Quantum Hall states in two-component Bose gases in synthetic gauge fields

1

Shunsuke Furukawa

Department of Physics, Univ. of Tokyo

YITP workshop "Novel Quantum States in Condensed Matter 2014"Dec. 1, 2014, Kyoto University

Collaborator: Masahito Ueda

Combination of spinor physics and quantum Hall physics

Phys. Rev. A 86, 031604(R) (2012)Phys. Rev. Lett. 111, 090401 (2013)Phys. Rev. A 90, 033602 (2014)

/33Outline 2

Background

Quantum Hall states in two-component Bose gases

- Quantum Hall effects in 2D electron systems in high magnetic fields

- Quantum Hall physics in ultracold atomic gases(Attempts for realization; theory for single-component Bose gases)

+B

-B

+B

•Non-Abelian spin-singlet state

•Bosonic integer quantum Hall stateprotected by a U(1) symmetry

•Stability of fractional quantum spin Hall states

/332D electron systems in magnetic fields 3

From D. Yoshioka, Quantum Hall Effect

Longitudinal resistivity Hall resistivity

According to classical Drude theory,

However, remarkable things happen at low T and high B!

/33Quantum Hall effects (QHE) 4

(insulator)

(From Willet et al., PRL, 1987)

Integer QHEvon Klitzing, Dorda,Pepper, PRL, 1980

Tsui, Stormer,Gassard, PRL,1982

Fractional QHE

'85

'98

•

•

cf. classical theory:

/33Quantum Hall effect (QHE) 5

(From Willet et al., PRL, 1987)

Integer QHE

von Klitzing, Dorda,Pepper, PRL, 1980

Tsui, Stormer,Gassard, PRL,1982

Possibility of a non-Abelian QH state

Willet et al., PRL, 1987

FractionalQHE

•

•

•

/33What occurs at QH plateaus? 6

A certain stable structure associated with an excitation gap

Integer QHE: Single-particle picture

Fractional QHE: Many-body effect... Landau levels

Origin of the excitation gap?

Massive degeneracy

What is the "stable structure"?

Twist in the many-body wave function, which cannot be characterized by a local order parameter

"Topological order" (Xiao-Gang Wen, 1989)

Quasiparticle fractionalizationFractional (and sometimes non-Abelian) statistics Gapless edge states

Surface of 2+1D topological field theory 1+1D conformal field theory (Witten,1989)

/33Possibility of QH states in cold atomic systems 7

•Different statistics (boson,fermion), hyper-fine spin states (1/2,1,2,..)

A variety of QH states

•Charge neutral Orbital degrees of freedom do not couple directly with a magnetic field

Synthetic gauge fields

•How to create synthetic gauge fields?

Classical rotation

Quantum-mechanical Berry phase

/33Mechanical rotation of atomic gases

Coriolis force (in the rotating frame of reference) Lorentz force

Effective magnetic field:

One-component Two-component

Schweikhard et al., PRL, 2004 (JIRA group)

8

/33Laser-induced gauge fields in cold atomic gases 9

Y.-J. Lin et al., Nature (2009)(Spielman group)

Berry connection

Adiabatic approximation

Spatially dependent eigenstates

Image after TOF

"dressed state"

Scalar Bose gas in a nearly uniform magnetic field

/33Scalar Bose gases in synthetic magnetic fields 10

BEC Vortex lattice IncompressibleQH statesB* B* B*

B*

# of particles

# of vortices= # of flux quanta

Filling factor

Review: Cooper, Adv. Phys. 57, 539 (2008)

NB. For bosons, lowest Landau level approximation is applicable beyond

/33Quantum Hall wave functions 11

Symmetric polynomial

•Single-particle states:

Many-body wave function:

•Laughlin stateExact zero-energy ground state of Hint

Laughlin, PRL, 1983Wilkin, Gunn, & Smith, PRL, 1998

magnetic length

•Contact interaction:

Interaction range << Inter-atomic distance

(in the lowest Landau level)

/33Read-Rezayi states @ 12

Read&Rezayi,PRB, 1999

k=1: Laughlin state

k=2: Moore-Read state

Exact zero-energy ground state for (k+1)-body interaction

Good overlaps with the ground states of 2-body int. for k=2,3,4(?),...

Exact diag. study: Cooper, Wilkin,& Gunn, PRL, 2001

Nucl. Phys. B, 1991

Divide N particles into k groups. Form a Laughlin state in each group. Symmetrize over different ways of grouping.

...

k

/33Quantum Hall states in scalar Bose gases (theory) 13

Numberof particles

Filling factor:

Number of flux quanta= Number of vortices

Non-Abelian anyonsReview: N. Cooper, Adv. Phys. 57, 539 (2008)

B*

/33Outline 14

Background

Quantum Hall states in two-component Bose gases

- Quantum Hall effects in 2D electron systems in high magnetic fields

- Quantum Hall physics in ultracold atomic gases(Attempts for realization; theory of single-component Bose gases)

+B

-B

+B

•Non-Abelian spin-singlet state

•Bosonic integer quantum Hall stateprotected by a U(1) symmetry

•Stability of fractional quantum spin Hall states

/33Two-component Bose gases in a synthetic magnetic field 15

Intra-component Inter-component

: Two hyper-fine spin states

87Rb

Moderate B Unique vortex structures Theory: Mueller & T.-L. Ho, PRL (2002); Kasamatsu, Tsubota, &Ueda, PRL (2003)Experiment: Schweikhard et al., PRL (2004)

Strong B Ideal setting for studying the roles of "spins" in quantum Hall physics

+B

Exact diag. using LLL states on a sphere and a torus

We mainly focus on the SU(2)-symmetric case:

/33"Charge gap" 16

Spikes at

Non-Abelianspin-singlet state

Halperin(221) state(exact GS)

Bosonic integer quantum Hall state

Torus calculation

S.F. and M. Ueda, Phys. Rev. A 86, 031604 (R) (2012)

(Indicator of incompressibility)

/332-component extension of Read-Rezayi series 17

Halperin (221) state Helv. Phys. Acta, 1984

Exact ground state for

Non-Abelian spin-singlet (NASS) states

• Exact ground states for (k+1)-body interactions

Ardonne & Schoutens, PRL, 1999

"SU(3)k states"

Group N particlesinto k groups

• Do they well describe the ground states of 2-body interactions as well?

...

k

/33Evidence of a NASS state @ 18

• Check topological degeneracy: On a torus, SU(3)2 state is known to show 6-fold GS degeneracy.

• Since every state is 3-fold degenerate (special property of a torus), it is sufficient to check 2-fold degeneracy at K=0.

Energy spectrum of Hint Squared overlap

for N= 8

for N=12

(k=2 case)

S.F. and M. Ueda, Phys. Rev. A 86, 031604 (R) (2012)

/33Dependence on torus aspect ratio 19

The spectral structure becomes more robust for larger N. Supports the stability of SU(3)2 state in the thermodynamic limit.

S.F. and M. Ueda, Phys. Rev. A 86, 031604 (R) (2012)

/33Bosonic Integer quantum Hall state @ 20

Senthil and Levin, Phys. Rev. Lett. 110, 046801 (2013)

- Condensation of composite bosonsQuantum Hall (QH) state

- Flux-attachment between different components

- Interaction-induced integer QH stateas opposed to the fermionic case

Flux

Compositebosons

Effective Chern-Simons theory:

K-matrix describes the coupling between particles and fluxes.

/3321

Example of a topological state protected by particle number conservation

Average densityDensity difference

Bosonic Integer quantum Hall state @

Diagonalization of K matrix -> Information of edge modes

ChargeSpin Eigenvalue +1:Eigenvalue -1:

Wave function

Senthil and Levin, Phys. Rev. Lett. 110, 046801 (2013)

Holomorphic and antiholomorphic factors

/3322

charge

spin

Numerical evidence of counter-propagating charge and spin edge modes(protected by a symmetry)

(Sterdyniak et al., PRB, 2012; Dubail et al., PRB, 2012)

Real-space entanglement spectrum

A

B

Logarithms of the eigenvalues of

-> Information of edge-state spectrum

ChargeSpin

S.F. and M. Ueda, Phys. Rev. Lett. 111, 090401 (2013)

/33Classification of symmetry-protected topological phases of bosons 23

Haldanechain

X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, Science 338, 1604 (2012); Phys. Rev. B 87, 155114 (2013).

Bosonic integerquantum Hallstate

/33Summary: Quantum Hall states in two-component Bose gases 26

2/34/32

Halperin (221) state

Non-Abelianspin-singlet state

Bosonic integerquantum Hall state

Synthetic magnetic field

Phys. Rev. A 86, 031604 (R) (2012)

Phys. Rev. Lett. 111, 090401 (2013)

chargespin

So far a rare example of a symmetry-protected topological phase of bosons in dimensions D>1

Y.-H. Wu and J. K. Jain, Phys. Rev. B 87, 245123 (2013)N. Regnault and T. Senthil, Phys. Rev. B 88, 161106 (2013)

Grass, Julia-Diaz, Barberan, and Lewenstein, PRA 86, 021603(R) (2012)Related numericalworks：

Combination of IQH and NASS ideasfor ?

/33Outline 27

Background

Quantum Hall states in two-component Bose gases

- Quantum Hall effects in 2D electron systems in high magnetic fields

- Quantum Hall physics in ultracold atomic gases(Attempts for realization; theory of single-component Bose gases)

+B

-B

+B

•Non-Abelian spin-singlet state

•Bosonic integer quantum Hall stateprotected by a U(1) symmetry

•Stability of fractional quantum spin Hall states

/33Antiparallel magnetic fields 28

Beeler et al., Nature 498, 201 (2013)(Spielman group)

Observation of a (classical) spin Hall effect

+B

-B

Q. Any exotic quantum states at low T and high B?

Spin-dependent Lorentz forces

y momentum

xm

om

en

tum

Dressed states

(Dressed-state basis)

/33Two-comp. Bose gases in antiparallel mag. fields 29

+B

-B

S. F. and M. Ueda,Phys. Rev. A 90, 033602 (2014)

/3330

+B

-B

Mean-field regime: the same phase diagram as the parallel-field case

Gross-Pitaevskii energy functional:

Mean-field vortex phase diagram in the parallel-field case: Mueller & T.-L. Ho, PRL,2002; Kasamatsu, Tsubota, &Ueda, PRL, 2003

S. F. and M. Ueda,Phys. Rev. A 90, 033602 (2014)

Two-comp. Bose gases in antiparallel mag. fields - 1

/3331

(Fractional) quantum spin Hall states are robust, even when .

+B

-B

(Exact diag. analysis)

• Two-comp. gases originating from two hyperfine states naturally has .

• Markedly contrasts with the parallel-field case

S. F. and M. Ueda,Phys. Rev. A 90, 033602 (2014)

Two-comp. Bose gases in antiparallel mag. fields - 2

/3332

Exact pairing ground state at

+B

-B

Pair creation op.

Novel quantum state at the transition point beyond which the system collapses.Is this universal?

S. F. and M. Ueda,Phys. Rev. A 90, 033602 (2014)

Two-comp. Bose gases in antiparallel mag. fields - 3