Topological protection, disorder, and interactions: Life and death at the surface of a topological superconductor

Topological protection, disorder, and interactions: Life and death at the surface of a topological superconductor Matthew S. Foster Rice University March 14 th, 2014 Collaborators: Emil Yuzbashyan (Rutgers), Hong-Yi Xie and Yang-Zhi Chou PRL 109, 246801 (2012); arxiv:1402.6797; In preparation

3D Topological Superconductor Volovik 88 Read and Green 00 Schnyder, Ryu, Furusaki, Ludwig 08 Kitaev 09 Quasiparticle bandstructure is fully-gapped, topologically twisted CI: spin SU(2) symmetry (spin singlet pairing) AIII: spin U(1) (e.g. spin triplet p-wave) DIII: no spin symmetry (e.g. 3 He B, Cu x Bi 2 Se 3? YPtBi?) Protected by time-reversal symmetry (in 3D)

Background: Overview Surface states of 3D bulk topological superconductors: Gapless Dirac ( Majorana ) quasiparticle surface bands With disorder: Quenched 2+1-D QCD (random non-abelian vector potentials) Idea: Test the limits of Topological protection : Disorder & interactions Disorder induces multifractal wavefunction fluctuations (critical delocalization) Multifractality, Chalker scaling: Disorder can enhance interactions Schnyder, Ryu, Furusaki, Ludwig 08; Kitaev 09 Feigelman, Ioffe, Kravtsov, Yuzbashyan 07 Feigelman, Ioffe, Kravtsov, Cuevas 10

Background: Surface states of 3D bulk topological superconductors: Gapless Dirac ( Majorana ) quasiparticle surface bands With disorder: Quenched 2+1-D QCD (random non-abelian vector potentials) Idea: Test the limits of Topological protection : Disorder & interactions Disorder induces multifractal wavefunction fluctuations (critical delocalization) Results and conclusions: Overview 1) Disorder-amplified interactions can sabotage topological protection (Class CI), inducing surface quantum Hall order (Class C). 2) Stable gapless surface states: AIII, DIII. Interacting fixed point for AIII. 3) Suppression of Altshuler-Aronov to surface spin, thermal conductance when TR is unbroken (AIII, DIII). Transport topological invariant? Foster and Yuzbashyan 12 Foster, Xie, Chou, Xie, Chou, Foster In preparation

Topological superconductor surface states Class CI Topologically-protected, gapless surface state (Bogoliubov) quasiparticles ν = 2k valleys, k = (1,2,3, ) Low energy surface Andreev state Hamiltonian: Schnyder, Ryu, Furusaki, Ludwig 08 Bernard and LeClair 02 Anomalous chiral symmetry (= physical time-reversal):

Topological superconductor surface states Fermion bilinears and time-reversal symmetry 1) Time-reversal even: Currents Spin SU(2) Valley Sp(2k) 2) Time-reversal odd: Densities Spin SU(2) Valley Sp(2k) 3) Time-reversal odd: Masses Invariant Valley (010..0) Physical interpretation of the mass? Class C Spin QHE

Dirty topological superconductor surface states Class CI Add TRI disorder: Valley Sp(2k) vector potentials only! Sources of : Impurities, vacancies External electric fields Edge, corner, dislocation potentials disorder Quenched 2+1-D QCD: Dirac fermions in a sea of frozen gauge fluctuations

Disorder & CFT ν = 2k valleys, k = (1,2,3, ), n replicas Clean CI surface state Hamiltonian: spin valley replica Invariant under combined spin x valley x replica SO(4nk) rotations: Conformal Field Theory: SO(4nk) 1 (free fermions) Class CI: Disorder couples only to valley Sp(2k) n Kac-Moody currents:

Disorder & CFT: Multifractal spectra CFT solution via conformal embeddings Reviewed in (e.g.) J. Fuchs, Affine Lie Algebras and Quantum Groups Winding number ν = # valleys Fractionalization : Level n valley KM sector localizes Nersesyan, Tsvelik, Wenger 94 Multifractal LDoS moments: Level k primary fields Exact Results Foster, Yuzbashyan 12 Mudry, Chamon, Wen 96 Caux, Kogan, Tsvelik 96 Foster (unpublished)

Disorder & CFT: Multifractal spectra Class AIII: TRI TSC with spin U(1) symmetry (e.g., p-wave spin triplet) Foster, Ludwig 08 Schnyder, Ryu, Furusaki, Ludwig 08 Disorder: Valley currents (non-abelian) Spin U(1) current (abelian) Hamiltonian: Spin current disorder strength: Exact Results Foster, Yuzbashyan 12 Mudry, Chamon, Wen 96 Caux, Kogan, Tsvelik 96 Foster (unpublished)

Topological protection? Disorder and interactions Extended, multifractal surface states: No Anderson localization = topological protection! BUT Add generic, weak interparticle interactions, consistent with bulk symmetries [time-reversal, spin SU(2) for CI]

Physical picture: Chalker scaling, multifractality, and interactions Chalker scaling: Overlapping peaks and valleys in multifractal eigenstates with nearby energies Chalker, Daniell 88 Chalker 90 Cuevas, Kravtsov 07 Anderson insulator: No overlap for nearby energies Feigelman, Ioffe, Kravtsov, Yuzbashyan 07 Feigelman, Ioffe, Kravtsov, Cuevas 10

Method 1: Scaling interactions with disorder Clean limit: DoS determines relevance of short-ranged interactions Clean Dirac: interactions irrelevant!

Method 1: Scaling interactions with disorder Clean limit: DoS determines relevance of short-ranged interactions Clean Dirac: interactions irrelevant! Dirty case: scaling dimension of disorder-averaged LDoS scaling dimension of disorder-averaged interaction Constraint: multifractal scaling dimension of second LDoS moment Compute exactly via CFT IQHP: Lee and Wang 96

Method 1: Scaling interactions with disorder Clean limit: DoS determines relevance of short-ranged interactions Clean Dirac: interactions irrelevant! Dirty case: scaling dimension of disorder-averaged LDoS scaling dimension of disorder-averaged interaction Maximally relevant interaction: Convexity property for a multifractal extended surface state: (independent dimensions!) Duplantier and Ludwig 1991 Wavefunction multifractality can amplify short-ranged interactions!

Method 2: Many valleys, WZW-FNLsM 1) Embed, fractionalize: SO(4nk) 1 Sp(2n) k C.f. Altland, Simons, Ziirnbauer 02 2) Bosonize: Sp(2n) k = principal chiral NLsM plus WZW term 3) 2+1-D version: Field becomes matrix in replicas and frequencies Finkelstein 83 4) Incorporate interactions (local in time and replicas) Under perturbative control for large winding numbers

Class CI (Spin SU(2) symmetry): Disorder and interactions Hamiltonian Interaction channels: Cooper pairing of surface quasiparticles (time-reversal invariance) Spin exchange (spin is conserved = hydrodynamic mode) Spin current-current Order parameters break time-reversal: Spin polarization Imaginary s-wave pairing mass Class C Spin QHE

Class CI: Disorder and interactions Hamiltonian CFT: Relevant! (Multifractal enhancement) Irrelevant Irrelevant Order parameters break time-reversal: Spin polarization Imaginary s-wave pairing mass Class C Spin QHE

Class CI: Disorder and interactions Foster, Yuzbashyan 12 Due to disorder, interactions are always relevant; flow to strong coupling Expect time-reversal breaks spontaneously. Interactions plus disorder can sabotage surface topological protection.

Class AIII (Spin U(1) symmetry): Disorder and interactions Hamiltonian Interaction channels: Cooper pairing of surface quasiparticles (time-reversal invariance) z-spin exchange (z-spin is conserved = hydrodynamic mode) z-spin current-current Order parameters break time-reversal: Spin polarization Imaginary s-wave pairing mass Class C Spin QHE

Class AIII: Disorder and interactions Hamiltonian CFT: Window of stability:

Class AIII: WZW-FNLsM (many valleys) Parameters Dimensionless inverse spin conductance Spin current disorder strength Cooper pairing of surface quasiparticles (time-reversal invariance) Spin exchange (spin is conserved = hydrodynamic mode) For (universal fixed point value) Landauer spin conductance is universal without interactions. True even with interactions? Yes, to one-loop. Higher loops? Tsvelik 95 Ostrovsky, Gornyi, Mirlin 06 Topological transport invariant? Xie, Chou, Foster In preparation

Class AIII: WZW-FNLsM (many valleys) CFT: Window of Stability Simplified interaction plane flow (λ = 1/k): Retain only BCS non-linearity (Anderson s theorem) Qualitatively the same as full WZW-FNLsM results for λ > 1/k 2 In both cases: New interaction-stabilized fixed point Foster, Xie, Chou In preparation

Class AIII: Disorder and interactions Foster, Xie, Chou In preparation CFT: For weak disorder, interactions irrelevant. Stable surface. CFT: Stronger disorder, interactions relevant. WZW-FNLsM: critical interacting fixed point. Stable surface is possible. Weakly-coupled (perturbatively accessible) for finite window of disorder

Class DIII (No spin symmetry): Disorder and interactions Hamiltonian Foster, Xie, Chou In preparation Interaction channel: Cooper pairing of surface quasiparticles (time-reversal invariance) CFT: Always Irrelevant! No multifractal enhancement Knowing behavior of average density of states is not enough! Class CI: Class DIII:

Background: Surface states of 3D bulk topological superconductors: Gapless Dirac ( Majorana ) quasiparticle surface bands With disorder: Quenched 2+1-D QCD (random non-abelian vector potentials) Idea: Test the limits of Topological protection : Disorder & interactions Disorder induces multifractal wavefunction fluctuations (critical delocalization) Results and conclusions: Review 1) Disorder-amplified interactions can sabotage topological protection (Class CI), inducing surface quantum Hall order (Class C). 2) Stable gapless surface states: AIII, DIII. Interacting fixed point for AIII. 3) Suppression of Altshuler-Aronov to surface spin, thermal conductance when TR unbroken (AIII, DIII). Transport topological invariant? Foster and Yuzbashyan 12 Foster, Xie, Chou, Xie, Chou, Foster In preparation

Numerical tests: Critical DOS, multifractal scaling Minimal case: 2 valley Dirac (Classes CI and AIII) CFT predictions: Global density of states Multifractal spectrum Numerical scheme: Momentum-space disordered Dirac fermion (avoids fermion doubling) Bardarson, Tworzydlo, Brouwer, Beenakker 07 Nomura, Koshino, Ryu 07 Y. Z. Chou, Foster arxiv:1402.6797

Numerical tests: Critical DOS, multifractal scaling Numerics suggests delocalization and conformal invariance are both topologically protected Non-trivial statement for classes AIII, DIII: Runaway to Gade regime? Guruswamy, LeClair, Ludwig 00 Runaway to diffusive metal? Senthil and Fisher 00 Assume no interactions. Why should we pin λ = 1/k? Universal Dirac Landauer conductance unchanged by disorder. Interpret λ as inverse (spin) conductance (as in localization theory) Tsvelik 95; Ostrovsky, Gornyi, Mirlin 06