VortexVortex StatesStates inin aa Non-AbelianNon-Abelian MagneticMagnetic FieldField
PredragPredrag NikoliNikolićć
GeorgeGeorge MasonMason UniversityUniversity InstituteInstitute forfor QuantumQuantum MatterMatter @@ JohnsJohns HopkinsHopkins UniversityUniversity
SESAPS November 10, 2016 Acknowledgments
US Department National Science Collin Broholm Wesley T. Fuhrman of Energy Foundation IQM @ Johns Hopkins Johns Hopkins
Zlatko Tešanović IQM @ Johns Hopkins
Vortex States in a Non-Abelian Magnetic Field 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
Vortex States in a Non-Abelian Magnetic Field 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 W. T. Fuhrman, J. Leiner, P. N, PRL 110, 176804 (2013) et.al, PRL 114, 036401 (2015)
Vortex States in a Non-Abelian Magnetic Field 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:
Vortex States in a Non-Abelian Magnetic Field 5/17 Triplet Pairing in Correlated TIs
TI ultra-thin film Kondo TI surface (SmB6) Phonon proximity effect 2D heavy fermions
exciton Cooper
Vortex States in a Non-Abelian Magnetic Field 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:
Vortex States in a Non-Abelian Magnetic Field 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:
Vortex States in a Non-Abelian Magnetic Field 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
Vortex States in a Non-Abelian Magnetic Field 9/17 Type-I Condensates
• TR-invariant spin “current”
Depleted (α=π/2) inside a θ-vortex • Spin current densities
Rashba S.O.C.
Vortex States in a Non-Abelian Magnetic Field 10/17 Type-I Vortices
• Conservation laws:
– no sources for , and – vortex is source/drain
• Neutrality: vortex quadruplets – vortices carry two “charges” – U(1) q (anti)vortex is bound to Ña vector (anti)vortex
Vortex States in a Non-Abelian Magnetic Field 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”
Vortex States in a Non-Abelian Magnetic Field 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)
Vortex States in a Non-Abelian Magnetic Field 13/17 The Hamiltonian of TI Surfaces
2D tight-binding model SU(2) gauge field on lattice links
– Singularity – SU(2) charge: Dirac point in E(k) at some energy g = τ z = ±1 helicity of spin-momentum – Dirac points can be gapped locking only in pairs (TR symmetry)
– SmB6 (100 surface): M is gapped (bulk TI)
Continuum limit Rashba spin-orbit coupling
Yang-Mills (magnetic) flux
Vortex States in a Non-Abelian Magnetic Field 14/17 Competing Orders
PN, PRB 94, 064523 (2016)
Vortex States in a Non-Abelian Magnetic Field 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 PN, PRB 87, 245120 (2013) – Particles per flux quantum ~ 1 Cyclotron: – Fractional TI 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
Vortex States in a Non-Abelian Magnetic Field 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
Vortex States in a Non-Abelian Magnetic Field 17/17 Surface Instabilities
• Weak-coupling orders exciton Cooper – Cooper: – Exciton:
• Stronger coupling at – The same triplets as in ultra-thin TI films
Vortex TR-invariant: lattice
TR-broken:
Vortex States in a Non-Abelian Magnetic Field 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.
Vortex States in a Non-Abelian Magnetic Field 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)
Vortex States in a Non-Abelian Magnetic Field 20/17