数値くりこみ群の発展と エンタングルメント

新潟大理 奥西 巧一

基研 16/12/2014

Numerical renormalization

group

computational cost/

entanglement geometry

Network structure

density matrix/ entanglement sepctrum

AdS/CFT holography

Quantum manybody systems

History ---numerical RG for quantum systems---

1992 DMRG White

1968 Wilson RG Baxter MPS/CTM 1971 Wilson’s NRG

Kondo problem 1987 VBS state/AKLT model

matrix product state(MPS)

2007 MERA Vidal

2003 iTEBD Maeshima,Vidal 2004 PEPS(anisotropic TPS)

MPS + variation: Ostlund-Rommer MPS ~ Schmitt decomp.

2000 TPS( tensor product state)+variation

Verstraete, Cirac

2014 TNR Evenbly

2007 TRG Levin Nave, Xiang

Wilson’s NRG (1971)

Kondo impurity problem

Add free electrons and project out the higher energy states

)( 111

NNNNNN cccctHH +

+++

Λ+

Λ +Λ+=

Λ(>1) is a cut off parameter, which controls the energy scale of the system.

1D quantum system with the boundary

Λ itself comes from log-discretization of fermi sea

k-space real-space

block spin transformation RG (~90)

trace out

rescale

BUT, wavefunction

≠

true ground state: nodeless

disconnected

S.R.White. Phys.Rev.Lett. 69, 2863(1992)

The best approximation (minimization of the deviation)

Singular value decomposition

MPS structure enables us to guess the total wavefunction

+

RLRiji

LijiRL vu ,,,,

,,, αα

ααλψψψ ∑∑ ⊗≈⊗=Ψ

jik

jkikji UU ,2*

,,*,, ααα λψψρ == ∑

Singular Value Decomposition

=

mmm

m

mnmn

m

nmn

m

vv

vv

uu

uu

aa

aa

1

1111

1

111

1

111

λ

λ

+Λ= VUA Λ 特異値、 U, V 特異ベクトル

2|||| XA − 2乗残差を最小化するようにXを作る SVDの小さいものを 捨てることと一致

IUUIVV == ++ ,

RLA ,Ψ= SVD ～ エンタングルメントの最大化 T. Xiang

～

量子情報 ： シュミット分解

MPS & Entanglement

jik

jkikji UU ,2*

,,*,, ααα λψψρ == ∑

: Entanglement spectrum )exp(2αα ελ −=

density matrix eigenvalue

∑== 22 lnln λλρρTrS

Entanglement entropy maximum

VBS state MPS

=

λ

エンタングルメントスペクトル（密度行列の固有値）

N→∞の固有値スペクトル 縮退構造は可積分性を反映

XXZ鎖（Δ=2）

∑ +++ ∆++=n

zn

zn

yn

yn

xn

xn SSSSSSH 111

q= exp(-2λ)

∏ +=

++++≈=

)1(....1][ 22

nqqqqTrZ ρ

1D quantum and 2D classical Trotter formula — path integral(summation) representation---

MHM

HM

NH oe eee ][lim

βββ −−

∞→− = 0],[ =oe HH

we

we

we

we

we

we

Wo Wo

Wo Wo

M.Suzuki, Prog.Theor.Phys.56.1454(1976)

1,,

,+

−= iioeM H

oe eWβ

x

τ

1D quantum ~ anisotropic limit of 2D classical

M, β ->∞

Transfer matrix

∑•

=all

Z

=T

Thermodynamic behaviors => the largest eigenvalue of T

row-to-row transfer matrix

W Bolztmann weight )(2121

221121212)( ssssssssessssW ′+′+′′+−=′′β

NNN TTrZ Λ≈= ∞→ )(lim

MPS form of the eigenvector

0lim Ψ=Ψ ∞→N

N T Max eigenvalue-eigenvector

0ΨNT

∑ += +

∏=Ψ}{

1,1

),(1

µµµ ii

N

issF

ii

),( 1, 1 ++= ii ssF

ii µµ

is

1+is 1+iµ

iµ

block spin variables µ

is1+is

1−is

2+is

He write the eigenvector as a matrix product form.

.

.

.

.

R.J. Baxter, J.Math.Phys (1968) J. Stat. Phys. 19, 461(1978)

A, B: CTM

Rindler vs DMRG

x

t

][/]ˆ[ˆ 22 KiKi eTreOTrO ππ=ΦΦ

S. Lukyanov-Zamolodochikov, Nucl. Phys. B 493, 571 (1997) Expectation vale of Sine-Gordon model

K: Lorentz boost operator

Rindler coordinate: constantly accelerating observer

Kie πρ 2=

Flat space-time

Cf. CTM/DM K: rapidity shift operator

uKe−~

Matrix -> Tensor

Direct variational computation(imaginary time evolution)

Higher Dimension

Higher order singular value decomposition

Projected entangled pair state(PEPS) extension of VBS state

Z.Xie etal, PRB 86, 045139(2012)

Slow-decay of singular value (no equivalence of matrix diagonalization)

Tensor product state(TPS) Nishino&KO,1997,2000

Maeshima, Vidal,….

Verstraete, Cirac cond-mat/0407066 2004

boundary law for MPS/TPS

S ~ (number of bond) x (EE of each bond)

Gapful!

1−∝ dL

area law for MPS/TPS

S ~ (number of bond) x (EE of each bond)

Gapless!

LLd log∝

LcS log6

=

1D : CFT

Calabrese-Cardy

MERA Multiscale entanglement renormalization ansatz

MERA network

disentangler=(entangler)-1

(unitary) Isometry (coarse graining)

Scaled layer structure

)( 9~8χOComputational cost

G.Vidal, Phys. Rev. Lett.99 220402(2007)

Ｔｒｅｅ network = block spin tranformation

Entanglement between two blocks is maintained by only single bond!

Role of disentangler

cutoff plays an essential role (if there is no cutoff, MERA network is just a unitary transformation).

A unitary matrix does not change physics!

A microscopic length scale is assigned to each layer.

This enables us to extract the scale dimensions

[ disentengler, isometry] ≠ 0

Unitary operation

MERA is a most proper Wilson RG!

Entanglement is maintained by the total network structure

Satisfies the area law of entanglement entropy

But, update of tensors can be local. disentangler is unitary!

Local update of tensors finite size sweep in DMRG

disentangler=(entangler)-1

(unitary) Isometry (coarse graining)

What is the role of the extra dimension in MERA?

MERA-net work is just a representation of the ground state wavefunction

•Tensor is a classical object?

•Role of time direction? •Other networks?

•beyond correspondence

Questions

continuous MERA

disentangler=(entangler)^-l Local unitary: K(u)

?

Continuous limit

beyond free field theory

IR(disentrangle) UV(entangled)

Variation should be take each RG steps

Haegeman,Osborne, Henri, Verstraete, PRL110, 100402(2013) Nozaki,Ryu,Takayanagi JHEP10(2012)193

Can entanglement define geometry? Distance between information involved in quantum fields

Tensor RG

RG transformation for tensor(Boltzmann weight)

Double line tensor = Fixed point

=

= ～ CTM ^ 4

～ ⇒

~MPS Tree tensor network

Tensor Network Rg: arXive:1412.0732 TNR has RG flow ⇒ are able to the critical point

Insert disentangler

Fixed point(double line tensor) あまり良くないと思う

Summary

Quantum circuit

e.g. FFT for free fermion A.J.Ferris, arXiv:1310.7605

Free fermion Fermi liquid

?