Vortex States in a Non-Abelian Magnetic Field Predrag Nikolić George Mason University Institute for Quantum Matter @ Johns Hopkins University SESAPS November 10, 2016
Acknowledgments Collin Broholm IQM @ Johns Hopkins Wesley T. Fuhrman Johns Hopkins US Department of Energy National Science Foundation Zlatko Tešanović IQM @ Johns Hopkins 2/17
Topological Train Tracks Z2 topology in 1D: Odd # of defects in a loop (measurable) Defects can be created or removed in pairs Invariant: the presence of a charm element 3/17
Superconductivity in Correlated TIs Electrons on the TI boundary + interactions Pairing in a SU(2) magnetic field Spin-triplets are enhanced by S.O.C. (Rashba: momentum-dependent Zeeman effect) Vortex lattices of spin supercurrents Incompressible quantum vortex liquids (non-abelian fractional TIs) PN, T.Duric, Z.Tesanovic PRL 110, 176804 (2013) W. T. Fuhrman, J. Leiner, P. N, et.al, PRL 114, 036401 (2015) 4/17
Rashba S.O.C. on a TI's Boundary Particles + static SU(2) gauge field SU(2) generators (spin projection matrices) Yang-Mills flux matrix ( magnetic for μ=0) D. Hsieh, et.al, PRL 103, 146401 (2009) Y. Zhang, et.al, Nature Phys. 6, 584 (2010) Rashba S.O.C. Dirac spectrum... Cyclotron: SU(2) flux: 5/17
Triplet Pairing in Correlated TIs TI ultra-thin film Phonon proximity effect Kondo TI surface (SmB6) 2D heavy fermions exciton Cooper 6/17
Pairing Channels The minimal TR-invariant topological band-insulator 2 spin states ( ) X 2 surface states ( short-range interactions ) Decouple all interactions 6 Hubbard-Stratonovich fields: Spin-singlet: Spin-triplet:, 7/17
Effective Theory The most generic Cooper-pair action Integrate out all fermion fields near the Cooper Mott transition Spin-triplets feel spin-orbit coupling Constructed from the SU(2) gauge symmetry (idealized) There is SU(2) magnetic flux: 8/17
Triplet Condensates & Vortices Rashba S.O.C.: momentum-dependent Zeeman A large-momentum mode has low energy PN, PRA 90, 023623 (2014) Landau-Ginzburg picture Helical spin-current T1 & T2 condensates T1 phase can be TR-invariant T1 breaks rotation and translation symmetries T1 has metastable vortex clusters & lattices 9/17
Type-I Condensates TR-invariant spin current Depleted (α=π/2) inside a θ-vortex Spin current densities Rashba S.O.C. 10/17
Type-I Vortices Conservation laws: no sources for, and source/drain vortex is Neutrality: vortex quadruplets vortices carry two charges U(1) q (anti)vortex is bound to Ñ a vector (anti)vortex 11/17
Type-I Vortex Structures Non-neutral clusters Domain wall Vortex lattice unit cell is a quadruplet square geometry a changes by np between singularities rigid (meta)stable structure one (n =1) vortex per SU(2) flux quantum 12/17
Stability of Vortex States Continuum: vortex cores are costly uniform states Do vortex lattices ever win? good candidates: metastable type-i structures tight-binding lattice systems: vortex cores are cheap (if small) entropy favors vortices (order by disorder, or vortex liquids) Microscopic lattice model triplet pairing of fermions with Rashba S.O.C. on a square lattice bilayer (triplet superconductivity in a TI film) 13/17
The Hamiltonian of TI Surfaces 2D tight-binding model SU(2) gauge field on lattice links Singularity Dirac point in E(k) at some energy Dirac points can be gapped only in pairs (TR symmetry) SmB6 (100 surface): M is gapped (bulk TI) SU(2) charge: g = τ z = ±1 helicity of spin-momentum locking Continuum limit Rashba spin-orbit coupling Yang-Mills (magnetic) flux 14/17
Competing Orders PN, PRB 94, 064523 (2016) 15/17
Vortex Lattice Melting Quantum fluctuations Positional fluctuations of SU(2) vortices Grow when the condensate is weakened (by tuning the gate voltage) Eventually, 1st order phase transition (preempts the 2nd order one) PN, T.Duric, Z.Tesanovic, PRL 110, 176804 (2013) Vortex liquid Particles per flux quantum ~ 1 Fractional TI PN, PRB 87, 245120 (2013) Cyclotron: SU(2) flux: Fractionalization by vortex lattice melting? Numerical evidence: N. Cooper, etc. U(1) bosonic quantum Hall Field theory indications: a generalization of Chern-Simons 16/17
Conclusions Electrons on the TI boundary + interactions Pairing in a SU(2) magnetic field Vortex lattices of spin supercurrents Quantum vortex liquids: non-abelian fractional TIs 17/17
Surface Instabilities Weak-coupling orders exciton Cooper Cooper: Exciton: Stronger coupling at The same triplets as in ultra-thin TI films TR-invariant: Vortex lattice TR-broken: 18/17
Type-II Condensates Spin current by charge current + spin texture Current densities & the Hamiltonian charge current + spin texture TR broken Rashba S.O.C. 19/17
Type-II Vortices Conservation laws: no sources for, and no sources for Vortex quadruplet not classically (meta)stable charge singularities bound to spin vortices (not antivortices) 20/17