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

Zlatko Tešanović IQM @ Johns Hopkins

Vortex States in a Non-Abelian 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 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 (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) proximity effect 2D heavy fermions

Cooper

Vortex States in a Non-Abelian Magnetic Field 6/17 Pairing Channels

• The minimal TR-invariant topological band- – 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 (preempts the 2nd order one) PN, T.Duric, Z.Tesanovic, PRL 110, 176804 (2013) • Vortex 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