Entanglement Spectroscopy and its application tothe fractional quantum Hall phases

N. Regnault

Ecole Normale Superieure Paris and CNRS,Department of Physics, Princeton Global Scholar

ÉCOLE NORMALE

S U P É R I E U R E

Acknowledgment

A. Sterdyniak (MPQ, Germany)

B. Estienne (Universite Pierre et Marie Curie, France)

Z. Papic (University of Leeds)

F.D.M. Haldane (Princeton University)

R. Thomale (Wurzburg, Germany)

M. Haque (Maynooth, Ireland)

A.B. Bernevig (Princeton University)

Topological phases

What is topological order ?

phases that can’t be described by a broken symmetry. Nolocal order parameter.

At least one physical (i.e. measurable) quantity related to atopological invariant (like the surface genus).

A system with a gapped bulk and gapped or gapless surface(or edge) modes.

Simplest example : the integer quantum Hall effect (quantizedHall conductance).

Since 2005, the revolution of topological insulators

2D TI :3D TI :

Making things harder : strong interactions

Most celebrate example : thefractional quantum Hall effect.

Alliance of a non-trivial bandstructure (Landau levels) and stronginteractions.

An exotic place : emergent fractionalcharges with fractional statistics ornon-abelian.

Rxx

RxyB

No classification of fractional phases (as opposed to thenon-interacting case).

Non-perturbative problem → variational methods andnumerical simulations.

No local order parameter → which phase is emerging ?

Outline

Entanglement Spectrum

Fractional Quantum Hall Effect

FQHE and Entanglement Spectrum

ES and Fractional Chern Insulators

Entanglement Spectrum

Entanglement spectrum (Li and Haldane 2008)

Start from a quantum state |Ψ〉Create a bipartition of the system intoA and B

Reduced density matrixρA = TrB |Ψ〉 〈Ψ|Entanglement Hamiltonian :ρA = e−Hent

A B1D:

A B

2D:

L

The eigenvalues of Hent are the entanglement energies {ξi}.Lower entanglement energies ' higher weights in ρA.

If O = OA +OB and , the ξi can be labeled by the OA

quantum numbers.

Entanglement entropy SA = −TrA [ρA ln ρA], area law forgapped systems (i.e. SA ∝ Ld−1).

Entanglement spectrum

Example : system made of two spins 1/2

BAEntanglement spectrum : ξ as a function of Sz,A(z projection of the spin A)

|↑↑〉

0

0.5

1

1.5

2

-1 -0.5 0 0.5 1

ξ

Sz,A

1√2

(|↑↓〉+ |↓↑〉)

0

0.5

1

1.5

2

-1 -0.5 0 0.5 1

ξ

Sz,A

1√4|↑↓〉+

√34 |↓↑〉

0

0.5

1

1.5

2

-1 -0.5 0 0.5 1

ξ

Sz,A

The counting (i.e the number of non zero eigenvalue) also providesinformation about the entanglement

The AKLT spin chain

A prototype of a gapped spin-1 chain.

HAKLT =∑j

~Sj · ~Sj+1 +1

3

∑j

(~Sj · ~Sj+1

)2

The ground state of the AKLT Hamiltonian is the valence bondstate.

A B}

S=1}

S=0

S=1/2 S=1/2

For an open chain, the two extreme unpaired spin- 12 correspond to

the edge excitations (4-fold degenerate ground state)

The AKLT spin chain, the Li-Haldane conjecture

0

5

10

15

20

25

30

35

-3 -2 -1 0 1 2 3 4

ξ

Sz,A

0

1

2

0 1

ξ

Sz,A

AKLT Heis.

0

5

10

15

20

25

30

35

-3 -2 -1 0 1 2 3 4

ξ

Sz,A

ξD

ES for an open chainwith 8 sites and lA = 4.Sz,A : z-projection of Atotal spin.

Reduced density matrix is 81× 81 but only two non-zeroeigenvalues for the AKLT model.

The trace has introduced an artificial edge → a spin- 12 edge

excitation. The ES mimics the edge spectrum of the model.

Away from the model state : An entanglement gap ∆ξ

between a low (entanglement) energy structure related themodel state and a high energy structure.∆ξ should stay finite at the thermodynamical limit if the twophases are in the same universality class.

Fractional Quantum Hall Effect

Fractional Quantum Hall effect

Landau levels (spinless case)

hwc

hwcN=0

N=1

N=2

Rxx

RxyB

Cyclotron frequency : ωc = eBm ,

Filling factor : ν = hneB = N

NΦ

Partial filling + interaction → FQHE

Lowest Landau level (ν < 1) :zm exp

(−|z |2/(4l2B)

)N-body wave function :Ψ = P(z1, ..., zN) exp(−

∑|zi |2/(4l2B))

What are the low energy properties ?Gapped bulk, Massless edge

Strongly correlated systems,emergence of exotic phases :fractionalcharges, non-abelian braiding.

The Fractional QHE

FQHE is a hard N-body problem :

a single Landau level (the lowest one for ν < 1, no spin)

the effective Hamiltonian is just the (projected) interaction !

H = PLLL∑i<j

V (~ri − ~rj)PLLL

(insert in V your favorite interaction plus screening effect,finite width, Landau level,...)

Two major methods :

variational method : find a wave functions describinglow energy physics (symmetries, CFT, model...)

numerical calculation : exact diagonalizations ondifferent geometries (sphere, plane, torus, ...), DMRG

Z

NF

L =+3z

L =+2z

L =-3z

L =-2z

L =0z

L =+1z

L =-1z

Nbr orb. ' NΦ

The Laughlin wave function

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

ΨL(z1, ...zN) =∏i<j

(zi − zj)3e−

∑i|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 excitations

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

Quasi-holes obey fractional statistics (fractional charge + flux)

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).

The torus geometry : topological degeneracy

N = 4 N=4 bosonsF

k = 0y

k = 1y

k = 2y

k = 3y

k =y

2 1 10 1 2

03

0

0.02

0.04

0.06

0.08

0.1

0 2 4 6 8 10 12 14 16

Energ

y

KT

y

Model interaction for theLaughlin state,N = 6,NΦ = 18

In the LLL, the one-body wf are :∑k∈Z e

2πLy

(ky+kNφ)(x+iy)e−

x2

2 e− 1

2

(2πLy

)2(ky+kNφ)2

The Laughlin ν = 1/m is m-fold degenerateon the torus.

Number of orbitals is Nφ.

Each orbital is labeled by its quantumnumber ky .

Invariant under the magnetic translations.

KTy = (

∑i ky ,i )modNΦ.

There is another quantum number (purelymany-body) related the center of massdegeneracy.

The Haldane’s exclusion principle

The number of quasihole states per momentum sector can bepredicted by a generalization of the Pauli’s principle.

For the Laughlin ν = 1/m, no more than 1 particle in mconsecutive orbitals (including periodic boundary conditionson the torus).

Example Laughlin ν = 1/3 state with 9 flux quanta

k =y

1 0 00 1 2

1 0 03 4 5

1 0 06 7 8 k =y

1 0 00 1 2

0 1 03 4 5

1 0 06 7 8

k =y

0 1 00 1 2

0 1 03 4 5

0 1 06 7 8k =y

1 0 00 1 2

0 1 03 4 5

0 1 06 7 8

Can be generalized to the Moore-Read or Read-Rezayi states(non-abelian excitations) : no more than k particles in k + 2consecutive orbitals.

The Haldane’s exclusion principle

Example : Finding back the 3-fold degeneracy of the Laughlinν = 1/3 with Nφ = 3× N = 9

k =y

1 0 00 1 2

1 0 03 4 5

1 0 06 7 8

K =0+3+6 mod 9=0 yT

k =y

0 1 00 1 2

0 1 03 4 5

0 1 06 7 8

K =1+4+7 mod 9=3 yT

k =y

0 0 10 1 2

0 0 13 4 5

0 0 16 7 8

K =2+5+8 mod 9=6 yT

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0 1 2 3 4 5 6 7 8 9

Ene

rgy

KTy

Laughlin GS

The Laughlin wave function : edge excitations

B B

One dimensional chiral modewith a linear dispersion relationE ' 2πv

L n

The degeneracy of eachmany-body energy level Et isgiven by the sequence1, 1, 2, 3, 5, 7, ....

Non-interacting case (i.e. IQHE)(a) E = 0

(b) E = 1

(c) E = 2

(d) E = 2

energ

y

momentum

t t

t t

The Laughlin wave function : edge excitations

B B

One dimensional chiral modewith a linear dispersion relationE ' 2πv

L n

The degeneracy of eachmany-body energy level Et isgiven by the sequence1, 1, 2, 3, 5, 7, ....

Interacting case (Laughlin ν = 13)

(a) E = 0

(b) E = 1

(c) E = 1

(d) E = 2

(e) E = 2

t

t t

tt

energ

y

momentum

FQHE and Entanglement Spectrum

Orbital entanglement spectrum

FQHE on a cylinder (Landau gauge) : orbitals are labeled by

ky , rings at position2πkyL l2B

Divide your orbitals into two groups A and B, keeping Norb,A

orbitals : orbital cut ' real space cut (fuzzy cut)

K =01234567....y

A B

Ky

A

1 0 1K =y 0 1 2} 01

4 5}B

016 73

1

Laughlin state N = 12, half cut

0

10

20

30

40

50

60

0 2 4 6 8 10 12 14 16 18

ξ

Ky,A

OES Laughlin N=12, NA=6 on a cylinder L=15

Ky

1 1 2 3 5 7 ...

Fingerprint of the edge mode (edge mode counting) can beread from the ES. ES mimics the chiral edge mode spectrum.

For FQH model states, nbr. levels is exp. lower than expected.

ES on various geometries

0

10

20

30

40

50

0 5 10 15 20 25 30 35

NF

(a)

ξ

Lz,A

0

10

20

30

40

50

0 5 10 15 20 25 30 35

ξ

Lz,A

(d)

NF

R=

0

10

20

30

40

50

0 2 4 6 8 10 12 14 16 18

ξ

(b)

Ky,A

0

10

20

30

40

50

60

0 2 4 6 8 10 12 14 16 18

ξ

Ky,A

OES Laughlin N=12, NA=6 on a cylinder L=15

Ky

1 1 2 3 5 7 ...

Different eigenvalues of ρA (shape of the ES) but the samenumber of non-zero eigenvalues (counting)

The counting IS the important feature. For model states inthe FQHE , exponentially lower than expected

Away from model states

Groundstate of the Coulomb interaction at ν = 1/3 for N = 12 ona sphere/thin annulus

ν = 1/3 Laughlin state

0

10

20

30

40

50

0 5 10 15 20 25 30 35

ξ

Lz,A

369

0 1 2 3 4 5 6

ξ

Lz,A

1 1 2 3 5 7 11

VS

GS of the Coulomb interaction

0

5

10

15

20

25

30

35

40

0 10 20 30 40

ξ

Lz,A

ξ

Lz,A

369

22 24 26 Dx

A low ent. energy structure identical to the Laughlin state..An entanglement gap that does not spread over the fullspectrum but protects the region mimicking the edge mode.An additional structure in the high energy part related to theneutral excitations (quasihole-quasielecton pairs).

How to cut the system ?

The system can be cut in different ways :

real spaceorbital (or momentum) spaceparticle space

Each way may provide different information about the system (ex :trivial in momentum space but not in real space)

NF

geometrical partition

particle partition

NF/2

edge physics

quasihole physics

NF

Real space partitioning :extracting the edge physics

Particle partitioning :extracting the bulk physics

Particle entanglement spectrum

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)

0

5

10

15

20

0 5 10 15 20Ky,A

ξ

ν = 1/3 Laughlin N = 8, NA = 4

0

5

10

15

20

0 5 10 15 20

ξKy,A

ξD

Coulomb GS at ν = 1/3 on a torus

Counting is the number of quasihole states for NA particles on thesame geometry → the fingerprint of the phase.

ES and Fractional Chern Insulators

Chern insulators

A Chern insulator is a zero magneticfield version of the QHE (Haldane,88).

Topological properties emerge fromthe band structure.

At least one band is a non-zero Chernnumber C , Hall conductanceσxy = e2

h |C |What about the strong interactingregime ? → Fractional Cherninsulators.

Neupert et al. PRL 106, 236804(2011), Sheng et al. Nat. Comm. 2,389 (2011), NR and BAB, PRX(2011)

e1

e2

0 1 2 3 4 5 6 0 1 2 3 4 5 6-10

-5

0

5

10

15

kxky

E(kx,k y)

C=-1

C=0

C=1

PES and FCI

Filling the lowest band ν = 1/3 plus strong interaction, do we geta Laughlin-like state or a charge density wave ?

FCI phase

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0 5 10 15 20

E (

arb

. u

nit)

Kx + Nx * Ky

(a)

0

5

10

15

20

0 5 10 15 20

ξ

Kx,A + Nx * Ky,A

CDW phase

1e-05

0.0001

0.001

0.01

0.1

1

0 2 4 6 8 10 12 14 16

E (

arb

. u

nit)

Ky

(a)

0

5

10

15

20

0 2 4 6 8 10 12 14 16

ξ

Ky,A

(b)

Energy spectrum with similar features (3-fold degenerategroundstate) but a different entanglement spectrum.

Conclusion

For many quantum phases, the ground state contains asurprisingly large amount of information about the excitations.

The entanglement spectroscopy is a way to probe (or extract)this information.

Seeing the bulk-edge correspondence.

Different partitions give access to different types ofexcitations.

Entanglement spectroscopy is a concrete tool, requiring onlythe ground state (example of the fractional Chern insulators).

What is the meaning of the counting exponential reduction ?Efficient description using a matrix product staterepresentation (but this is another story...).

Conclusion

Counting is great !

Conclusion

Counting is great !(you just have to be careful...)