matrix product state for (cft-) fractional quantum hall statesesicqw12/talks_pdf/estienne.pdf ·...
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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
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 2 / 24
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.
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 3 / 24
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...
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 4 / 24
Where does this MPS structurecome from ?
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 5 / 24
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〉
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 6 / 24
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〉
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 7 / 24
The auxiliary space |α〉 :CFT Hilbert space
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 8 / 24
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
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 9 / 24
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.
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 11 / 24
MPS matrix elementshow to compute them
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 12 / 24
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.
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 14 / 24
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)
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 16 / 24
NumericsWhat is the MPS worth ?
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 17 / 24
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
Laughlin ν=1/3Moore-Read (k=2,r=2)
Gaffnian (k=2,r=3)Read-Rezayi k=3
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
Laughlin ν=1/3Moore-Read (k=2,r=2)
Gaffnian (k=2,r=3)Read-Rezayi k=3
Sparsity of the
MPS matrix
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 18 / 24
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
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 19 / 24
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
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 20 / 24
Conclusion
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 21 / 24
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,. . .
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 22 / 24
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
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 23 / 24
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
Benoit Estienne (LPTHE) Matrix Product State for FQHS Dresden 11/2012 24 / 24