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

CAQMP2019 （ 2019.07.16-08.08@ISSP, Kashiwa ） Introduction 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

Magnetic/Non-Magnetic Transition (Deconfined-Critical Phenomena) Senthil, et al, (2004): SU(2) broken. SU(2) not broken. "Deconfined Criticality" Rotation not broken. Rotation broken. A new type of phase transition. Pair of spinons are unbound as we approach the QCP NK and Tanabe (2007) ＝ Q/J --- proximity of SU(N) model to DCP SU(N) JQ-Model Sandvik (2007)    1 1   --- first direct numerical study =  − −  −     H J S S Q S S S S of DCP i j i j k l    4 4 ( ) = ( ) ij p i j k l , , , Lou, et al (2009) --- higher representation

See Anders Sandvik's lecture and talk on July 22. Large-scale calculation is necessary Harada et al (2013) System-size-restricted finite-size scaling yields strongly size dependent estimates of scaling dimensions

See A. Sandvik and and W.-N. Guo Dangerously Irrelevant Field 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. Monte Carlo calculation of RG flow diagram H. Shao, W.-N. Guo, A. W. Sandvik: arXiv:1905.13640

Negative Sign Problem Monte Carlo Method is a general solution for the problem. ( ) 3 spins on a triangle  =   Z  ( Feynman Paths) ( )    ( )   →  = Metropolis P ( )   However, what if ρ （ π ）＜０ ? 3  ( ) ( )   = − J 0

See M. Imada's lecture, and T. Misawa's talk Variational Monte Carlo Jastrow: PR98 (1955); Gutzwiller: PRL10 (1963); Ceperley, Chester, and Kalos: PRB16 (1977); Sorella: PRB64 (2001); Tahara and Imada: JPSJ77 (2008) 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

Tensor Network Variational Wave-Functions

TN Ansatz for Quantum State ( )     =    Cont , , , T S S S  TN 1 2 N   S , S , , S =  =  =  1 2 N S 1 S 1 S 1 1 2 N T 8 T 7 virtual index T 5 T 6 T 2 physical T T T 4 3 1 (real) index   S S S S N 3 1 2 The state can be specified by only n tensors (i.e., nd k =O(N) variables) compared to d N variables necessary in general.

Variational Tensor Network Simplest TN = (Direct Product State) ) (      =   DPS T i i i i  Always yields some sort of mean-field approximation. Something more complicated but still manageable is necessary.

Nishino may talk about this DMRG S. R. White, PRL 69, 2863 (1992) --- Variational Tensor Network (d=1) (0) Initialization (1) Hamiltonian of sub-systems (2) Hamiltonian of the whole system (3) Ground state (Doable with O(χ 3 )) (5) Renormalization (4) SVD

Vanderstraeten's talk may be related to this Variational Tensor Network (d ≥ 2) P. Corboz: PRB94, 035133 (2016) For corner transfer matrix method, ( contains pair Hamiltonian ) see Nishino and Okunishi, JPSJ65 (1996)

[2] Tensor Network and Neural Network

AI shock 2016.03 Alpha-Go defeated the world go-champion Lee Sedol multi-layer neural network http://www.huffingtonpost.jp/2016 /03/12/alpha-go_n_9444998.html

reinforced learning Neural network = variational principle Why not using NN for quantum many body problems? Corleo and Troyer 2016 Restricted Boltzmann machine Monte Carlo sampling of the energy and its gradient wrt a, b and W , for gradient- descent method. Negative-sign problem!

See Nomura's talk and Imada's lecture Neural network for fermions 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 (Google) the network structure (data-flow graph). Automatic differentiation (AD) does this job, and it is supported by some libraries. (Facebook)

Automatic differentiation (AD) Our calculation (TN included) is always of the form Just a chain rule, but typically very tedious to program.

See Hai-Jun Liao, Lei Wang, and Tao Xiang AD for TN calculation Non-trivial technical problem, Optimize the energy wrt the elementary tensor T related to matrix diagonalization . Forward calculation A -> U and D Hai-Jun Liao, Jin-Guo Liu, How about degeneracy? Lei Wang, and Tao Xiang, Is truncation of D differentiable? arXiv:1903.09650

[3] Basic Properties of Tensor Network States

Classical lattice models are TN ( ) S S 3 = S S 3 S T 4 3 S 4 p 4 S S p T 1 2 p ( ) + + + S K S S S S S S S S S e 1 2 2 3 3 4 4 1 2 2 S S 1 1   =   Z Cont p T   Ising p 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 Foong & Kanno: PRL 72 1148 (1994)    A S log m A log d E e e A Ω m = A ... dim. of Hilbert space of A d  ... average with the invariant Haar measure For a randomly picked-up quantum state,  "entanglement entropy" "volume"

States of interest are not typical "Area Law" A L ( ) ( ) ( ) − − =  d 1 d 1 d S O L or O L log L O L E ( ) d(>1)-dim. free boson/fermion −   = d 1 S a A O L with a finite gap E 1 ( ) = + S E log L const gapless fermionic chain 2 3 Finite-dimensional quantum states are "area law" rather than "volume law" very special

PEPS satisfies area law ( ) ( )    S E 2 # of bonds on the boundary log D ( ) ( ) = = dim. of a bond 1 D O

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

Tensor Network States as Prototype of New Quantum States AKLT state is a MPS AKLT state ... essentially equivalent to the Haldane   1 0 state of S=1 AFH.    =   T   1   0 0   0 1 1      = T   0   1 0  2  + i i 1   0 0      = T   − 1   0 1

Kitaev and Laumann, "Topological phases and quantum computation" (Oxford) Toric Code Square-Lattice Kitaev Model The simplest solvable model discussed heavily in the context of quantum information, topological phase, Majorana fermion, etc. Z p ( )   = − −  H J X J Z J , J 0 x p z p x z X p   p A p B     x X 1 p i  i p     z Z 1 p i  i p

Toric Code State is a TNS  ( )              =   x P Cont T   p i  i i       , i   i i p A i + 1 X ( )      =  p x T 0 or 1 P  i p 2 i   =  =  =        1, iff 1 ,   =   z     j i j i    =  =    1 ,   =   1   j   =    =    1  i i

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

Loop Gas State = Critical Ising ... fully-polarized state in (111) direction Exactly at the critical point.

See H.Y.Lee's seminar KSL is essentially LGS ... ψ0=LGS ψ1 ψ2 KHM gr. st. # of prmtrs. 0 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

[5] Tensor Network based Real-Space Renormalization Group

Simplest RG with TN --- TRG Levin & Nave (2007); Gu, Levin, Wen (2008); Schuch, et al, (2007) Contraction SVD

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