Vortices and vortex states of Rashba Vortices and vortex states of Rashba spin-orbit coupled condensatesspin-orbit coupled condensates

Predrag NikolićPredrag Nikolić

George Mason UniversityGeorge Mason University

Institute for Quantum Matter @ Johns Hopkins UniversityInstitute for Quantum Matter @ Johns Hopkins University

March 5, 2014March 5, 2014

P.N, T.Duric, Z.Tesanovic, Phys.Rev.Lett. 110, 176804 (2013)P.N, T.Duric, Z.Tesanovic, Phys.Rev.Lett. 110, 176804 (2013)

Support: NSF, DOESupport: NSF, DOE

• 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?

Overview

2/14Vortices and vortex states of Rashba spin-orbit coupled condensates

Introduction & motivation

Vortices and vortex states of Rashba spin-orbit coupled condensates 3/14

• Particles + static SU(2) gauge field in 2D

– Quantum spin-Hall effect (conserved S z)

.....

.....

– Rashba S.O.C. ⇒ Dirac spectrum (S z not conserved)

Yang-Mills flux matrix(“magnetic” for μ=0)

SU(2) generators(spin projection matrices)

Motivation

4/14Vortices and vortex states of Rashba spin-orbit coupled condensates

• 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

• What if a 2D S.O.C. superfluid is uniform? – unconventional continuous transitions (not Kosterlitz-Thouless)

P.N, Phys.Rev.B87, 245120 (2013)

P.N, J.Phys: Cond.Mat.25, 025602 (2013).flux matrixLevi-Civita tensor anti-commutators

Type-I condensates

5/14Vortices and vortex states of Rashba spin-orbit coupled condensates

• Spin current without charge current

• Spin current densities & the Hamiltonian

Rashba S.O.C. ⇒

TR-invariant

Type-I vortices

6/14Vortices and vortex states of Rashba spin-orbit coupled condensates

• Conservation laws:

– no sources for , and

– vortex is source/drain

• Neutrality: vortex quadruplets – vortices carry two “charges”

– U(1) θ (anti)vortex is bound to ∇α vector (anti)vortex

Type-I vortex structures

7/14Vortices and vortex states of Rashba spin-orbit coupled condensates

• 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”

Type-II condensates

8/14Vortices and vortex states of Rashba spin-orbit coupled condensates

• Spin current by charge current + spin texture

• Current densities & the Hamiltonian

Rashba S.O.C. ⇒

– charge current + spin texture– TR broken

Type-II vortices

9/14Vortices and vortex states of Rashba spin-orbit coupled condensates

• Conservation laws:

• Vortex quadruplet – not classically (meta)stable – charge singularities bound to spin vortices (not antivortices)

– no sources for , and

– no sources for

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)

10/14Vortices and vortex states of Rashba spin-orbit coupled condensates

VL

PDW

Lattice model numerics

Vortices and vortex states of Rashba spin-orbit coupled condensates

• Main competitors for the ground state: – pair density wave (PDW) states usually win, but found only when looked for! – SU(2) vortex lattices (type-I structures) always found in unconstrained minimization, sometimes win?

11/14

Similar to:

W.S.Cole, S.Zhang, A.Paramekanti, N. Trivedi,

PRL 109, 085302 (2012)

Vortex lattice wins?(tight S=1 Cooper pairs)

Competing orders

Vortices and vortex states of Rashba spin-orbit coupled condensates

• Competition for 1% of free energy (F)

– PDW map (by ordering wavevectors):

commensurate pair density waves

vortex lattices

incommensurate plane waves

12/14

Vortex unbinding transitions at T>0

Vortices and vortex states of Rashba spin-orbit coupled condensates

• Vortex quadruplets are ubiquitous – dipole unbinding: Kosterlitz-Thouless – new universality classes for quadruplets

• Coulomb gas renormalization group

non-KT

KT ?

KT

C0 – confined quadrupoles (low-T)

D1 – dipoles, confined Q, deconfined φ

D2 – dipoles, confined φ, deconfined Q

D0 – deconfined (vortex plasma, high-T)

KT:

non-KT:

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OBSOLETE

Conclusions

• 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

• Unconventional universality class for thermal unbinding of vortex quadruplets

• Elementary vortex excitations are quadruplets – vortices carry two kinds of charges

Vortices and vortex states of Rashba spin-orbit coupled condensates 14/14

Type-I vortices (S=1)

Vortices and vortex states of Rashba spin-orbit coupled condensates

• Conservation laws:

– no sources for , and

– no sources for

• Upon coarse-graining (rapid α oscillations):

Vortex unbinding transitions at T>0

Vortices and vortex states of Rashba spin-orbit coupled condensates

• Coulomb gas renormalization group

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

y±

– vortex fugacity

K,K' – superfluid stiffness

Type-I condensates

Vortices and vortex states of Rashba spin-orbit coupled condensates

• Uniform superfluid order parameter

• Spin current densities & the Hamiltonian

Rashba S.O.C. ⇒

• 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 – Non-Abelian gauge fields by Kekule deformations?

Physical realizations?

Vortices and vortex states of Rashba spin-orbit coupled condensates

P.N, T.Duric, Z.Tešanović,arXiv:1109.0017

P.N, Z.Tešanović,arXiv:1208.0608 & 1210.7821