Entanglement Spectrum in Real Space in the FQHE

Ivan Rodriguez

SISSA

Work together with

J. Slingerland S. Simon

( I.R., J.Slingerland and S.Simon Phys. Rev. Lett. 108 (2012) )

H=∑i=1

N

H 0( r⃗ i)+V (r1 , ... , rN ) General potential

Suppose you have a gapped Hamiltonian describing a topol. system (FQHE, top. Insul.) :

Very often solvable by numerical methods and small N (in FQHE 10-16 part.).

H=∑i=1

N

H 0( r⃗ i)+V (r1 , ... , rN ) General potential

Very often solvable by numerical methods and small N (in FQHE 10-16 part.).

Entanglement Spectrum (ES):

In particular given a QH g.state (Laughlin, Moore&Read, Jain) by computing the ES it's possible to obtain the excitations of the system for up to 100 particles with a laptop.

Conjectured by Li and Haldane (PRL 101, 2008)and proved in many systems.

Suppose you have a gapped Hamiltonian describing a topol. system (FQHE, top. Insul.) :

New tool to obtain the excitations of :H

I'll show you a powerful method to obtain the ES given the g.state wavefunction.

|Ψ >=∑ic i |ΨA>i |Ψ B>i

ρA=Tr Bρρ=∣Ψ⟩ ⟨Ψ∣=∣Ψ A ⟩ i∣ΨB ⟩ i j ⟨ΨB∣ j ⟨Ψ A∣

H A

A B Pure state (Laughlin,Moore Read, Jain)

Reduce density matrix

∝ e−H A

: Effective Hamiltonian of A

Spectrum of = Entanglement SpectrumH A

We can obtain the excitations in A from the ground state !! | >

conjecture

r Ar⃗1 , r⃗ 2 , .. , ⃗r N A

; ⃗rN A+1 , .. , r⃗N

r⃗ i : Position of the i-th particle

Real Space Partition (RSP)

r Ar⃗1 , r⃗ 2 , .. , ⃗r N A

; ⃗rN A+1 , .. , r⃗N

r⃗ i : Position of the i-th particle

Real Space Partition (RSP)

r⃗1 , r⃗ 2 , .. , ⃗r N A; ⃗rN A+1 , .. , r⃗N

r⃗ i : Position of the i-th particle

Particle Partition (PP)

(Schoutens, Haque, Zozulya. Rezayi PRB 2007; Sterdyniak, Regnault, Bernevig PRL 106 2011 )

A=TrB ∝e−H A

H ARSES

: describes particles on a disc: N A

Boundary excitations

FQHE Boundary excitations: CFT (Wen, Cappelli et al., Fradkin and Lopez. )Real Space ES (RSES)

r A

A=TrB ∝e−H A

H ARSES

: describes particles on a disc: N A

Boundary excitations

H APES

: describes particles on a plane: N A

Bulk excitations

FQHE Boundary excitations: CFT (Wen, Cappelli et al., Fradkin and Lopez. )Real Space ES (RSES)

Particle ES (PES)

r A

L̂z= L̂ zA+ L̂ z

B=L z

Symmetry

N=N A+N B=const

Non relativistic

ρA=∑N A , Lz

A

ρAN A , L z

A

∝e−∑N A , Lz

A

H AN A , Lz

A

ρAN A , L z

A

Rank of = # of low energy states of CFT !!H AN A , L z

A

L̂z= L̂ zA+ L̂ z

B=L z

Symmetry

N=N A+N B=const

Non relativistic

ρA=∑N A , Lz

A

ρAN A , L z

A

∝e−∑N A , Lz

A

H AN A , Lz

A

ρAN A , L z

A

Rank of = # of low energy states of CFT !!H AN A , L z

A

ρA=Tr Bψψ̄= ∑N A , L z

A

ρAN A , Lz

A

ρAN A , L z

A

=??

ρAN A(Z A ; Z ' A)=∫dZ Bψ(Z A ; Z B) ψ̄(Z ' A ; Z B)

ψ( z1 , .. , zN )=ψ( z1 , .. , zN A, zN A+1 , .. zN )=ψ(Z A ; Z B) Z A=( z1 , .. , zN A

)

Z B=( z N A+1 , .. , z N )

ρAN A :

ψ(z1 , .. , zN )=∑λ

cλmλ( z1 , .. , zN )

λ=(l1 , .. , l N ):∑il i=L z

ρAN A , L z

A

:

L̂ zmλ (z1 , .. , z N )=L zmλ (z1 , .. , z N )

λ=(l1, .. , l N A, l N A+1 , .. , l N )=λA∪λB

LzA+Lz

B=Lz

ψ(Z A ; Z B)=∑λ

cλmλ(Z A ; Z B)=∑L zA :

( ∑λA , λB

cλ AλBmλA

L zA

(Z A)mλ B

L zB

(Z B))=∑L zA

ξLzA

(Z A ; Z B)

ψ(z1 , .. , zN )=∑λ

cλmλ( z1 , .. , zN )

λ=(l1 , .. , l N ):∑il i=L z

ρAN A , L z

A

:

L̂ zmλ (z1 , .. , z N )=L zmλ (z1 , .. , z N )

ψ(z1 , .. , zN )=∑λ

cλmλ( z1 , .. , zN )

λ=(l1 , .. , l N ):∑il i=L z

ρAN A , L z

A

:

λ=(l1, .. , l N A, l N A+1 , .. , l N )=λA∪λB

LzA+Lz

B=Lz

ψ(Z A ; Z B)=∑λ

cλmλ(Z A ; Z B)=∑L zA :

( ∑λA , λB

cλ AλBmλA

L zA

(Z A)mλ B

L zB

(Z B))=∑L zA

ξLzA

(Z A ; Z B)

ρA(Z A ; Z ' A)=∑N A

ρAN A(Z A ; Z ' A)=∑

N A , LzA

∫ dZ B ξL zA

(Z A ; Z B) ξ̄LzA

(Z ' A ; Z B)= ∑N A , L z

A

ρAN A , L z

A

(Z A ; Z ' A)

L̂ zmλ (z1 , .. , z N )=L zmλ (z1 , .. , z N )

ψ(z1 , .. , zN )=∑λ

cλmλ( z1 , .. , zN )

λ=(l1 , .. , l N ):∑il i=L z

ρAN A , L z

A

:

λ=(l1, .. , l N A, l N A+1 , .. , l N )=λA∪λB

LzA+Lz

B=Lz

ψ(Z A ; Z B)=∑λ

cλmλ(Z A ; Z B)=∑L zA :

( ∑λA , λB

cλ AλBmλA

L zA

(Z A)mλ B

L zB

(Z B))=∑L zA

ξLzA

(Z A ; Z B)

ρA(Z A ; Z ' A)=∑N A

ρAN A(Z A ; Z ' A)=∑

N A , LzA

∫ dZ B ξL zA

(Z A ; Z B) ξ̄LzA

(Z ' A ; Z B)= ∑N A , L z

A

ρAN A , L z

A

(Z A ; Z ' A)

(ρA)Z AZ ' AN A , L z

A

=∫ dZ BξZ AL zA

(Z B) ξ̄Z ' AL zA

(Z B)ThusRSES:

PES:

Z A , Z ' A∈D; Z B∈DcZ A , Z ' A , Z B∈ plane

L̂ zmλ (z1 , .. , z N )=L zmλ (z1 , .. , z N )

# of states with particles and fluxes in the LLL= finite≤ N B N B

ρAN A , Lz

A

(∞×∞)

(ρA)Z AZ ' AN A , L z

A

=∫ dZ BξZ AL zA

(Z B) ξ̄Z ' AL zA

(Z B) P={Z ALzA

Z B ,Z ' ALzA

Z B , ..}overlap matrix of

Two important results:

1) Rank( ) = # of indep. states in P = finiteρAN A , L z

A

Example FQHE:

but Rank( ) = finiteρAN A , Lz

A

Rank( )ρAN A , Lz

A

(I.R., J.Slingerland, S.Simon PRL 2012)

(ρA)Z AZ ' AN A , L z

A

=∫ dZ BξZ AL zA

(Z B) ξ̄Z ' AL zA

(Z B) P={Z ALzA

Z B ,Z ' ALzA

Z B , ..}overlap matrix of

(I.R., J.Slingerland, S.Simon PRL 2012, A.Sterdyniak et al. PRB; Dubail, Read, Rezayi PRB)

2) Rank of PES = Rank of RSES

# low energy states in = # low energy states inH APES H A

RSES

(I.R., J.Slingerland, S.Simon PRL 2012)

# of states with particles and fluxes in the LLL= finite≤ N B N B

ρAN A , L z

A

(∞×∞)

Two important results:

1) Rank( ) = # of indep. states in P = finiteρAN A , L z

A

Example FQHE:

but Rank( ) = finiteρAN A , Lz

A

Rank( )ρAN A , Lz

A

but its rank is finite ! ∞×∞

ξZ ALzA

(Z B) j=1 , .. , d

{Z A}∞ {Z A}d={Z A1 , .. , Z A

j , .. , Z Ad }

ξZ AjL zA

(Z B)=ξ jL zA

(Z B)

(ρA)Z A , Z ' AN A , L z

A

=∫ d Z B ξZ AL zA

(Z B)̄ξZ ' AL zA

(Z B)

Finite dimensional approx. of :

(ρ̃A )Z AZ ' AN A , L z

A

(d×d ) ⊂ (ρA )Z AZ ' AN A , L z

A

(∞×∞)∃a)

(ρ̃A )Z AZ ' AN A , L z

A

=Rank ( ) Rank ( )(ρA )Z AZ ' AN A , L z

A

(I'll obtain it in FQHE in a moment)

∫d Z B ∑=1

r

b)

{Z B}∞ {Z B}r={Z B1 , .. , Z B

r }

ρAN A , L z

A

a) + b) (ρ̃A)ijN A , Lz

A

=∑l=1

r

ξiL zA

(Z Bl)ξ̄ j

L zA

(Z Bl) (d×d )

M il=ξiL zA

(Z Bl) (d×r)Let's define:

(ρ̃A)ijN A , L z

A

=M ilM lj+

a) + b) (ρ̃A)ijN A , Lz

A

=∑l=1

r

ξiL zA

(Z Bl)ξ̄ j

L zA

(Z Bl) (d×d )

M il=ξiL zA

(Z Bl) (d×r)Let's define:

a) + b) (ρ̃A)ijN A , Lz

A

=∑l=1

r

ξiL zA

(Z Bl)ξ̄ j

L zA

(Z Bl) (d×d )

(ρ̃A)ijN A , L z

A

=M ilM lj+

M il=ξiL zA

(Z Bl) (d×r)Let's define:

Rank( ) = Rank( ) = Rank( ) ρ̃AN A , L z

A

M

Eigen( ) Eigen( ) = SVD ( )ρ̃AN A , L z

A

MρAN A , L z

A

≃

ρAN A , L z

A

=∫d Z BξiL zA

(Z Bl ) ξ̄ j

LzA

(Z Bl )→M ij=ξi

LzA

(Z Bj ) ! ! !

ρAN A , L z

A

(large )M

All info in:

M ! ! !

∞→d

(I.R., J.Slingerland, S.Simon 2012)

FQHE: 2d interacting electrons in perpendicular B.

Application to FQHE:

∣l z ∣

rr∼√l

r∼√ l

Angular momentum Real Space

z=x+i y

l z ~zl e−∣z∣

2/4 , l=0,1 , ..

∣z∣=r

{Z Ai }d ; {Z B

j }r= ??

ρ̃AN A , L z

A

=M +M M ij=ξiL zA

(Z Bj )=ξ

L zA

(Z Ai , Z B

j )

Eigen( ) Eigen( )ρAN A , L z

A

≃

Rank( ) = Rank( ) M ρAN A , L z

A

(large M)

M +M

Z Ai=( z1

i , .. , zN A

i)

{Z Ai }d ; {Z B

j }r= ??

LAi=(l1

i , .. , l N A

i) \ l 1

i+..+lN A

i =LzA

∣z∣∼√l

ρ̃AN A , L z

A

=M +M

1st) Points where is high (Metropolis algorithm)

M ij=ξiL zA

(Z Bj )=ξ

L zA

(Z Ai , Z B

j )

Smart choice for (the same holds for ) :{Z Ai }d

∣ψ(Z A , Z B)∣2

2nd)

Z Ai

{Z Bj }r

Eigen( ) Eigen( )ρAN A , L z

A

≃

Rank( ) = Rank( ) M ρAN A , L z

A

(large M)

M +M

Z Ai=( z1

i , .. , zN A

i)

{Z Ai }d ; {Z B

j }r= ??

LAi=(l1

i , .. , l N A

i) \ l 1

i+..+lN A

i =LzA

∣z∣∼√l

ρ̃AN A , L z

A

=M +M

1st) Points where is high (Metropolis algorithm)

M ij=ξiL zA

(Z Bj )=ξ

L zA

(Z Ai , Z B

j )

Smart choice for (the same holds for ) :{Z Ai }d

∣ψ(Z A , Z B)∣2

2nd)

Z Ai

{Z Bj }r

You don't need to generate the whole set of to obtain the rank and eigen.LAi

Important comment:

Eigen( ) Eigen( )ρAN A , L z

A

≃

Rank( ) = Rank( ) M ρAN A , L z

A

(large M)

M +M

d , r∼2 rank (M )

:accurate eigenvalues d~2 rank M , r≫1

If M d×r :clear ranks

Range where you find nice ranks and eigenvalues in FQHE:

Property:

ψ( z1 , z 2, .. , z N ) = ∑L zA

(∑λA ,λBcλ Aλ BmλA

L zA

( z1 , .. , zN A) mλB

L zB

( zN A+1 , .. , zN ))Angular momentum expansion:

λA=(l1 , .. , l N A) \ l1+..+l N A

=l zA λB=(l N A+1 , .. , lN ) \ lN A+1+..+lN=L z

Bandwith

ξL ' z

A

( z1 , .. , zN )=∫ d ϕe2π i L ' z

Aϕψ( z1 e

−2π iϕ , .. , zN Ae−2π i ϕ , zN A+1 , .. , zN )

mλA

L zA

( z1e2π i ϕ , .. , z N A

e2π i ϕ) = e2π i ϕL zA

mλ A

L zA

(z 1 , .. , z N A)

LzA+Lz

B=Lz

ξL ' z

A

(Z B) In FQHE:

∑λA , λB

mλA

L ' zA

( z1 , .. , zN A)mλB

L zB

( zN A+1 , .. , zN )

Property:

ψ( z1 , z 2, .. , z N ) = ∑L zA

(∑λA ,λBcλ Aλ BmλA

L zA

( z1 , .. , zN A) mλB

L zB

( zN A+1 , .. , zN ))Angular momentum expansion:

λA=(l1 , .. , l N A) \ l1+..+l N A

=l zA λB=(l N A+1 , .. , lN ) \ lN A+1+..+lN=L z

Bandwith

ξL ' z

A

( z1 , .. , zN )=∫ d ϕe2π i L ' z

Aϕψ( z1 e

−2π iϕ , .. , zN Ae−2π i ϕ , zN A+1 , .. , zN ) =

mλA

L zA

( z1e2π i ϕ , .. , z N A

e2π i ϕ) = e2π i ϕL zA

mλ A

L zA

(z 1 , .. , z N A)

LzA+Lz

B=Lz

∑L zA

(∑λA , λBcλ AλBmλA

LzA

( z1 , .. , zN A) mλB

L zB

( zN A+1 , .. , zN )) ∫ d ϕ e2π i ϕ(L ' zA−L z

A) =

L' zA+L z

B=L zwith

ξL ' z

A

(Z B) In FQHE:

Some examples:

a) Laughlin

b) Moore-Read

c) 2/5 Jain state

Rank (H ARSES) vs. Lz

A

N=70 N A=35 Rank of the RSES (=PES) for and

a) Laughlin = 1/2 : 1 /2 z1 , .. , zN =∏i j

z i−z j2 e

−∑i

∣z i2∣/4

Rank (ρAN A , L z

A=5)= low energy spec. of (H A

N A , L zA=5)

Rank (H ARSES) vs. Lz

A

N=70 N A=35 Rank of the RSES (=PES) for and

a) Laughlin = 1/2 : 1 /2 z1 , .. , zN =∏i j

z i−z j2 e

−∑i

∣z i2∣/4

Boundary excitations of the system

Rank (H ARSES) vs. Lz

A

# excit: 1 1 2 3 5 7 11 15 ...

Laughlin boundary excitations: Chiral boson Spec.

0 1 2 3 4 5 6 7 ...Lz :

N=70 N A=35 Rank of the RSES (=PES) for and

a) Laughlin = 1/2 : 1 /2 z1 , .. , zN =∏i j

z i−z j2 e

−∑i

∣z i2∣/4

Boundary excitations of the system

Energies of the RSES (large M) for and N A=9N=18

Linear energy spectrum.

LzA

a) Laughlin = 1/2 : 1 /2 z1 , .. , zN =∏i j

z i−z j2 e

−∑i

∣z i2∣/4

H ARSES vs. Lz

A

b) Pfaffian = 1 : Pfaf z1 , .. , z N =Pfaf 1z i−z j ∏i j z i−z je

−∑i

∣z i2∣/4

1,1,3,5,10,16,28,... 1,2,4,7,13,21,35,...

Ranks of the RSES for N=40,

N A=20Ranks of the RSES for N=42,

N A=21

LzA

H ARSES vs. L z

A H ARSES vs. L z

A

νc) Jain = 2/5 : Energies of the RSES

(Same excitations obtained by Jain PRB 84 2011 )

H ARSES vs. Lz

AN A=11N=22 ,

Inside a branch: 1,2,5,10,...

N A=12N=24 ,H ARSES vs. Lz

A

Inside a branch: 1,2,5,10,...

CONCLUSIONS:

- rank = number of independents = finite

- . Entanglement encoded in .

- rank RSES = rank PES .

- I applied the method to Laughlin and Moore&Read (for big systems) obtaining the expected behavior.

- I showed that the RSES for the 2/5 Jain state has a rich spectrum that can be interpreted in terms of composite fermions (more branches than in Wen theory).

ρAL zA , N A

ξiL zA , N A(Z B)

ρAL zA , N A→ρ̃A

L zA , N A=M +M M (d x r )

- overlap matrix of ρAL zA , N A { ξi

L zA , N A(Z B)}∞