Introduction Naoki KAWASHIMA (ISSP) 2019.07.16 Sponsors Institute - - PowerPoint PPT Presentation

Introduction

CAQMP2019 （2019.07.16-08.08@ISSP, Kashiwa）

Naoki KAWASHIMA (ISSP) 2019.07.16

Sponsors

Institute for Solid State Physics, University of Tokyo MEXT Project "Challenge of Basic Science" MEXT Project "Creation of new functional Devices and high-

performance Materials to Support next-generation Industries "

PCoMS "Program for Training Researchers for the Next Generation" (MEXT，Japan)

[1] Various Numerical Methods for Quantum Many-Body Problems

Quantum Monte Carlo

For a given Hamiltonian we want to compute expectation value of QMC is classical Monte Carlo in the space of world-line configurations

( )

( ) , , ,

1 1 4 4

i j i j k l ij p i j k l

H J S S Q S S S S

=

   =  − −  −       

 

＝Q/J

Magnetic/Non-Magnetic Transition (Deconfined-Critical Phenomena)

SU(2) broken. Rotation not broken. SU(2) not broken. Rotation broken.

NK and Tanabe (2007)

• -- proximity of SU(N) model

to DCP Sandvik (2007)

• -- first direct numerical study
• f DCP

Lou, et al (2009)

• -- higher representation

Senthil, et al, (2004): "Deconfined Criticality" A new type of phase transition. SU(N) JQ-Model

Pair of spinons are unbound as we approach the QCP

Large-scale calculation is necessary

System-size-restricted finite-size scaling yields strongly size dependent estimates of scaling dimensions Harada et al (2013)

See Anders Sandvik's lecture and talk on July 22.

Dangerously Irrelevant Field

See A. Sandvik and and W.-N. Guo

• H. Shao, W.-N. Guo, A. W. Sandvik:

arXiv:1905.13640 Monte Carlo calculation of RG flow diagram Classical 3D XY model with Z6 anisotropy It may be a good idea to think about the classical counterpart. One of the causes of difficulty may be the presence of the dangerously irrelevant field.

Negative Sign Problem

Monte Carlo Method is a general solution for the problem.

( ) ( ) ( ) ( )

       



 =  → =



P Z

Paths) Feynman (

Metropolis However, what if ρ（π）＜０ ?

( ) ( )

3 

− = J  

3 spins on a triangle

Variational Monte Carlo

the Hartree-Fock-Bogoliubov type wave function the quantum-number projector Gutzwiller-Jastrow factor X(β) : a matrix linear in F Computation of Pf(X) requires O(N^3) cpu time See M. Imada's lecture, and T. Misawa's talk

Jastrow: PR98 (1955); Gutzwiller: PRL10 (1963); Ceperley, Chester, and Kalos: PRB16 (1977); Sorella: PRB64 (2001); Tahara and Imada: JPSJ77 (2008)

Tensor Network Variational Wave-Functions

TN Ansatz for Quantum State

( )

  

 =  =  =

 =

1 1 2 1 , , , 1 TN

1 2 1 2

, , , Cont

S S N S S S S

N N

S S S T  

  



physical (real) index virtual index

1

S

2

S

3

S

N

S  

8

T

6

T

5

T

7

T

4

T

3

T

2

T

1

T

The state can be specified by only n tensors (i.e., ndk =O(N) variables) compared to dN variables necessary in general.

Variational Tensor Network

Simplest TN = (Direct Product State)

( )



=  

  

 

i i i i

T DPS Something more complicated but still manageable is necessary. Always yields some sort of mean-field approximation.

DMRG

• -- Variational Tensor Network (d=1)

(0) Initialization (1) Hamiltonian of sub-systems (2) Hamiltonian of the whole system (3) Ground state (4) SVD (5) Renormalization (Doable with O(χ3)) Nishino may talk about this

• S. R. White, PRL 69, 2863 (1992)

Variational Tensor Network (d≥2)

• P. Corboz: PRB94, 035133 (2016)

(

contains pair Hamiltonian) Vanderstraeten's talk may be related to this For corner transfer matrix method, see Nishino and Okunishi, JPSJ65 (1996)

[2] Tensor Network and Neural Network

AI shock

http://www.huffingtonpost.jp/2016 /03/12/alpha-go_n_9444998.html

2016.03 Alpha-Go defeated the world go-champion Lee Sedol multi-layer neural network

Neural network

Corleo and Troyer 2016 Restricted Boltzmann machine

reinforced learning = variational principle

Monte Carlo sampling of the energy and its gradient wrt a, b and W, for gradient- descent method.

Why not using NN for quantum many body problems?

Negative-sign problem!

Neural network for fermions

See Nomura's talk and Imada's lecture Extension to multi-layer (deep) neural networks was also investigated.

• G. Carleo, Y. Nomura, and M. Imada, Nat. Commun. 9, 5322 (2018)

Libraries for neural networks

Back-propagation is tedious in NN coding. If this part can be taken care of by some software, the user only needs to design the network structure (data-flow graph). Automatic differentiation (AD) does this job, and it is supported by some libraries.

(Google) (Facebook)

Automatic differentiation (AD)

Our calculation (TN included) is always of the form

Just a chain rule, but typically very tedious to program.

AD for TN calculation

See Hai-Jun Liao, Lei Wang, and Tao Xiang Hai-Jun Liao, Jin-Guo Liu, Lei Wang, and Tao Xiang, arXiv:1903.09650 Optimize the energy wrt the elementary tensor T

How about degeneracy? Is truncation of D differentiable?

Non-trivial technical problem, related to matrix diagonalization . Forward calculation A -> U and D

[3] Basic Properties of Tensor Network States

Classical lattice models are TN

      =

p p T

Z Cont

Ising

1

S

2

S

3

S

4

S

1

S

2

S

3

S

4

S

p

T

p

( )

( )

1 4 4 3 3 2 2 1 3 4 2 1

S S S S S S S S K S S S S p

e T

+ + +

=

Contraction techniques for quantum states can be used for solving classical problems

TN for Quantum States --- PEPS

   

i



• F. Verstraete, J. I. Cirac, cond-mat/0407066

Nishino, Hieida, et al., Prog. Theor. Phys. 105 (2001). Nishio, Maeshima, Gendiar, Nishino, cond-mat/0401115

Entanglement of typical state

log log

A E e e

S m A d   

Foong & Kanno: PRL 72 1148 (1994)



... average with the invariant Haar measure ... dim. of Hilbert space of A

A

d m =

Ω A

"entanglement entropy" "volume"



For a randomly picked-up quantum state,

( ) ( ) ( )

d d d E

L O L L O L O S  =

− −

log

• r

1 1

( )

const L SE + =

2

log 3 1

( )

1 −

=  

d E

L O A a S

A

L

Finite-dimensional quantum states are very special

"Area Law"

States of interest are not typical

d(>1)-dim. free boson/fermion with a finite gap gapless fermionic chain

"area law" rather than "volume law"

PEPS satisfies area law

( ) ( )

1 bond a

• f

dim. O D = =

( ) ( )

D SE log boundary

• n the

bonds

• f

# 2   

[4] Tensor Network States as Prototype of New Quantum States

AKLT state is a MPS

i



1 + i



        =         =         =

 −  

1 1 1 2 1 1

1 1      

T T T

Tensor Network States as Prototype of New Quantum States

AKLT state ... essentially equivalent to the Haldane state of S=1 AFH.

p

Z

p

X

Toric Code

Square-Lattice Kitaev Model

Kitaev and Laumann, "Topological phases and quantum computation" (Oxford)

( )

1 1 ,      − − =

   

    p i z i p p i x i p z x B p p z A p p x

Z X J J Z J X J H  

The simplest solvable model discussed heavily in the context of quantum information, topological phase, Majorana fermion, etc.

Toric Code State is a TNS

 

i A p x p

P           





   

1 2 1  = = + 

j i j i z j p x p

X P     

    

( )

           =  =  =  =   = = = =

 

1 , 1 , 1 iff 1, 1

• r

       

    i i i

i

T

 (

)

 

 

i i i

i i i

T 

  



 =

,

Cont

Kitaev Honeycomb Model

Exactly solvable model for gapless spin liquid

Kitaev, Ann. Phys. 321 (2006) 2 See Hyun-Yong Lee's seminar. The fact that exact solution is described in terms of Majorana fermions makes its physical picture hard to comprehend.

Loop Gas State = Critical Ising

Exactly at the critical point.

... fully-polarized state in (111) direction

KSL is essentially LGS ...

ψ0=LGS ψ1 ψ2 KHM gr. st. # of prmtrs. 1 2 E/J

• 0.16349
• 0.19643
• 0.19681
• 0.19682

ΔE/E 0.17 0.02 0.00007

• Best accuracy by numerical calculation is achieved

with only two tunable parameters

See H.Y.Lee's seminar

[5] Tensor Network based Real-Space Renormalization Group

Simplest RG with TN --- TRG

Levin & Nave (2007); Gu, Levin, Wen (2008); Schuch, et al, (2007)

SVD Contraction

Low-rank approximation

Eckhard-Young-Mirsky theorem: We can obtain the best LRA by truncating the small singular values.

HOTRG

• Z. Y. Xie et al, PRB86 (2012)

Corboz, Rice, Troyer, PRL 113 (2014) See S. Morita's talk and T. Sakurai's talk

C(X) =

Solution (Corboz, Rice, Troyer, PRL 113 (2014))

• 1. QR-decomp.
• 2. SVD + truncation
• 3. Projectors
• 4. LRA

Improvement of TRG



Optimization condition for u, v and w RG transformation: converges faster when D increased can get rid of local entanglement

Evenbly and Vidal: Phys. Rev. Lett. 115, 180405 (2015)

Removal of Ent. Loops

By pinching the "information path", we can split the remaining loop, and remove them at the next contraction.

It is essential that this line is thin.



Evenbly and Vidal: PRL115, 180405 (2015) Also see, Yang, Gu, and Wen: PRL118, 110504 (2017), and Harada: PRB97_045124 (2018)

[6] Tensor Network for Machine Learning

Machine learning

Stoundenmire and Schwab: Advances in Neural Information Processing Systems 29, 4799 (2016) b-dim. test training m= 10 0.05 0.05 m= 20 0.02 0.02 m=120 0.0097 0.0005

MPS and DMRG-like optimization can be used for pattern recognition.

Tensor Ring for Classification

COIL100 2D image classification task 128 x 128 x 3 x 7200 bits "open chain" "ring" Ring decomposition shows better performance.

maximum bond dim. average bond dim. tolerated error score （%) (large training) score （%) (small training)

Zhao, Cichocki ら arXiv:1606.05535 KNN classifier (K=1) applied to the image specifier core (Z4).

... but can we do that easily?

"Entanglement loop" problem

In TRG algorithm, a loop structure emerges in the tensor network. Such a loop entanglement is

• redundancy. (No effect on the outside.)

However, once such a loop appears, it never vanishes.

Alternating Least Square (ALS)

Z1 Z2 Z3 Z4

(1) random initial tensors Zi (2) for i=1,2,3,4, update Zi by 2

T

Z1 Z2 Z3 Z4

－

min

Zi (3) repeat until the error converges

However, ALS is trapped by local loops.

Ring Decomposition by Index Splitting

( )

, , i pq i I p q

  = 𝑓𝑗𝜚

When the given tensor T is a CDL, i.e., it must have the following form:

T

U V W

=

This tells us how we can

• btain the ring decomp.

H.-Y. Lee and N.K. arXiv:1807.03862

4 16 16 16 16 4 4 4

2D Ising Model above Tc

[7] Applications

Application --- kagome

Okuma, et al: Nat. Comm. (2019) Experiments + TN (PEPS + CTM) Characterization of magnetic plateaus. Effect of DM interaction.

• T. Okubo and Z. Hiroi may talk about this.
• Standard test case for the frustrated systems
• Nature of the ground state is heavily debated

Gapped/Gapless? What kind of spin liquid?

TN for dynamical systems

"Directed percolation" (Wikipedia) System with essential anisotropy (ν⊥≠ν||) Related topic: Kenji Harada (17th morning) 1/z≒0.63 Rényi entropy can be estimated! TEBD with TN representation

TNRG for surface physics

• -- Boundary TRG (BTRG)

Surface physics can be dealt with.

• S. Iino's talk at symposium,

also W.-N. Guo will talk about QMC approach on surface physics

surface magnetization scaling dimensions

Iino, Morita, NK, arXiv:1905.02351

Towards QCD phase diagram

Phase diagram of Finite-density QCD (early universe, neutron star, etc) ・ 1+1D relativistic fermion + U(1) gauge field (Schwinger model) ・ the next target may be 2+1D SU(2) gauge model

See Kuramashi's talk 2D φ4 model 4D Ising model

Summary

■Quantum Monte Carlo

• deconfined criticality
• dangerously irrelevant field

■Variational Monte Carlo

• many variables
• fermions

■Tensor Network

• Real space RG
• Variational optimization
• Removal of loop entanglement
• automatic differentiation

■Machine Learming

• Image compression
• Pattern recognition

■New Types of Quantum States

• AKLT
• Toric code
• Kitaev SL
• frustrated spins (Kagome, J1-J2, etc)

There'll be many other interesting topics.

Enjoy the event!