Matrix Product State for (CFT-) Fractional Quantum Hall Statesesicqw12/Talks_pdf/Estienne.pdf ·...

Matrix Product State for(CFT-) Fractional Quantum Hall States

Benoit Estiennecollaboration with Z. Papic, N. Regnault and B.A. Bernevig

LPTHEUniversite Pierre et Marie Curie, CNRS

Paris

Dresden 11/2012

Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 1 / 24

1 Motivations

2 Where does this MPS structure come from ?Site dependant MPSSite independant MPS

3 The auxiliary space : CFT Hilbert spaceThe U(1) partThe neutral part

4 MPS matrix elements, and how to compute themThe U(1) partThe neutral part

5 Numerics : does the MPS work ?Hilbert space sizeOverlaps

6 Summary and prospects


Motivations

Fractional quantum Hall states are notoriously difficult to studynumerically. With bigger system sizes :

test the existence (or absence) of a gap

probe the (non-abelian) braiding of quasi-particles

measure the size of quasi-holes (for Read-Rezayi)

collapse of the entanglement spectrum

correlation functions (guiding center structure factor, . . . )

New numerical techniques are highly desirable.


Matrix Product State for FQH trial wave-function

Zaletel and Mong (arXiv :1208.4862) : MPS for Laughlin and MR

|Ψ〉 =∑{mi}

Tr (Bm1Bm2 · · ·Bmn) |m1 · · ·mn〉

where the MPS matrices Bm can be computed analytically (underlyingCFT is non interacting)

Why is this formalism interesting ?

Many quantities (correlation functions, entanglement spectrum,...) can becomputed in the (small) auxiliary space.

⇒ what about FQHS involving interacting CFTs ?relevant for Read-Rezayi states, Halperin, NASS, Gaffnian, Jack states,

generalized parafermionic states, etc...


Where does this MPS structurecome from ?


Start with a trial wavefunction given by a CFT correlator

Ψ(z1, · · · , zN) = 〈V (z1) · · ·V (zN)〉 =∑

mi=0,1

c(m1,··· ,mn) |m1, · · · ,mn〉

with electron operator V (z) in some chiral 1 + 1 CFT .

Insert a complete basis of states∑α1,··· ,αN−1

〈0|V (z1)|α1〉〈α1|V (z2)|α2〉 · · · 〈αN−1|V (zN)|0〉

Project to |m1, · · · ,mn〉

One gets an infinite MPS (on any genus 0 geometry)

c(m1,··· ,mn) = Tr (Bm1 [1]Bm2 [2] · · ·Bmn [n])

Site dependent matrices :

〈α′|B0[j ]|α〉 = δα′,α, 〈α′|B1[j ]|α〉 = δ∆α′ ,∆α+h+j〈α′|V (1)|α〉


Site independant MPS

Uniform background charge ⇒ site independant MPS

B0 = e− i√

qϕ0 , B1 = V0 e

− i√qϕ0

where

ϕ0 is the bosonic zero mode (B0 shifts the electric charge by 1/q)

V0 is the matrix of the electron operator

〈α′|V0|α〉 = δ∆α′ ,∆α+h〈α′|V (1)|α〉

What is required for a numerical implementation ?

build the basis |α〉 (auxiliary space) + truncation scheme

compute the matrix elements 〈α′|Bm|α〉


The auxiliary space |α〉 :CFT Hilbert space


The U(1) Hilbert space

The CFT factorizes H = Hneutral ⊗HU(1)

as a neutral CFT times a U(1) chiral free boson.

ϕ(w) = ϕ0 − ia0 log(w) + i∑n 6=0

1

nanw

−n

Primary states |Q〉 are defined by their U(1) charge Q

a0|Q〉 = Q|Q〉, an|Q〉 = 0 for n > 0

The Hilbert space is simply a Fock space

Descendants are obtained with the lowering operators a†n = a−n, n > 0

|Q, µ〉 =n∏

i=1

a−µi |Q〉, a0|Q, µ〉 = Q|Q, µ〉

with µ1 ≥ µ2 ≥ · · · ≥ µn > 0


The neutral Hilbert spaceLn (modes of the stress-energy tensor) obey the Virasoro algebra

[Ln, Lm] = (n −m)Ln+m +c

12n(n2 − 1)δn+m,0

Primary fields |∆〉 are annihilated by the positive modes

L0|∆〉 = ∆|∆〉, Ln|∆〉 = 0 n > 0

Descendant states : lowering operators L†n = L−n, n > 0

|∆, λ〉 = L−λ1L−λ2 · · · L−λn |∆〉

Two issues :

these states are not orthogonal

they might not even be independant !

⇒ No closed formula, has to be implemented numerically.Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 10 / 24

Building up an orthonormal basis

The level of a descendant |∆, λ〉 is just the size of the partition|λ| =

∑j λj .

e.g. at level 2 we have two states : L2−1|∆〉, L−2|∆〉.

At each level we compute the overlap matrix between descendants〈∆, λ′|∆, λ〉.

e.g. at level 2 we have

(4∆(2∆ + 1) 6∆

6∆ 4∆ + c2

)if positive definite, Gramm-Schmidt

if states with vanishing norm (null-states), they have to be discarded

if states with negative norm (non-unitary CFT), the sign can beabsorbed in a transformation matrix

From CFT the number of null-vectors is known : check for numerics.


MPS matrix elementshow to compute them


The U(1) part

CFT factorization : V (z) = Φ(z)⊗ : e i√qϕ(z) :

where Φ(z) is a primary field in the neutral CFT.

The matrix elements of the vertex operator

〈Q ′, µ′| : e iβϕ(1) : |Q, µ〉 = δQ′,Q+β Aµ′,µ

can be easily computed through the commutation relation[am , e

iβϕ(z)]

= β e iβϕ(z)

Aµ′,µ =∏j≥1

m′j∑r=0

mj∑s=0

(−1)s

r !s!

(β√j

)r+s

δm′j+s,mj+r

√(mj + r − s)!mj !

(mj − s)!

Works for quasiholes too.Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 13 / 24

The neutral part

Matrix element for an arbitrary primary field Φ(h)(z) are of the form

〈∆′, λ′|Φ(h)(1)|∆, λ〉

They can be computed (in principle) using

[Lm − L0,Φ(h)(1)] = mh Φ(h)(1)

where h is the conformal dimension of Φ(h)(z).For instance

〈∆′|L1Φ(h)(1)|∆〉 = (h + ∆′ −∆)〈∆′|Φ(h)(1)|∆〉〈∆′|L2Φ(∆)(1)L−1|∆〉 = (2h + ∆′ −∆− 1)(h −∆′ + ∆)〈∆′|Φ(h)(1)|∆〉

⇒ But no analytical closed formula, has to be implementednumerically.


For more complicated CFTs (parafermions, W algebras, N = 1 susy,S3...), much more involved... but it can be done. For k = 3 Jack states :

The underlying algebra is the W3 algebra :

[Ln, Lm] = (n −m)Ln+m +c

12n(n2 − 1)δn+m,0

[Ln,Wm] = (2n −m)Wn+m

[Wn,Wm] =16

22 + 5c(n −m)Λn+m +

c

360n(n2 − 1)(n2 − 4)δn+m,0

+(n −m)

[1

15(n + m + 2)(n + m + 3)− 1

6(n + 2)(m + 2)

]Ln+m

The matrix elements can be computed using :

〈α′| [Wn,Ψ(1)] |α〉 = Cn(∆,∆α′ ,∆α)〈α′|Ψ(1)Wn|α〉

− 6ω(∆ + 1)

∆(5∆ + 1)

∑m≥1

[〈α′|L−mΨ(1)|α〉+ 〈α′|Ψ(1)Lm|α〉

]Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 15 / 24

Truncation of the auxiliary CFT basis

The auxiliary space (i.e. the CFT Hilbert space) basis is of the form

|Q, µ〉 ⊗ |∆, λ〉

The natural cut-off is the level |λ|+ |µ| ≤ P.

P = 0 recovers the thin-torus limit (Jack root partition)

The cλ are either exact or zero at a given truncation level

In finite size the truncated MPS becomes exact for P large enough

But the overlap is extremely good way before this P(N)


NumericsWhat is the MPS worth ?


Hilbert space VS auxiliary space

0

5

10

15

20

25

30

35

0 5 10 15 20 25 30

log

2(d

im)

Number of particles

Laughlin ν=1/3Moore-Read (k=2,r=2)

Gaffnian (k=2,r=3)Read-Rezayi k=3

Hilbert space size, as a function of N

0

2

4

6

8

10

12

14

16

18

0 2 4 6 8 10

log

2(d

imB

1)

truncation level P



Auxiliary space size, as a function of P

1e-07

1e-06

1e-05

0.0001

0.001

0.01

0.1

0 2 4 6 8 10

no

n z

ero

/ d

im2

truncation level P



Sparsity of the

MPS matrix


Overlaps for the ν = 1/3 Laughlin state

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

0 2 4 6 8 10

1-|

<Ψ

exact|Ψ

MP

S>

|2

truncation level P

Laughlin state ν=1/3 on the cylinder

N=12N=13N=14

0.001

0.01

0.1

1

0 2 4 6 8 10

1-|

<Ψ

exact|Ψ

MP

S>

|2truncation level P

Laughlin state ν=1/3 on the sphere

N=12N=13N=14


Overlaps for the k = 3 Read-Rezayi state

1e-12

1e-10

1e-08

1e-06

0.0001

0.01

1

0 2 4 6 8 10

1-|

<Ψ

exact|Ψ

MP

S>

|2

truncation level P

Read-Rezayi state k=3 on the cylinder

N=15N=18N=21

0.001

0.01

0.1

1

0 2 4 6 8 10

1-|

<Ψ

exact|Ψ

MP

S>

|2truncation level P

Read-Rezayi state k=3 on the sphere

N=15N=18N=21


Conclusion


Summary and prospects

MPS formalism available for (CFT type) FQH trial wavefunctions

On any genus 0 geometry : sphere, plane, annulus, cylinder...

but works much better on the cylinder !

For a large class of CFT- FQH trial wf :I LaughlinI k = 2 Jack states (incuding Moore-Read and Gaffnian)I k = 3 Jack states (including Read-Rezayi)

Prospects : large system sizes

expectation values/correlations of local observables

entanglement spectrum

non-abelian braiding, gap,. . .


Density-density correlation function

0

0.005

0.01

0.015

0.02

0.025

0.03

0.035

0.04

0.045

0 2 4 6 8 10 12 14 16

g(r)

x/lB

|P|max=4|P|max=6|P|max=8

N = 50 Gaffnian (cylinder)for comparison N ≤ 18 for Jacks

0

0.01

0.02

0.03

0.04

0.05

0.06

0 2 4 6 8 10 12 14

g(r)

x/lB

|P|max=4|P|max=6|P|max=8

N = 45 RR (cylinder)for comparison N ≤ 30 for Jacks


Entanglement spectrum

0

5

10

15

20

25

30

-2 0 2 4 6 8 10 12 14 16

ξ

|P|

OES for N = 40 ν = 1/3 Laughlin (cylinder) with a truncation P = 15for comparison N ≤ 17 for Jacks


Matrix Product State for (CFT-) Fractional Quantum Hall Statesesicqw12/Talks_pdf/Estienne.pdf ·...

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