Adiabatic connection of Fractional Chern Insulators to ... · A. Sterdyniak, N. Regnault, & GM,...

Adiabatic connection of Fractional Chern Insulators to Fractional Quantum Hall States

Gunnar MöllerCavendish Laboratory, University of Cambridge

Cavendish Laboratory

November 21, 2012, University of Leeds

A. Sterdyniak, N. Regnault, & GM, Phys. Rev. B 86, 165314 (2012)

Th. Scaffidi & GM, Phys. Rev. Lett. 109, 246805 (2012)

GM & N. R. Cooper, Phys. Rev. Lett. 103, 105303 (2009)

Gunnar Möller University of Leeds, November 21st, 2012

• Motivation: physical realizations of lattices with high flux density

• Relationship between different realizations of topological band structures:

• Particle entanglement spectrum as a probe for quantum Hall states in lattices

Outline

‣ Adiabatic continuation from Fractional Chern Insulators to Fractional Quantum Hall states

Gunnar Möller University of Leeds, November 21, 2012

Magnetic flux through periodic potentials

• magnetic field in presence of crystal potentials in solid state systems

0 a n 1

• cold atoms in `rotating’ optical lattices: 0 ∼ a n 1

0 a n 1

• Simulating Aharonov-Bohm flux by complexhopping in tight binding optical lattices 0 a n 1

• Simulating Aharonov-Bohm flux by Berry phases in real space

0 a n 1“Optical Flux Lattices”

Simulation of Landau-gauge (continuum) 0 ∼ L1/2

Cold A

0 a n 1

Cold A

0 a n 1

Cold A

0 a n 1

• Chern bands: Berry flux in reciprocal space

0 a n 1

Cold A

Solid State

Cold A

0 a n 1

• Chern bands: Berry flux in reciprocal space

0 a n 1

Cold A

Solid State

Cold A

Cold Atoms I: Optical Lattices with Complex Hopping

optical lattice + Raman lasers

H = −J

b†αbβe

iAαβ + h.c.+

nα(nα − 1)− µ

⇒ possibility to simulate Aharonov-Bohm effect of magnetic field by imprinting phases for hopping via Raman transitions

Aαβ = 2πnV

⇒ Bose-Hubbard with a magnetic field (! Lorentz force)

particle density vortex/flux density interaction nV U/JnV

J. Dalibard, et al. Rev. Mod. Phys. 83, 1523 (2011)

experimental realisation:

Cold Atoms I: Optical Lattices with Complex Hopping

‘anti-magic’ optical lattice for Yb

Gerbier & Dalibard, NJP 2010

Staggered Flux ‘Rectified’ Flux

Realized by group of I. Bloch:Phys. Rev. Lett. 107, 255301 (2011). Experimental realization outstanding

Can tune flux density!

Cold Atoms II: Berry phases of optically dressed states

spacially varying optical coupling of N internal states (consider N=2)

−∆ ΩR(r)

ΩR(r) ∆

Yblocal spectrum En(r) and dressed states: |Ψr

J. Dalibard, F. Gerbier, G. Juzeliunas, P. Öhberg, RMP 2011

−∆ ΩR(r)

ΩR(r) ∆

adiabatic motion of atoms in space in the optical potential generates a Berry phase

for N=2, consider Bloch sphere of unit vector

−∆ ΩR(r)

ΩR(r) ∆

adiabatic motion of atoms in space in the optical potential generates a Berry phase

nφ =1

8πijkµνni∂µnj∂νnk

n = Ψr|σ|Ψr

Berry flux generated:

Anφ d

2r =Ω

Total flux quanta = # times Bloch vector wraps sphere

Cold Atoms II: Optical Flux Lattices

periodic optical Raman potentials are conveniently located by standing wave Lasersyielding an optical flux lattice of high flux density, here nϕ=1/2a2 (fixed)

(a) Local direction of unit Bloch vector (b) Local density of Berry Flux n nφ

Nigel Cooper, PRL (2011); N. Cooper & J. Dalibard, EPL (2011)

Cold Atoms II: Optical Flux Lattices

periodic optical Raman potentials are conveniently located by standing wave Lasersyielding an optical flux lattice of high flux density, here nϕ=1/2a2 (fixed)

(a) Local direction of unit Bloch vector (b) Local density of Berry Flux n nφ

Nφ = 1

Nigel Cooper, PRL (2011); N. Cooper & J. Dalibard, EPL (2011)

Fractionalization in Chern bands? Chern #1 Bands = FQHE ?

Proposition: correlated states reproduce the physics of FQHE

H =− t1

a†rar + h.c.

− t2

a†rare

iφrr + h.c.

− t3

a†rar + h.c.

nr(nr − 1)

Characteristic example: The Haldane Model

• tight binding model in real space on hexagonal lattice• with fine-tuned hopping parameters: obtain flat lower band, e.g. values [D. Sheng, PRL (2011)]

t1 = 1, t2 = 0.60, t2 = −0.58 and φ = 0.4π

F.D.M. Haldane, PRL (1988), Neupert et al. PRL (2011)

hαβ(k)unβ(k) = n(k)u

nα(k)

A(n,k) = −i

un∗α (k)∇ku

nα(k)

Bloch states

k a†k,αhαβ(k)ak,β

• diagonalize Hamiltonian by Fourier transform

Topological (flat) bands in two-dimensions

• study Berry curvature in nth band:

Berry connection:

B(k) = ∇k ∧A(k)Berry curvature:

C = 12π

BZ d2kB(k)Chern number:

Flux Lattices vs Chern Bands

= tight binding model in real space with topological flat bandsCBsreal space reciprocal space

real space reciprocal space

= tight binding model in real space with topological flat bandsCBsreal space reciprocal space

= tight binding model in real space with topological flat bandsCBs

= tight binding model in reciprocal space with topological flat bands

N. R. Cooper, R. Moessner, PRL (2012) [arxiv:1208.4579]

Flux Lattice

absorption of photons = momentum + state transfers

= tight binding model in real space with topological flat bandsCBs

= tight binding model in reciprocal space with topological flat bands

N. R. Cooper, R. Moessner, PRL (2012) [arxiv:1208.4579]

Flux Lattice

absorption of photons = momentum + state transfers

Overview of underlying band structures

Lattices with homogeneous flux(Hofstadter bands)

Flux Lattices(vanishing overall flux)

Flux Lattices

insertion of flux quanta through plaquettes [see Wu et al]

(vanishing overall flux)

Flux Lattices

Magnetic Fields & Landau Levels

Flux Lattices

+ Interactions

Interaction driven phases in these bands

Lattices withhomogeneous flux

Flux Latticesinsertion of flux quanta

through plaquettes

FQHE in lattices

FQHE ?

through plaquettes

FQHE in lattices

FQHE ?

through plaquettes

FQHE in lattices

FQHE ?

Strongly correlated states from the Hofstadter spectrum

• Hofstadter spectrum provides bands of all Chern numbers [Avron et al.]

Strongly correlated states from the Hofstadter spectrum

• Hofstadter spectrum provides bands of all Chern numbers [Avron et al.]

• Interactions stabilize fractional quantum Hall liquids in these bands!

• CF Theory: GM & N. R. Cooper, PRL (2009)

• Near rational flux density: LL’s with additional pseudospin indexR. Palmer & D. Jaksch PRL 2006L. Hormozi et al, PRL 2012

Interrelations between different interacting problems

Flux Lattices

insertion of flux quanta through plaquettes

FQHE in lattices

FQHE ?

Flux Lattices

FQHE in lattices

FQHE ?

addition of oscillatorymagnetic field density

leaves LLL intact:should be adiabatic

Flux Lattices

FQHE in lattices

FQHE ?

adiabatically turning onlattice confinement

Flux Lattices

FQHE in lattices

FQHE ?

adiabatic continuation(Wannier or Bloch basis)

adiabatically turning onlattice confinement

X.-L. Qi (2011), Wu et al (2012), Scaffidi & Möller PRL (2012),Liu & Bergholtz PRB (2013)

Numerical evidence for “Fractional Chern Insulators”

• existence of a gap & groundstate degeneracy [checkerboard lattice]• chern number of groundstate manifold

[D. Sheng]

• existence of a gap & groundstate degeneracy [checkerboard lattice]• chern number of groundstate manifold

[D. Sheng]

• Finite size scaling of gap

“Fractional Chern Insulators (FCI)” [N. Regnault & A. Bernevig, PRX ’11]

• Particle Entanglement Spectra : count of excitations matches FQHE (here - Laughlin state)

• existence of a gap & groundstate degeneracy [checkerboard lattice]

• chern number of groundstate manifold [D. Sheng]

• Strong numerical evidence for QHE physics, but no clear organising principle for different lattice models

• Finite size scaling of gap

“Fractional Chern Insulators (FCI)” [N. Regnault & A. Bernevig, PRX ’11]

• Particle Entanglement Spectra : count of excitations matches FQHE (here - Laughlin state)

• existence of a gap & groundstate degeneracy [checkerboard lattice]

• chern number of groundstate manifold [D. Sheng]

Understanding Fractional Quantum Hall states

φm ∝ zme−|z|2/40

• Single particle states are analytic functions in symmetric gauge A =1

2r ∧ B

• Many particle states are still analytic functions - can write explicitly!

zj = xj + iyj

Ψν= 1m

(zi − zj)me−

i |zi|

2/40e.g. Laughlin:

φm ∝ zme−|z|2/40

2r ∧ B

zj = xj + iyj

Ψν= 1m

(zi − zj)me−

i |zi|

2/40e.g. Laughlin:

• Wavefunctions are nice! Can understand many features

" Quasiparticle excitations: charge / statistics" Incompressibility

" Correlations / You name the observable...

φm ∝ zme−|z|2/40

2r ∧ B

zj = xj + iyj

Ψν= 1m

(zi − zj)me−

i |zi|

2/40e.g. Laughlin:

• Wavefunctions are nice! Can understand many features

" Quasiparticle excitations: charge / statistics" Incompressibility

" Correlations / You name the observable...

Can we construct wavefunctions

forChern Insulators?

Mapping from FQHE to FCI: Single Particle Orbitals

FCIFQHE

• Proposal by X.-L. Qi [PRL ’11]: Get FCI Wavefunctions by mapping single particle orbitals

FCIFQHE

• Idea: use Wannier states which are localized in the x-direction• keep translational invariance in y (cannot create fully localized Wannier state if C>0!)

|W (x, ky) =

f(x,ky)kx

|kx, ky

FCIFQHE

• Qi’s Proposition: using a mapping between the LLL eigenstates (QHE) and localized Wannier states (FCI), we can establish an exact mapping between their many-particle wavefunctions

|W (x, ky) =

f(x,ky)kx

|kx, ky

FCIFQHE

• Qi’s Proposition: using a mapping between the LLL eigenstates (QHE) and localized Wannier states (FCI), we can establish an exact mapping between their many-particle wavefunctions

|W (x, ky) =

f(x,ky)kx

|kx, ky

Wannier states in Chern bands

• Some formalism

α = 1 α = 2

α = 3

c†k,α =1√N

eik·(R+δα)c†R,α

nα(k)

• construction of a Wannier state at fixed ky

A(n,k) = −i

un∗α (k)∇ku

nα(k)

Hamiltonian

Eigenstates

Berry connection

|W (x, ky) =χ(ky)√

e−i kx0 Ax(px,ky)dpx × eikx

θ(ky)2π × e−ikxx|kx, ky

Fourier transform

• Some formalism

α = 1 α = 2

α = 3

c†k,α =1√N

nα(k)

• construction of a Wannier state at fixed ky in gauge with

A(n,k) = −i

un∗α (k)∇ku

nα(k)

Hamiltonian

Eigenstates

Berry connection

Fourier transform

|W (x, ky) =χ(ky)√

‘Parallel transport’ of phase

Berry connection indicates changeof phase due to displacement in BZ

Ay = 0

• Some formalism

α = 1 α = 2

α = 3

c†k,α =1√N

nα(k)

A(n,k) = −i

un∗α (k)∇ku

nα(k)

Hamiltonian

Eigenstates

Berry connection

Fourier transform

|W (x, ky) =χ(ky)√

‘Parallel transport’ of phase ‘Polarization’

ensures periodicity of WF in ky # ky + 2!

Ay = 0

• Some formalism

α = 1 α = 2

α = 3

c†k,α =1√N

nα(k)

A(n,k) = −i

un∗α (k)∇ku

nα(k)

Hamiltonian

Eigenstates

Berry connection

Fourier transform

|W (x, ky) =χ(ky)√

Ay = 0

ky-dependent phase factor, or `gauge’

More on gauge of Wannier orbitals: Y-L Wu, A. Bernevig, N. Regnault, PRB (2012)

Fourier transform

|W (x, ky) =χ(ky)√

Ay = 0

• or, more simply we can think of the Wannier states as the eigenstates of the position operator

Xcg|W (x, ky) = [x− θ(ky)/2π]|W (x, ky)

• role of polarization: displacement of centre of mass of the Wannier state

Xcg = limqx→0

∂qxρqx

θ(ky) =

0Ax(px, ky)dpx

H =− t1

a†rar + h.c.

− t2

a†rare

iφrr + h.c.

− t3

a†rar + h.c.

nr(nr − 1)

An example: The Haldane Model

• tight binding model on hexagonal lattice• with fine-tuned hopping parameters: obtain flat lower band

t1 = 1, t2 = 0.60, t2 = −0.58 and φ = 0.4π

Conventions for numerical evaluation

Real Space Reciprocal Space

G1 = 2πex/L1 sin(γ)

G2 = 2π[− cot(γ)ex + ey]/L2

v1 = sin(γ)ex + cos(γ)ey

v2 = ey

nx = 0 Lx − 1

Ly − 1

ny = 0

unβ(k+ LiGi) = un

β(k)• choose ‘periodic’ gauge with Bloch functions:

A few remarks:

Anx(q1, q2) = log [un∗

α (q1, q2)unα(q1 + 1, q2)]• use discretized Berry connection

0Ax(px, ky)dpx →

q1(kx)

Anx(q1, q2)

• discretize its integrals by the rectangle rule

Qi’s Mapping

• Can introduce a canonical order of states with monotonously increasing position:

Ky = ky + 2πx = 2πj/Ly

j = ny + Lyx = 0, 1, ..., Nφ − 1

ky = 2πny/Ly

• Increase in position for ky # ky + 2! = Chern-number C, as

∂kyXcg|x = − 1

∂θ(ky)

∂ky=

0B(px, ky)dpx

Y-L Wu, A. Bernevig, N. Regnault, PRB (2012)

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