)ν l E ɛc lm ln E (0) E = ɛ c e /N(0)ν l (1) Spherical harmonics: orthogonality relation dωk 4π Y lm(ˆk)y l m (ˆk) = δ ll δ mm (14) 3

4 if E ɛ c (note: ɛ c functions as a somewhat arbitrary cutoff for the integral); bound state with energy E < ɛ F instability: lowest bound state for strongest attractive channel l (ν l < ν l for l l ). Symmetry aspect: bound state parity distinguished by l: ( 1) l l = 0,, 4,... l = 1, 3, 5,... even parity, spin singlet odd parity, spin triplet. Examples: electron-phonon interaction: ν 0 (ξ, ξ ) < 0 ɛ D ξ, ξ ɛ D V k k = 0 otherwise () 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: V k k = V (ξ, ξ )(ˆk ˆk ) ν > 0 ɛ c ξ, ξ ɛ c V (ξ, ξ ) = 0 otherwise (3) but ν(ˆk ˆk ) = ν[1 ˆk ˆk ] = }{{} 8πν Y 00 (ˆk)Y 00(ˆk ) 8π 3 ν =ν 0 >0 }{{} =ν 1 <0 +1 m= 1 Y 1m (ˆk)Y 1m(ˆk ) (4) no bound state in l = 0, S = 0 (repulsive) channel; bound state in (attractive) l = 1, S 1 channel: odd parity spin triplet p-wave. 1.. Generalized BCS theory we introduce a general form of a BCS Hamiltonian: H BCS = k,s ξ k c ks c ks + 1 k,k we restrict to the BCS scattering channel and the Pauli exclusion principle requires s 1,s,s 3,s 4 V k,k ;s 1 s s 3 s 4 c ks 1 c ks c k s 3 c k s 4 (5) V k,k ;s 1 s s 3 s 4 = k, s 1 ; k, s V k, s 3 ; k, s 4. (6) V k,k ;s 1 s s 3 s 4 = V k,k ;s s 1 s 3 s 4 = V k, k ;s 1 s s 4 s 3 = V k, k ;s s 1 s 4 s 3. (7) instability discussed by decoupling through generalized mean field like b k in Eq.5: c k sc k s = b k,ss + (c k sc k s b k,ss ) with b k,ss = c k sc k s (8) 4

5 inserting leads to mean field Hamiltonian H mf = 1 ξ k (c ks c ks c ks c ks ) 1 where k,s k,k k,s 1,s [ k,s1 s c ks1 c ks + k,s1 s c ks1 c ks ] + K, (9) K = 1 V k,k ;s 1 s s 3 s 4 c ks 1 c ks c k s 3 c k s ξ k. (30) s 1,s,s 3,s 4 and the self-consistency equations k,ss = k,ss or b k,ss characterize BCS state k,s 3 s 4 V k,k ;ss s 3 s 4 b k,s3 s 4 and k,ss = k,s k s 1 s V k,k;s 1 s s sb k,s s 1. (31) Bogolyubov quasiparticle spectrum: matrix formulation H mf = k C kêkc k + K, (3) with C k = c k c k c k c k to be diagonalized into where, Ê k = 1 ξ k σ 0 k and k = k ξ k σ 0 A k = a k a k a k a k H = k k, k, k, k,. (33) A kêka k + K (34) E k σ 0 0 and Ê k = (35) 0 E k σ 0 Bogolyubov transformation with unitary matrix û k v k Û k = C k = ÛkA k and Ê k = Û kêkûk (36) v k and ÛkÛ k = Û kûk = 1. with û k û k = (E k + ξ k ) σ 0 Ek (E k + ξ k ) and v k = k Ek (E k + ξ k ) (37) and the quasiparticle energy E k = ξk + k with k = 1 ( tr ) k k. (38) Quasiparticle spectrum with excitation gap (electron-hole hybridization at the Fermi energy): 5

6 E electron like hole like k electron like k F hole like Self-consistence equation (gap equation): k,s1 s = k,s 3 s 4 V k,k ;s 1 s s 3 s 4 ( ) k,s 4 s 3 Ek tanh E k k B T (39) and BCS coherent ground state: Ψ BCS = k,s,s {u k,ss + v k,ss ĉ ksĉ ks } 0 (40) gap matrix parametrization: structure of the mean field: b k,ss = φ(k)χ ss +φ( k) φ(k) = φ( k) even parity, spin singlet 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, ) ( = with even scalar gap function, ψ(k) = ψ( k), 0 ψ(k) ψ(k) 0 ) = iˆσ y ψ(k). (4) odd parity - spin triplet: ( k, k = k, k, k, ) ( dx (k) + id = y (k) d z (k) d z (k) d x (k) + id y (k) ) = i (d(k) ˆσ) ˆσ y, (43) with odd vector gap function, d(k) = d( k) Note: spin configuration d S because d x ( ) id y (( + ) + d z ( + ) (44) excitation gap: 6

7 { } we use k = 1 tr k k : k k = ψ(k) ˆσ 0 k = ψ(k) even parity - spin singlet k k = d(k) ˆσ 0 + i(d(k) d(k) ) ˆσ k = d(k) odd parity spin triplet note for unitary states : d(k) d(k) = 0. (45) New parametrization: rewrite interaction to separate even and odd parity: V k,k ;s 1 s s 3 s 4 = J 0 k,k ˆσ0 s 1 s 4 ˆσ 0 s s 3 + J k,k ˆσ s1 s 4 ˆσ s s 3, (46) leads to gap equations for even parity, ψ(k) = (J 0 k,k 3J k,k ) ψ(k ( ) ) Ek tanh }{{} E k k k B T = v s k,k (47) for odd parity, where d(k) = (J 0 k,k + J k,k ) d(k ( ) ) Ek tanh }{{} E k k k B T = v t k,k (48) v s,t k,k = l ν s,t l (ξ k, ξ k ) +l m= l Y lm (ˆk)Y lm (ˆk ) (49) note sum over l restricts to given parity ( 1) l and ν s,t l (ξ, ξ ) with the usual restrictions linearized gap equation: T T c, k 0 and E k = ξ k, case: even parity ψ(k) = k ν s k,k ψ(k ) ξ k ( ) ξk tanh k B T ɛc = N(0) ν s k,k ψ(k ) k,f S dξ 1 ( ) ξ 0 ξ tanh k B T }{{} = ln(1.14ɛ c /k B T ) eigenvalue equation (λ: dimensionless eigenvalue defining T c ) (50) λψ(k) = N(0) ν s k,k ψ(k ) k,f S with k B T c = 1.14ɛ c e 1/λ (51) analogous for odd parity λd(k) = N(0) v t k,k d(k ) k,f S (5) where k,f S angular average on Fermi surface 7

8 largest eigenvalue determines highest T c superconducting instability Symmetry operations: symmetries of the normal state: orbital rotation O(3): ĝ k, s = R g k, s with R g ; rotation matrix of element ĝ O(3) spin rotation SU(): ĝ k, s = s D(g) ss k, s ; ˆD(g) = exp[i Ŝ θ g ] with ĝ SU() time reversal ˆK: ˆK k, s = s ( iσ y) ss k, s ; ˆK = i ˆσy Ĉ with Ĉ complex conjugation ( ˆK K = {Ê, ˆK}. inversion Î: Î k, s = k, s with Î I = {Ê, Î} gauge U(1): ˆΦ k, s = e iφ/ k, s with ˆΦ U(1) Symmetry operations on gap function: Fermion exchange: ˆ k = ˆ T k orbital rotation: ĝ ˆ k = Rgk spin rotation: ĝ ˆ k = ˆD(g) ˆ k ˆD(g) time reversal: ˆK ˆ k = ˆσ y ˆ kˆσ y inversion: Î ˆ k = ˆ k gauge: ˆΦ k = e iφ ˆ 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 ĝψ(k) = ψ(r g k) ĝd(k) = d(r g k) Spin rotation ĝψ(k) = ψ(k) ĝd(k) = R g d(k) Time-reversal Kψ(k) = ψ ( k) Kd(k) = d ( k) Inversion Îψ(k) = ψ( k) Îd(k) = d( k) U(1)-gauge Φψ(k) = e iφ ψ(k) Φd(k) = e iφ d(k) Conventional pairing state: most symmetric pairing state l = 0, S = 0 ψ(k) = ψ 0 Unconventional pairing state: l 0, S = 0, 1 8

9 Examples of unconventional pairing states: cuprate high-temperature superconductors: quasi-two-dimensional ψ(k) = 0 (k x k y) with l =, S = 0 d-wave ; excitation gap with line nodes: k = 0 k x k y. 3 He B-phase : d(k) = 0 k with l = 1, S = 1 p-wave ; excitation gap without nodes (isotropic): k = 0 k. 3 He A-phase : d(k) = 0 kẑ(k x ± ik y ) p-wave (-fold degenerate); excitation gap point nodes: k = 0 k x ± ik y. gap nodes influence low-temperature thermodynamic properties, e.g. specific heat T 3/ e /k BT C(T ) T 3 T nodeless point nodes line nodes 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 1 1, ultrasound absorption,... ) (53). Generalized Ginzburg-Landau theory of superconductivity Ginzburg-Landau theory for nd -order phase transitions based on concept of spontaneous symmetry breaking key quantity: order parameter which grows continuously from zero crossing the transition temperature into the order phase.1. Conventional Ginzburg-Landau theory order parameter: gap function ˆ k or pair mean field b,ss free energy expansion in order parameter η = η(r, T ) for T T c : F [η, A; T ] = Ω d 3 r [ a(t ) η + b(t ) η 4 + K(T ) Πη + 1 ] ( A) 8π, (54) with Π = i + e c A and a(t ) a (T T c ), b(t ) b(t c ) = b > 0 and K(T ) = K(T c ) = 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 + bη η KΠ Πη = 0 e c K {η Πη + ηπ η } 1 ( A) = 0 }{{} 4π }{{} =B =j/c (55) 9

10 1. equation: uniform case for phase transition. equation: London equation with 0 T > T c 0 = a(t )η + bη η η = a(t ) (56) T T c b ( B) = 4π c n s density of superfluid electrons λ L 8e c K η B B = 1 λ B, (57) L = 3πe c K η = 4πe n s mc (58) 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() K I U(1) order parameter: η is a scalar under O(3) and SU() because of pairing in fully symmetric channel l = 0, S = 0 time reversal : U(1) gauge : Kη = η gradient and vector potential: Π invariant under SU() Φη = ηe iφ (59) orbital rotation: time reversal : inversion: U(1) gauge : ĝπ = R g Π KΠ = Π = i e c A ÎΠ = Π ΦΠ = Π + e c φ (60) scalar combinations: η η, (η η) and (Πη) (Πη) as well as ( A) broken symmetry at phase transition: U(1) with G = O(3) SU() K I.. Ginzburg-Landau theory for unconventional pairing linearized gap equation: λψ(k) = N(0) v s k,k ψ(k ) k,f S λd(k) = N(0) v t k,k d(k ) k,f S (61) generally degenerate solutions for eigenvalues λ: largest λ highest T c 10

11 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 l + 1 (dimension of representation) assume even parity spin singlet Cooper pairs with l 0: scalar gap function ψ(k) = m η m(r)ψ m (k) where {ψ m (k)} are basis function of the irreducible representation D l and η m is the order parameter generalized scalar free energy involves invariant terms of η m, ηm and Π example O(3) I D 4h : discrete rotation symmetry D 4h 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: ĝd(k) = R g d(r g k) inversion: Îd(k) = d( k) = d(k) (6) basis functions for the irreducible representations of D 4h : Γ E C 4 C C C I S 4 σ h σ v σ d basis function used names A 1g ψ = 1 s-wave A g ψ = k x k y (kx ky) g-wave B 1g ψ = kx ky d x y -wave B g ψ = k x k y d xy -wave E g ψ = {k x k z, k y k z } d-wave A 1u d = ˆxk x + ŷk y p-wave A u d = ˆxk y ŷk x p-wave B 1u d = ˆxk x ŷk y p-wave B u d = ˆxk y + ŷk x p-wave E u d = {ẑk x, ẑk y } p-wave there are 4 non-degenerate and 1 -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) = η x k x k z + η y k y k z or d(k) = η x ẑk x + η y ẑk y (63) with the scalar free energy of the order parameter η = (η x, η y ) and vector potential A: 11

12 F [η, A; T ] = d 3 r [ a(t ) η + b 1 η 4 + b {η x η y + η xη y } + b 3 η x η y +K 1 { Π x η x + Π y η y } + K { Π x η y + Π y η x } +K 3 {(Π x η x ) (Π y η y ) + c.c.} + K 4 {(Π x η y ) (Π y η x ) + c.c.} +K 5 { Π z η x + Π z η y } + 1 ] ( A) 8π (64) where a(t ) = a (T T c ), b 1 > 0, 4b 1 b + b 3 > 0 and K 1,...,5 > 0. uniform superconducting phase: 3 phases possible for symmetry reasons (table of basis functions) Phase ψ(k) d(k) broken symmetry A k z (k x ± ik y ) ẑ(k x ± ik y ) U(1), K broken time reversal symmetry B k z (k x ± k y ) ẑ(k x ± k y ) U(1), D 4h D h broken rotation symmetry C k z k x, k z k y ẑk x, ẑk y U(1), D 4h D h broken rotation symmetry b / b A C B b b + b = 0 4b + b + b = b 3 / b 1 Which phase is most stable from a microscopic view point? energy (weak coupling), Consider T = 0 condensation 1

13 E cond = H H =0 = 1 (ξ k E k ) + 1 k,s k,s 1,s k,s 1 s k,s s 1 E k ɛc = N(0) dξ (ξ ɛc ξ + k k,f S ) + k dξ ξ + k k,f S (65) N(0) k k,f S Gap structure important for stability: simple discussion assuming spherical Fermi surface: determine k k,f S Phase ψ(k) k,f S d(k) k,f S A /15 /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 = D 4h 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 = D h K (1, 1), (1, 1) B-phase η = (1, 0), (0, 1) C-phase coupling to lattice strain ɛ µν : invariant terms in free energy F ɛ η = d 3 r [{ γ 1 (ɛ xx + ɛ yy ) + γ 1ɛ zz } η + γ (ɛ xx ɛ yy )( η x η y ) + γ 3 ɛ xy (η xη y + η x η y) ] with γ i real coefficients. This free energy has to be supplemented by the elastic energy: F el = d 3 r µ 1,...,µ 4 1 C µ 1 µ 4 ɛ µ1 µ ɛ µ3 µ 4. B-phase couples to the strain ɛ xy uniaxial distorting along [110] or [1 10] C-phase couples to the strain ɛ xx ɛ yy uniaxial distorting along [100] or [0] A-phase does not coupling to anisotropic strain not nematic (66) (67) 13

14 Single domain phase through cooling under application of uniaxial stress to sample. Magnetic phase through broken time reversal symmetric as in A-phase: G = D 4h define Cooper pair angular moment: L = i ψ(k) (k k )ψ(k) k,f S (68) and analog for odd parity state with ψ(k) = η x k z k x + η y k z k y L = i ẑ { ηxη y kzk x k,f S ηyη x kzk y } k,f S i (η x η y ηyη x )ẑ (69) Categories of time reversal symmetry breaking phases (G.E. Volovik and L.P. Gor kov): Ferromagnetic phase: L 0 example: A-phase (see above) Antiferromagnetic phase: L = 0 example: s + id x y -wave state ψ(k) = η s + iη d (kx ky) (ηs iη d (kx k y))iη d k y k z k,f S L = i (ηs iη d (kx ky))iη d k z k x k,f S = 0 (70) (ηs iη d (kx ky))iη d k x k y k,f S from a group theoretical point of view: components of L are basis functions of irreducible representations of point group example D 4h : {L x, L y } E g and L z A g order parameter of A-phase: η = {η x, η y } E g : E g E g = A 1g A g B 1g B g ; the decomposition of this Kronecker product contains A g which is connected with L z ; thus the L z -component can be finite. order parameter of s + id-wave phase: η = {η s, η d } A 1g B 1g : (A 1g B 1g ) (A 1g B 1g ) = A 1g B 1g ; 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 spontaneous 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) 14

15 U(1) gauge rotation around z-axis ˆΦψ(k) = e iφ ψ(k) ˆΘψ(k) = e iθ ˆΦ ˆΘ 1 = Ê for φ = θ (71) ψ(k) thus there is a conserved charge: ˆQ = ˆLz ˆN/: changes of angular momentum and charge are coupled removing a Cooper pair (N N ) changes the angular momentum of the system by ± ẑ: 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) η x (x) = η x ( x) (η x, η y ) ( η x, η y ) (7) η y (x) = η y ( x) Ginzburg-Landau equation give simplified solution: η x (x) = η 0 tanh(x/ξ) and η y (x) = iη 0 with η 0 = a (T T c ) 4b 1 b + b 3 (73) supercurrent density: j = c F/ A, j x = 8πe [ K 1 η xπ x η x + K η yπ x η y + K 3 η xπ y η y + K 4 η yπ y η x + c.c. ] j y = 8πe [ K 1 η yπ y η y + K η xπ y η x + K 3 η yπ x η x + K 4 η xπ x η y + c.c. ] (74) j z = 8πeK 5 {η xπ z η x + η yπ z η y + c.c.}. with A x = A z = 0 we find j x (x) = 0 no current flows through the edge η x j y (x) = 16πeK 3 η y i x + which enters the London equation c 4πλ A y = 16πe η0 ξ cosh (x/ξ) }{{} = j y (0) (x) + c 4πλ A y. (75) A y x 1 λ A y = 4π c j(0) y (x) (76) where j y (0) spontaneous current parallel to the edge on a width ξ = K 1 /a(t ); there is a Meissner screening current which compensates j y (0) such that the magnetic flux induced is 15

16 located close to the surface only and penetrates over the London penetration depth λ, 1 λ = 3π e c (K 1 + K ) η 0 (77) observation of spontaneous magnetic fields by zero-field µsr, measures internal magnetic field j y B z 0 ξ λ ξ λ x 0 0 x spread in sample, enhancement of magnetism in superconducting phase: Sr RuO 4, PrOs 4 Sb 1, (U,Th)Be 13, SrPtAs, Re 6 Zr,... no direct observation of edge currents so far Possible realizations: Sr RuO 4 : d(k) = 0 ẑ(k x ± ik y ) URu Si : ψ(k) = 0 k z (k x ± ik y ) 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: ˆK k = k k, k form even-parity spin-singlet pair inversion: Î 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: the term H conserves H H + H = H + k s,s g k ĉ ks σ ss ĉ ks (79) 16

17 (a) inversion symmetry, if g k = g k (b) time reversal symmetry, if g k = g k (80) examples: case (a): g k = µ B H Zeeman field, leads to spin splitting of the Fermi surface (majority / minority spin Fermi sea) case (b): g k = αẑ 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 H c of orbital depairing is usually much lower than the limiting field H 3 p H c (T = 0) = Φ 0 πξ 0 and H p (T = 0) = H c(0) 4πχp (8) where H c (0) is the thermodynamic critical field at T = 0 and χ p is the Pauli spin susceptibility. coupling terms to the free energy expansion due to magnetic field H for order parameters ψ(k) = j η jψ i (k) and d(k) = µ,j η µj ˆµk j nd -order coupling to H for suppression (paramagnetic limit) F () H = β µ,ν { H µ H ν ηj δ µν + ηµjη } νj H ψ(k) k,f S + H d(k) k,f S (83) j with β > 0 (this terms gives correction to spin susceptibility, Yosida) 1 st -order coupling to H for the structure of state F (1) H = iβ λ,µ,ν ɛ λµν H λ η µjη νj ih d(k) d(k) k,f S (84) 3 Paramagnetic limit: comparison of spin polarization and condensation energy: H c(0) 8π = χp H H p = Hc 4πχp (81) 17

18 with ɛ λµν completely antisymmetric tensor Assuming H ẑ 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) 0. This is a non-unitary state with k k = d(k) ˆσ 0 + i(d(k) d(k) ) ˆσ (85) which is not equal to ˆσ 0 : on the two split Fermi surfaces. Note, the A 1 -phase of Helium in a magnetic field is non-unitary with 0 and = 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 g k = µ,j g µj ˆµk j nd -order coupling to g k for suppression spin-triplet pairing F () g = β µ,ν { gµj η νj (g µj η µj ) (g νj η νj ) } g k d(k) k,f S (86) j,j vskip 0. cm 1 st -order coupling to g k yields parity-mixing F (1) g = β µ,j g µj (η µjη s + η µj η s) g k d(k) ψ(k) k,f S + 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 g k have the same symmetry properties and the gap matrix is given by k = (ψ(k) + d(k) ˆσ)iˆσ y k k = ( ψ + d )ˆσ 0 + {ψ d + ψd } ˆσ (88) which means the mixed-parity state is non-unitary with a different gap on the two spin-split Fermi surfaces. 18