Entanglement entropy inquantum Hall states

Kareljan SchoutensInstitute for Theoretical Physics

University of Amsterdam

IPAM - Feb 2007

References

with M. Haque (Dresden), O. Zozulya (UvA),cond-mat/0609263, PRL 98 (2007) 060401;in preparation

Outline

1. Interpreting entanglement2. Bipartite entanglement entropy and area law3. Many-body entanglement4. Laughlin on the sphere5. Orbital partitioning — topological order6. Particle partitioning — exclusion statistics

• 1935 — Schrödinger introduces `Verschränkung’ as key notion in quantum mechanics.

Interpreting entanglement

``… I would not call that one but rather thecharacteristic trait of quantum mechanics, theone that enforces its entire departure fromclassical lines of thought. By the interaction thetwo representatives [the quantum states] havebecome entangled.’’

• 1935 — Schrödinger introduces `Verschränkung’ as key notion in quantum mechanics

• since 1980s — entanglement recognized as key resource for quantum communication and quantum computation

• since 1990s — entanglement measures considered for characterizing nature of quantum states of matter

Interpreting entanglement

• measure of mutual entanglement between parts of quantum system

- system partitioned into A and B blocks- B degrees of freedom traced out:- entropy defined as

• example: two spins 1/2

Bipartite entanglement entropy

!

" = #$

!!BAtr=

!

" =1

2#$ ± $#( )

!

SA

= "tr #Aln#

A[ ] = SB

!

SA

= 0 2ln=AS

• for spatial partitioning: expect in general that entanglement entropy will be proportional to area separating the A and B blocks, leading to

• exceptions to the area law, as well as sub-leading corrections, are tell-tale of the quantum many-body state

Area law for bipartite entanglement entropy

1!"

d

AALS

Bipartite entanglement entropies of a quantummany-body state can shed light on

• quantum criticality

• topological order

• correlations

• …

Many-body entanglement

1D critical systems

Holzhey-Larsen-Wilczek 1994 Vidal-Latorre-Rico-Kitaev 2003

Calabrese-Cardy 2004

Entanglement and criticality

2D critical systems: subleading logarithm in area law

Fradkin-Moore 2006

)ln(3

AAL

cS =

!

SA = 2 fs (L /a)+"c ln(L /a)

Topological order in a 2D system is captured by asub-leading term in the dependence of spatialentanglement entropy on the radius LA of the A block

Topological entanglement entropy

)( 1!+!=

AAALOLS "#

The topological entanglement entropy is related o thetotal quantum dimension according to

!

" = lnD

!

D = di

2

i"

Kitaev-Preskill 2006, Levin-Wen 2006

Subleading terms in entanglement entropy forparticle partitioning hold clue tocorrelation effects and exclusion statistics.

Entanglement and correlations

!

nA, n

B= N " n

A

!

SA" ln

N# +1

nA

$

% &

'

( ) * "

1

N

m "1

mnA(n

A"1)+O(

1

N2)

For the ν=1/m Laughlin states

How entangled is a fqH state?

• entanglement among spatial regions?

• entanglement among constituent particles?

!

"(z1,K, zn ) = (zi # zj )m

i< j

$ e#

|zi |2

4i

%

• N fermions in spherical geometry, filling factor ν=1/m• monopole at center provides magnetic field; total flux

• eigenstates of orbital angular momentum localized on latitude lines → Lowest Landau Level orbitals

• -th orbital, localized at distance from north pole• spherical projection onto plane gives standard Laughlin wavefunction

Laughlin wave functions

!

N" = m(N #1)

!

" l

!

"(z1,K, zn ) = (zi # zj )m

i< j

$ e#

|zi |2

4i

%

!

l

!

l

!

l = 0,1,...,N"

• orbital partitioning A-block — orbitals

B-block — orbitals

• boundary between orbitals and located

at ; expect asymptotic behavior

Orbital partitioning

!

l = lA,... N"

!

l = 0,... lA"1

!

r = lA

!

lA"1

!

lA

!

SlA" #$ +% l

A

!

B!

A

LLL orbitals

!

l = 2

!

l = 3

!

l "1

!

l

!

r = l

!

" r

!

|" |2

LLL orbitals

!

l = 2

!

l = 3

!

l "1

!

l

!

r = l

!

" r

!

|" |2

block A block B

!

"1

l

!

"1

Orbital partitioning

extrapolating

for fixed at m=3

Orbital partitioning -- extrapolating N → ∞

!

N"#

!

lA

Orbital partitioning -- extracting γ

best fit to

10.060.0 ±=!

!

SlA" #$ +% l

A

!

" 3 = log 3 = 0.55

from TQFT

• orbital partitioning A-block — orbitals

B-block — orbitals

• combination

gives topological entanglement entropy if

Orbital partitioning, II

!

l = lA,... l

B"1

!

l = 0,... lA"1

!

SB" S

A" S

AB# $

!

0 << lA

<< lB

<< N"

!

B

!

A

• if sufficient accuracy can be achieved, the method allows the extraction of the total quantum dimension from finite-size wave-functions

• particularly interesting for wavefunctions obtained from exact diagonalization of realistic potentials:

- direct probe of (universal) topological order- alternative to overlap with model wavefns

• work in progress [Haque, Zozulya, KjS]

Perspective

!

B!

A

• particle partitioning A-block — particles

B-block — particles

• maximal entropy for fermions in orbitals

• correlations in the many-body state lead to reduced value of

Particle partitioning, I

!!"

#$$%

& +'

A

A

n

NS

1ln

(

!

N" +1!

nA

!

(N " nA)

!

nA

!

SA

• example: m=3, N particles, 3N-2 orbitals, 1-particle states carry orbital angular momentum

→ possible states at of fermions:

→ total number of states

→ of these the multiplet at is absent, reducing the total number of states to

Particle partitioning, II

!

L = (3N " 4), (3N " 6), ..., 1(0)

!

nA

= 2

!

1

2(3N " 2)(3N " 3)

!

1

2(3N " 2)(3N " 3)"[2(3N " 4)+1]=

1

2(3N " 4)(3N " 5)

!

L = 3(N "1) /2

!

L = 3N " 4

• general: using quasi-hole counting results [Read-Rezayi,1996] we improved the upper bound

• interpretation in terms of exclusion statistics: the reduced number of states is precisely equal to the number of ways particles can be put into orbitals, observing a minimal distance of m between adjacent occupied orbitals

Particle partitioning, III

!!"

#$$%

& '''+(

A

A

A

n

nmNS

)1)(1(1ln

)

!

nA

!

N" +1

!!"

#$$%

& +'

A

A

n

NS

1ln

(

particle partitioning for particles

[ free fermion bound, improved bound]

Particle partitioning, IV

!

nA

= 2

• further interpretation: comparing with the free fermion bound we have a expansion

• in this concrete case, the leading correction term can be traced to the short distance behavior of the 2-body correlator

• more generally, the leading term is a global measure for the leading correlations in a state of interacting fermions

Particle partitioning, V

!

SA" ln

N# +1

nA

$

% &

'

( ) * "

1

N

m "1

mnA(n

A"1)+O(

1

N2)!

1/N

!

g2(r)

!

1/N

• for the Moore-Read state at ν=1/m, m even

rigorous upper bounds

Particle partitioning, VI

!

SA" ln

N# +1

nA

$

% &

'

( ) * "

1

N2

3

4nA(n

A"1) (n

A" 2)+O(

1

N3) for m = 2

"1

N

m " 2

mnA(n

A"1)+O(

1

N2) for m > 2

conclusions

Bipartite entanglement entropy with orbital partitioning revealstopological order in a given LLL wave function. Practical use ofthis hinges on the accuracy that can be achieved with finitesize wavefunctions.

Bipartite entanglement entropy with particle partitioning revealscorrelations and exclusion statistics properties satisfied by fqHwave functions.