Vortices and vortex states of Rashba spin-orbit coupled condensates
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1 Vortices and vortex states of Rashba spin-orbit coupled condensates Predrag Nikolić George Mason University Institute for Quantum Johns Hopkins University March 5, 2014 P.N, T.Duric, Z.Tesanovic, Phys.Rev.Lett. 110, (2013) Support: NSF, DOE
2 Overview Introduction & motivation Vortex structures in Rashba S.O.C. condensates Infinite 2D systems of bosons (or many flux quanta in a trap) Excitations in uniform TR-invariant & TR-broken states Vortex lattices Stability of vortex lattices: a microscopic model Numerical mapping of the phase diagram (preliminary) Vortex unbinding T>0 transitions New universality classes? 2/14
3 Introduction & motivation Particles + static SU(2) gauge field in 2D SU(2) generators (spin projection matrices) Yang-Mills flux matrix ( magnetic for μ=0) Quantum spin-hall effect (conserved S z)... Rashba S.O.C. Dirac spectrum (S z not conserved)... 3/14
4 Motivation Are there superfluids with a vortex lattice of spin currents? by Rashba S.O.C. external gauge field with flux Interesting because quantum vortex lattice melting in 2D: preempts any 2nd order transition (by quantum Lindemann crit.) can yield a topological vortex liquid with fractional excitations Rashba S.O.C. naturally non-abelian P.N, Phys.Rev.B 87, (2013) Levi-Civita tensor anti-commutators flux matrix P.N, J.Phys: Cond.Mat. 25, (2013). What if a 2D S.O.C. superfluid is uniform? unconventional continuous transitions (not Kosterlitz-Thouless) 4/14
5 Type-I condensates Spin current without charge current TR-invariant Spin current densities & the Hamiltonian Rashba S.O.C. 5/14
6 Type-I vortices Conservation laws: no sources for, and source/drain vortex is Neutrality: vortex quadruplets vortices carry two charges U(1) θ (anti)vortex is bound to α vector (anti)vortex 6/14
7 Type-I vortex structures Non-neutral clusters Domain wall Vortex lattice unit cell is a quadruplet square geometry α changes by nπ between singularities rigid (meta)stable structure one (n =1) vortex per SU(2) flux quantum 7/14
8 Type-II condensates Spin current by charge current + spin texture Current densities & the Hamiltonian charge current + spin texture TR broken Rashba S.O.C. 8/14
9 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) 9/14
10 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 quantum well) 10/14
11 Lattice model numerics Main competitors for the ground state: Similar to: pair density wave (PDW) states W.S.Cole, S.Zhang, A.Paramekanti, N. Trivedi, usually win, but found only when looked for! PRL 109, (2012) SU(2) vortex lattices (type-i structures) always found in unconstrained minimization, sometimes win? VL PDW 11/14
12 Competing orders Competition for 1% of free energy (F) PDW map (by ordering wavevectors): Vortex lattice wins? (tight S=1 Cooper pairs) commensurate pair density waves vortex lattices incommensurate plane waves 12/14
13 Vortex unbinding transitions at T>0 Vortex quadruplets are ubiquitous non-kt dipole unbinding: Kosterlitz-Thouless new universality classes for quadruplets Coulomb gas renormalization group KT? OBSOLETE C0 confined quadrupoles (low-t) D1 dipoles, confined Q, deconfined φ D2 dipoles, confined φ, deconfined Q D0 deconfined (vortex plasma, high-t) KT KT: non-kt: 13/14
14 Conclusions Elementary vortex excitations are quadruplets vortices carry two kinds of charges Unconventional universality class for thermal unbinding of vortex quadruplets There are (meta)stable SU(2) vortex lattices at least in tight-binding lattice models non-abelian fractional Tis by quantum melting of a vortex lattice 14/14
15 Type-I vortices (S=1) Conservation laws: Upon coarse-graining (rapid α oscillations): no sources for no sources for, and
16 Vortex unbinding transitions at T>0 Coulomb gas renormalization group y± vortex fugacity K,K' superfluid stiffness F, F' integrals over relative positions of 4 vortices in a quadruplet (UV cut-off: vortex core size) IR-divergent as power-laws at partial unbinding transitions
17 Type-I condensates Uniform superfluid order parameter Spin current densities & the Hamiltonian Rashba S.O.C.
18 Physical realizations? Perturb a QCP by the spin-orbit effect New phases emerge due to relevant scales TI quantum well, or cold atoms Finite-momentum triplet condensation is a Zeeman effect Vortex lattice of spin supercurrents shaped by the spin-orbit SU(2) flux Quantum vortex lattice melting fractional TI? Strained graphene P.N, T.Duric, Z.Tešanović, arxiv: P.N, Z.Tešanović, arxiv: & Non-Abelian gauge fields by Kekule deformations?
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