entanglement spectrum in real space in the fqheesicqw12/talks_pdf/rodriguez.pdf · h=∑ i=1 n...
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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)}∞
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