From Fractional Quantum Hall EffectTo Fractional Chern Insulator

N. Regnault

Laboratoire Pierre Aigrain,Ecole Normale Superieure, Paris

Acknowledgment

Y.L. Wu (PhD, Princeton)

A.B. Bernevig (Princeton University)

Motivations : Topological insulators

An insulator has a (large) gap separating a fully filled valence bandand an empty conduction band

Atomic insulator : solidargon

Semiconductor : Si

How to define equivalent insulators ? Find a continuoustransformation from one Bloch Hamiltonian H0(~k) to anotherH1(~k) without closing the gap

Vacuum is the same kind of insulator than solid argon with agap 2mec

2

Are all insulators equivalent to the vacuum ? No

Motivations : Topological insulators

What is topological order ?

cannot be described by symmetry breaking (cannot useGinzburg-Landau theory)some physical quantities are given by a “topological invariant”(think about the surface genus)a bulk gapped system (i.e. insulator) system feeling thetopology (degenerate ground state, cannot be lifted by localmeasurement).naive picture : normal strip versus Moebius stripa famous example : Quantum Hall Effect (QHE)

TI theoretically predicted and experimentally observed in the past4 years missed by decades of band theory

2D TI :3D TI :

Motivations : FTI

A rich physics emerge when turning on strong interaction in QHE

What about Topological insulators ?

Time-reversal breaking

Time reversal invariant

QHE (B field)

Chern Insulator

3D TI

QSHE

no interaction

FQHE

FCI

FTI ?

FQSHE ?

strong interaction

Outline

Fractional Quantum Hall Effect

Fractional Chern Insulator

FQHE vs FCI

Fractional Quantum Hall Effect

Landau level

Rxx

RxyB

hwc

hwcN=0

N=1

N=2

Cyclotron frequency : ωc = eBm

Filling factor : ν = hneB = N

Nφ

At ν = n, n completely filled levelsand a energy gap ~ωc

Integer filling : a (Z) topologicalinsulator with a perfectly flat band/ perfectly flat Berry curvature !

Partial filling + interaction →FQHE

Lowest Landau level (ν < 1) :zm exp

(−|z |2/4l2

)N-body wave function :Ψ = P(z1, ..., zN) exp(−∑ |zi |2/4)

The Laughlin wave function

A (very) good approximation of the ground state at ν = 13

ΨL(z1, ...zN) =∏i<j

(zi − zj)3e−

Pi|zi |24l2

x

ρ

The Laughlin state is the unique (on genus zero surface)densest state that screens the short range (p-wave) repulsiveinteraction.

Topological state : the degeneracy of the densest statedepends on the surface genus (sphere, torus, ...)

The Laughlin wave function : quasihole

Add one flux quantum at z0 = one quasi-hole

Ψqh(z1, ...zN) =∏i

(z0 − zi ) ΨL(z1, ...zN)

ρ

x

Locally, create one quasi-hole with fractional charge +e3

“Wilczek” approach : quasi-holes obey fractional statistics

Adding quasiholes/flux quanta increases the size of the droplet

For given number of particles and flux quanta, there is aspecific number of qh states that one can write

These numbers/degeneracies can be classified with respectsome quantum number (angular momentum Lz) and are afingerprint of the phase (related to the statistics of theexcitations).

Fractional Chern Insulator

Interacting Chern insulators

A Chern insulator is a zero magnetic field version of the QHE(Haldane, 88)

Basic building block of 2D Z2 topological insulator (half of it)

Is there a zero magnetic field equivalent of the FQHE ? →Fractional Chern Insulator

Several proposals for a CI with nearly flat band (K. Sun et al.,Neupert et al., E. Tang et al.) that may lead to FCI

But “nearly” flat band is not crucial for FCI like flat band isnot crucial for FQHE (think about disorder)

The checkerboard lattice model

t1 -t2

t2

-t2

t2

K. Sun, Z.C. Gu, H. Katsura, S. DasSarma, Phys. Rev. Lett. 106, 236803(2011).

H1 =∑

k(c†kA, c†kB)h1(k)(ckA, ckB)T

h1(k) =∑

i di (k)σi

dx(k) =4t1 cos(φ) cos(kx/2) cos(ky/2)

dy (k) = 4t1 sin(φ) sin(kx/2) sin(ky/2)

dz(k) = 2t2(cos(kx)− cos(ky ))

φ, t1 and t2 parameters (plus secondNNN) can be optimized to get anearly flat band

The flat band limit

0 0.2

0.4 0.6

0.8 1 0

0.2 0.4

0.6 0.8

1

-2-1 0 1 2 3 4

E/t1

kx Lx / 2 πky Ly / 2 π

E/t1

E

k

dD

The checkerboard lattice modelhas a nearly flat valence band.

δ � Ec � ∆ (Ec being theinteraction energy scale)

We can deform continuously theband structure to have aperfectly flat valence band

and project the system onto thelowest band, similar to theprojection onto the lowestLandau level

Two body interaction and the checkerboard lattice

Our goal : stabilize a Laughlin-like state at ν = 1/3.A key property : the Laughlin state is the unique densest statethat screens the short range repulsive interaction.

Hint =∑<i ,j>

ninj

A nearest neighbor repulsionshould mimic the FQHinteraction.

We give the same energypenalty to any of theseRed-Blue clusters

Sheng et al. NatureCommunications 2, 389 (2011),Neupert et al. Phys. Rev. Lett.106, 236804 (2011)

The ν = 1/3 filling factor

An almost threefold degenerate ground state as you expect for theLaughlin state on a torus (here lattice with periodic BC)

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0 5 10 15 20 25 30 35

(E -

E1)

/ t1

Kx + Nx Ky

N=6N=10N=12

But 3fold degeneracy is not enough to prove that you haveLaughlin-like physics there.

FCI vs FQHE

0.24

0.26

0.28

0.3

0.32

0.34

0.36

0.38

0.4

0 5 10 15 20 25

Ene

rgy

kx * Ny + ky

FCI (ν = 1/3, N = 8)

0

0.2

0.4

0.6

0.8

1

80 90 100 110 120 130 140

Ene

rgy

kx * NΦ + ky

FQHE on torus

The degeneracy is not perfect. The Berry curvature density is notflat and thus we don’t have an exact magnetic translation algebra(see S. A. Parameswaran, R. Roy, S. L. Sondhi, arXiv :1106.4025)

Gap

Many-body gap can actually increase with the number ofparticles due to aspect ratio issues.

Finite size scaling not and not monotonic reliable because ofaspect ratio in the thermodynamic limit.

The 3-fold degeneracy at filling 1/3 in the continuum existsfor any potential and is not a hallmark of the FQH state. Onthe lattice, 3-fold degeneracy at filling 1/3 means more thanin the continuum, but still not much

0

0.02

0.04

0.06

0.08

0.1

0 0.05 0.1 0.15 0.2

∆/t 1

1/N

Ny=3Ny=6Ny=9

Gap vs size

0.001

0.01

0.1

0 0.05 0.1 0.15 0.2

δ/∆

1/N

Ny=3Ny=6Ny=9

spread of the GS manifold

Quasihole excitations

The form of the groundstate of the Chern insulator at filling1/3 is not exactly Laughlin-like. However, the universalproperties SHOULD be.The hallmark of FQH effect is the existence of fractionalstatistics quasiholes.In the continuum FQH, Quasiholes are zero modes of a modelHamiltonians - they are really groundstates but at lower filling.In our case, for generic Hamiltonian, we have a gap from a lowenergy manifold (quasihole states) to higher generic states.

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

0 5 10 15 20 25 30

E /

t 1

Kx + Nx * Ky

N = 9, Nx = 5,Ny = 6The number of states belowthe gap matches the one of

the FQHE !

Direct comparison

Question : “Why not using overlaps ?”Possible answers :

high overlaps do not guarantee that you are in the same phase

we don’t know how to write the Laughlin state on such alattice

we don’t know if there is an exact hamiltonian for theLaughlin state on such a lattice

we cannot compute an overlap but we can compute anentanglement spectrum...

Particle entanglement spectrum

Particle cut : start with the ground state Ψ for N particles,remove N − NA, keep NA

ρA(x1, ..., xNA; x ′1, ..., x

′NA

)

=

∫...

∫dxNA+1...dxN Ψ∗(x1, ..., xNA

, xNA+1, ..., xN)

× Ψ(x ′1, ..., x′NA, xNA+1, ..., xN)

“Textbook expression” for the reduced density matrix.

6

8

10

12

14

16

18

-25 -20 -15 -10 -5 0 5 10 15 20 25

ξ

LzA

Laughlin ν = 1/3 state N = 8,NA = 4

The particle ES reveals thefingerprint of the phase.

This information that comes fromthe bulk excitations is encodedwithin the groundstate !

If the FCI at ν = 1/3 is aLaughlin-like phase, we shouldobserve the Laughlin counting.

Particle entanglement spectrum

8

10

12

14

16

18

20

22

24

-5 0 5 10 15 20 25 30 35 40

ξ

Kx + Nx * Ky

PES for N = 12, NA = 4, 58905 states below the gap as expected

From topological to trivial insulator

One can go to a trivial insulator, adding a +M potential on A sitesand −M potential on B sites. A perfect (atomic) insulator ifM →∞.

0

0.02

0.04

0.06

0.08

0.1

0 0.2 0.4 0.6 0.8 1 1.2 1.4

∆/t 1

M/4t2

N=8N=10

Energy gap

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2∆ P

ES

M/4t2

N=8, NA=3N=10, NA=3N=10, NA=4N=10, NA=5

Entanglement gap

Sudden change in both gaps at the transition (M = 4t2)

Emergent Symmetries in the Chern Insulator

In FCIs, there is in principle no exactdegeneracy (apart from the latticesymmetries).

But both the low energy part of theenergy and entanglement spectra exhibitan emergent translational symmetry.

The momentum quantum numbers of theFCI can be deduced by folding the FQHBrillouin zone.

FQH : m = GCD(N,Nφ = Nx × Ny ),FCI : nx = GCD(N,Nx),ny = GCD(N,Ny )

FQHE BZ

K =(0,...,m-1)x

K =(0,...,m-1)y

FCI BZ

K =(0,...,n -1)x

K =(0,...,n -1)y

x

y

FOLDING

FQHE vs FCI

Some questions about the FCIs

What is specific about the checkerboard lattice model ?

What are the important ingredients ?

Any evidence for other fractions ?

p/2p + 1 series ? Composite fermions ?Moore-Read / Read-Rezayi states

Is there any FQHE feature missing on the FCI side ?

Four other models

b1

b2

a1

a2

a3 t1

t2eiφ

+M

−M

Haldane model (2 bands)

x

yV

t2

−t2

+M

−M

−it1it1

t1

−t1

1/2 HgTe (2 bands)

e1

e2

Kagome (3 bands)

1 2

345

6b2

b1

Ruby lattice (6 bands)

FCI on the Haldane model

ν = 1/3 on the Haldane model with short range repulsion.

b1

b2

a1

a2

a3 t1

t2eiφ

+M

−M

We can tune t1, t2 and Φ to make aflat band model (Neupert et al.)

0 5 10 15 20 25

kx +Nx ky

0.00

0.01

0.02

0.03

0.04

0.05

E−E

0

(N,Nx ,Ny ) =(8,6,4)

(N,Nx ,Ny ) =(10,6,5)

Exactly the same features than ν = 1/3 on the checkerboardlattice (but ...)

FCI on the Haldane model

How does the flatness of the Berry curvature affects the FCIspectrum ?Look at N = 8, ν = 1/3, tuning Φ tracking σB, the deviation tothe average Berry curvature (1/2π)

0.1 0.2 0.3 0.4 0.5 0.6 0.7

σB

0.000

0.005

0.010

0.015

0.020

0.025

0.030

∆E

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Energy gap

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

σB

-0.005

0.000

0.005

0.010

0.015

0.020

0.025

W

0.15

0.30

0.45

0.60

0.75

0.90

1.05

1.20

1.35

Energy spread

0.1 0.2 0.3 0.4 0.5 0.6 0.7

σB

0.0

0.5

1.0

1.5

2.0

2.5

3.0

∆ξ

0.15

0.30

0.45

0.60

0.75

0.90

1.05

1.20

1.35

Ent. gap

The flatness of the Berry curvature helps but not as much asexpected

Benchmarking the FCIs

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

1/N

-0.04

-0.03

-0.02

-0.01

0.00

0.01

0.02

0.03

∆E

Ny =3

Ny =4

Ny =5

Ny =6

Gaps for the Haldane model

0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16

1/N

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

∆E

Ny =3

Ny =4

Ny =5

Ny =6

Gaps for the Kagome model

both the Haldane and 1/2 HgTe modelsshow a weak FCI phase at ν = 1/3

Checkerboard, Kagome and ruby modelsexhibit a clear Laughlin like phase.

Two versions of the Kagome model : theone with the flattest band (with NNN) isthe weakest FCI.

A flat band model is not a guarantee toget a good FCI

0 5 10 15 20 25 30 35

kx +Nx ky

10

12

14

16

18

20

ξ

PES Kagome N = 12,NA = 5

The Moore-Read state

Prototype of the non-abelian state that may describe theincompressible fraction ν = 5/2. In the FQHE, the MR state canbe exactly produced using a three-body interaction.What about the FCI ?

Two types of nearest neighbor3 particle cluster :

Blue-Blue-Red

Red-Red-Blue

We give the same energypenalty to any of these clusters

The Moore-Read state

Energy spectrum at ν = 1/2 for N = 12,Nx = 6,Ny = 4

0.15

0.16

0.17

0.18

0.19

0.2

0.21

0 5 10 15 20 25

Ene

rgy

kx * Ny + ky

6-fold almost degenerate groundstate (2 at kx = ky = 0, 4 atkx = 0, ky = 4), as predicted by the FQHE to FCI mapping !

The Moore-Read state

Particle entanglement spectrum at ν = 1/2 forN = 12,Nx = 6,Ny = 4,NA = 5

8

10

12

14

16

18

20

22

0 5 10 15 20 25

ξ

Kx + Nx * Ky

1277 states in each momentum sector below the dotted line.

The Moore-Read state

Energy spectrum at ν = 1/2 + one flux quantum forN = 12,Nx = 5,Ny = 5

0.125

0.13

0.135

0.14

0.145

0.15

0.155

0.16

0.165

0 5 10 15 20 25

Ene

rgy

kx * Ny + ky

7 states in each momentum sector below the dotted line.Can we stabilize the MR state with a two-body interaction ? Workin progress...

The Read-Rezayi states

RRk=3

RRk=4

MR

Lg

We can cook-up an interactionfor each RR states.

works like a charm for theRR k = 3.

not completely clear forthe RR k = 4. Finite sizeeffect ?

MR do not show up in theHaldane and 1/2 HgTemodels

Strong MR phase in theKagome and ruby models.

Particle-hole symmetry

An important feature of the FQHE (with a two body interaction) :ν = 1/3↔ ν = 2/3

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

-2 0 2 4 6 8 10 12 14 16 18

Ene

rgy

kx * Ny + ky

N = 6,Nx = 6,Ny = 3ν = 1/3

4.7

4.75

4.8

4.85

4.9

4.95

5

-2 0 2 4 6 8 10 12 14 16 18

Ene

rgy

kx * Ny + ky

N = 12,Nx = 6,Ny = 3ν = 2/3

The p-h symmetry is absent from the FCI model, it might berestored at large N

FCI are not as boring as expected

Some questions about the FCI

What is specific about the checkerboard lattice model ? notreally

What are the important ingredients ? Don’t know

Any evidence for other fractions ?

p/2p + 1 series ? Composite fermions ? TODOMoore-Read / Read-Rezayi states yes !

Is there any FQHE feature missing on the FCI side ?particle-hole symmetry

Conclusion

Fractional topological insulator at zero magnetic field exists asa proof of principle.

There is a counting principles for the groundstate and theexcitations.

Beyond ν = 1/3 : ν = 1/5, “ν = 5/2” (at least with threebody interaction). Other fractions ?

What are the good ingredients for an FCI ?

First time entanglement spectrum is used to find informationabout a new state of matter whose ground-state wavefunctionis not known.

Entanglement spectrum powerful tool to understand stronglyinteracting phases of matter.

Roadmap : find one such insulator experimentally, 2D-3Dfractional topological insulators ?