2021/12/08「富岳」成果創出加速プログラム 物質・材料系課題 合同研究会

人工ニューラルネットワークを用いた量子多体ソルバー
野村悠祐

理研CEMS

共同研究者:
Andrew S. Darmawan, Youhei Yamaji, Masatoshi Imada,Giuseppe Carleo, Nobuyuki Yoshioka, and Franco Nori

YN, A. Darmawan, Y. Yamaji, and M. Imada, Phys. Rev. B 96, 205152 (2017)
G. Carleo, YN, and M. Imada, Nat. Commun. 9, 5322 (2018)
YN J. Phys. Soc. Jpn. 89, 054706 (2020) [Editorʼs choice]
YN J. Phys.: Condens. Matter 33, 174003 (2021) [special topic “Emerging Leaders 2020”]
YN*, N. Yoshioka*, and F. Nori, Phys. Rev. Lett. 127, 060601 (2021)
YN and M. Imada, Phys. Rev. X 11, 031034 (2021)

Quantum many-body wave functions H = E<latexit sha1_base64="y6n3eg99vNWUTo+3BsgQPtc6XDM=">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</latexit>

BCS wave function
BCS<latexit sha1_base64="CT62Cuu4QZA/MGhreYs3MBGMZxc=">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</latexit>

↵

=

Y
k

(uk + vk c† k" c†k# ) 0

Laughlin wave function↵

Laughlin =i>j<latexit sha1_base64="5WPFr4cbd0nP9dySMHjnszc1Ap0=">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</latexit>

conventional superconductivity

Y

zi

zj

m

exp

"

1 X
|zi |224lB
i#fractional quantum Hall eﬀect

J. Bardeen, L. N. Cooper, and J. R. Schrieﬀer,
Phys. Rev. 108, 1175 (1957).

R. B. Laughlin, Phys. Rev. Lett. 50, 1395 (1983)

RVB wave function
RVB<latexit sha1_base64="pE3Q9WeyYqUs0WtzDG9cC0JCg98=">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</latexit>

↵

= PG

⇣X
i,j

"# † †
fijci" ci#⌘Ne /2 ↵0quantum spin liquid
P. Anderson, Materials Research Bulletin 8, 153 (1973).

Accurate representation of wave functions : challenge in physics as well as in quantum chemistry2

Machine Learning
Modeling unknown relationship between datay(x) = F[ (x), x]<latexit sha1_base64="jk55SSHw3gm0NIWxd+6hvlRNxnI=">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</latexit>x, y(x) : data<latexit sha1_base64="qKZhg8U0H80fzxHjn5PKsZiAwTg=">AAACenichVHLSsNAFD2Nr1pfVREEN8GiKEq5kYLiQgQ3Lm1rVbBSkjitwTQJSVqswR/wB1y4UhBRP8ONP+DCTxCXFdy48DYNiIp6Q2bOPXPPnTMzmmMank/0FJM6Oru6e+K9ib7+gcGh5PDIlmfXXF0UdNu03R1N9YRpWKLgG74pdhxXqFXNFNva4VprfbsuXM+wrU2/4Yi9qlqxjLKhqz5TpeRYUNTK8tHJvFyUGzNRMltKpihNYcg/gRKBFKLYsJPXKGIfNnTUUIWABZ+xCRUef7tQQHCY20PAnMvICNcFTpBgbY2rBFeozB7yWOFsN2Itzls9vVCt8y4m/y4rZUzRI91Qkx7ojp7p/ddeQdij5aXBs9bWCqc0dDqef/tXVeXZx8Gn6k/PPspYCr0a7N0JmdYp9La+fnzWzC/npoJpuqQX9n9BT3TPJ7Dqr/pVVuTOkeAHUL5f90+wtZBWKK1kM6nVlegp4pjAJGb4vhexinVsoMD7BrjALe5i79KkNCvNtUulWKQZxZeQMh+1w5Fv</latexit>

F ：functional (nonlinear)<latexit sha1_base64="1NabSqKGSCYygjGMZcve+1h8Gvs=">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</latexit><latexit

Application to quantum many-body problems

H| i = E| i<latexit sha1_base64="6U6vIKPNQAk5aO44uIcfIDT5Gqc=">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</latexit>

Hamiltonian<latexit sha1_base64="SqhNy17yX9iTNkSLOhMb/MnzaJU=">AAACanichVG7SgNBFD1Z3/GRqI1iE4wRq3BXBMVKsLGMj5hAEsLuOtEl+2J3E9TgD9hZCaZSEBE/w8YfsPATRLsINhbebBZEg3qHmTlz5p47Z2ZUx9A9n+gpIvX09vUPDA5Fh0dGx2Lx8Yldz665mshqtmG7eVXxhKFbIuvrviHyjisUUzVETq2ut/dzdeF6um3t+EeOKJnKvqVXdE3xmco1imolcXhSjicpTUEkuoEcgiTCyNjxGxSxBxsaajAhYMFnbECBx60AGQSHuRIazLmM9GBf4ARR1tY4S3CGwmyVx31eFULW4nW7pheoNT7F4O6yMoEUPdItteiB7uiZPn6t1QhqtL0c8ax2tMIpx06ntt//VZk8+zj4Uv3p2UcFK4FXnb07AdO+hdbR14/PW9urW6nGPF3RC/u/pCe65xtY9TftelNsNRHlD5B/Pnc32F1My5SWN5eSa0vhVwxiBrNY4Pdexho2kEE2cHeGCzQjr9KENC3NdFKlSKiZxLeQ5j4BUu6MYg==</latexit><latexit

xEigenstate (vector with exponential large dimension)：Basis of the Hamiltoniany(x) ：Amplitude of quantum many-body wave function [Ψ(x)]<latexit sha1_base64="ZWRX6ARps9h5jVhypRnBBlW1y34=">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</latexit>

Extract essential pattern of wave functions → accurate representations with a ﬁnite # of parameters
3

Using artiﬁcial neural network to solve quantum many-body problems
G. Carleo and M. Troyer Science 355, 602 (2017)

Mag. Field (bias term) bj

RBM wave function
(

z

)=

X

exp

⇣X

ai iz +i{hj }Interaction W ij

Xzi Wij hj +

i,j

=e

ERBM ( z ,h)Xb j hjj⌘(Boltzmann weight)<latexit sha1_base64="8H4cSdC/BW2uFmY8GOScJhp65Bc=">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</latexit>

z

=

z
1,

z
2, . . . ,zN

hj = ±1Mag. Field (bias term) ai: real space spin conﬁg.
: spin of hidden units

G. Carleo and M. Troyer Science 355, 602 (2017)

Learning quantum states: Optimization of RBM using loss function (Energy)
Quantum correlations among physical spins via artiﬁcial neural network
Can represent any state with inﬁnite M (# of hidden units): universal approximation
(

z

)=e

P

i ai

z
i

⇥

Y
j

⇣

2 cosh bj +Xi

Wij iz⌘4

Extensions to various systems
Boson system

Fermion system

Spin system (localized Mott insulator)
Itinerant electrons in solids

without frustration（Heisenberg model）
Carleo and Troyer (2017), …

（Hubbard model)

Electron-phonon coupling

YN et al., (2017), …

（Holstein model)

with frustration（J1-J2 Heisenberg model）
YN and Imada (2021), …

YN JPSJ (2020)

Editorʼs Choice

Molecules（H2, LiH, …）Han et al., Choo and Carleo, …

Bose-Hubbard model（itinerant）
Saito (2017), …

From benchmark to “true” applications to challenges in physics(e.g., frustrated spin systems, fermion system, in which QMC suﬀers from sign problem)※ Why is the machine learning powerful？
→ Exact quantum-classical mapping using deep Boltzmann machine (c.f. numerical mapping using RBM)
G. Carleo, YN, and M. Imada, Nat. Commun, 9. 5322 (2018)5

Extensions to various systems

Boson system

Fermion system

Spin system (localized Mott insulator)

Itinerant electrons in solids

without frustration（Hubbard model)（Heisenberg model）
Carleo and Troyer (2017), …

Electron-phonon coupling

YN et al., (2017), …

（Holstein model)

with frustration

YN JPSJ (2020)（J1-J2 Heisenberg model）

Editorʼs Choice

YN and Imada (2021), …

Molecules（H2, LiH, …）Han et al., Choo and Carleo, …

Bose-Hubbard model（itinerant）
Saito (2017), …

YN, A. Darmawan, Y. Yamaji, and M. Imada, PRB 96, 205152 (2017)
See also Luo and Clark PRL (2019), …

6

RBM+PP wave function
restricted Boltzmann machine + pair-product
YN, A. Darmawan, Y. Yamaji, and M. Imada, PRB 96, 205152 (2017)

RBM+PP wave function
Mag. Field (bias term) bj<latexit sha1_base64="+nTOG/gFhzOwIwM6nSbaQOXzsfE=">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</latexit>(x) = N (x) pair (x)neural-network (RBM) PP state (geminal)bosonic

fermionic

Interaction W ij

exact (x)Approximating<latexit sha1_base64="usXPFMd8KZAiUbZut7hmSZPm3vU=">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</latexit>

pair (x)Mag. Field (bias term) ai

by N (x)G. Carleo and M. Troyer Science 355, 602 (2017)<latexit sha1_base64="2k2QA7r1Rh6BPZUG/jcQgHSAS/o=">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</latexit>

Deﬁnition of neural-network partN (x) =

Yk

2cosh bk +

X
i

Wik i

!

2i = 2ni"

1 = ±12i 1 = 2ni#

1 = ±1(number of visible variables) = 2Nsite (Hubbard model case)7

Application to 2D Hubbard model
8x8 square lattice, half-ﬁlling (periodic anti-periodic)
YN, A. Darmawan, Y. Yamaji, and M. Imada, PRB 96, 205152 (2017)
TNVMC data: H.-H. Zhao et al., PRB 96, 085103 (2017).

RBM

RBM+PP

RBM+PP

TNVMC

α = (# hidden units)/(# physical spins)

RBM+PP substantially improves accuracy compared to RBM
Using combination saves # of parameters (important when simulate large system size)

8Extensions to various systems

Boson system

Fermion system

Spin system (localized Mott insulator)
Itinerant electrons in solids

without frustration（Heisenberg model）
Carleo and Troyer (2017), …

（Hubbard model)

Electron-phonon coupling

YN et al., (2017), …

（Holstein model)

with frustration

YN JPSJ (2020)（J1-J2 Heisenberg model）
YN and Imada (2021), …

Editorʼs Choice

Molecules（H2, LiH, …）Han et al., Choo and Carleo, …

Bose-Hubbard model（itinerant）
Saito (2017), …

YN and M. Imada, Phys. Rev. X 11, 031034 (2021)
See also Liang et al., PRB (2018), Choo et al., PRB (2019), Ferrari et al., PRB (2019),
Westerhout et al, Nat. Commun. (2020), Szabó and Castelnovo PRR (2020), …

9Application to J1-J2 Heisenberg model
YN and M. Imada, Phys. Rev. X 11, 031034 (2021)

J1-J2 Heisenberg model on 2D square lattice（J1＞０, J2＞０, J1 and J2 compete）

Unsettled phase diagram
longstanding challenge in physics since 1980ʼsordered phase（”solid”）

ordered phase（”solid”）

Quantum spin liquid around J2=0.5 ?
Properties of quantum spin liquid if exist ?

0.5

Chandra, Doucot 1988, Read Sachdev 1989,
Capriotti et al., 2001, Zhang et al., 2003,
Jiang et al., 2012, Hu et al., 2013,
Gong et al., 2014, Morita et al., 2015,
Wang, Sandvik 2018, Ferrari, Becca, 2020, …

(J1=1)

10

RBM+PP wave function for spin systems
restricted Boltzmann machine + pair-productYN and M. Imada, Phys. Rev. X 11, 031034 (2021)

RBM+PP wave function<latexit sha1_base64="IHVEeSygcsvl3xZB8yFXJmmo+c0=">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</latexit>( ) = N ( ) ⇥ PG pair ( )neural-network (RBM) Gutzwiller-projected PP state

Improving performance by symmetry (imposing quantum numbers)
Choo et al., PRL (2018), Ferarri et al., PRB (2019), YN JPSJ (2020), …

spin-parity (+ : S even, ­ : S odd)

total momentum

wave func. w/o symmetry

YN, A. Darmawan, Y. Yamaji, and M. Imada,Ground state : S=0(even), K=0

PRB 96, 205152 (2017)

Excited states : other quantum numbers

11Benchmarks
Number hidden units = 16 Nsite

Ground-state energy (J2=0.5, 10x10 lattice)

Excitation energy (6x6 lattice)

Green: using neural network­0.49476(1)Neural quantum states [1]­0.49516(1)CNN [2]­0.49521(1)

VMC [3]­0.495530

DMRG [4]­0.49575(3)RBM+fermion wave func. [5]­0.497549(2) VMC+2nd order Lanczos [3]­0.497629(1) RBM+PP (present study)
more accurate[1] Szabó and Castelnovo, PRR 2020 [2] Choo et al., PRB 2019[3] Hu, et al., PRB 2013 [4] Gong et al., PRL 2014 [5] Ferrari et al., PRB 2019

YN and M. Imada, Phys. Rev. X 11, 031034 (2021)

RBM+PP achieves state-of-the-art accuracy not only for ground state but also for excited states
12

Zero-temperature phase diagram and excitation structure in spin liquid phase
Ground-state analysis

Two sides of the same physics

Excited-state analysis

Zero-temperature phase diagram（two analyses agree)

YN and M. Imada, Phys. Rev. X 11, 031034 (2021)

Unconventional excitation spectra in quantum spin liquid phase (J2=0.5)
c.f. Neel antiferromagnets (J2=0)

H. Shao et al., PRX 7, 041072 (2017)
YN and M. Imada, Phys. Rev. X 11, 031034 (2021)

Gapless Dirac-dispersion at high-symmetry k points

13

Fractionalization in quantum spin liquid phase

Spin excitation (S=1)

Spinon excitation (S=1/2)

observed in experiments

Excitations behind the phenomena

YN and M. Imada, Phys. Rev. X 11, 031034 (2021)

Spin excitations are formed by multiple excitations of two (or more) spinons (e.g., two red, two black, ...)

“true” applications to longstanding problem, revealing nodal spin liquid with fractionalization
YN and M. Imada, Phys. Rev. X 11, 031034 (2021)

14

Summary
Boson system

Fermion system

Spin system (localized Mott insulator)
Itinerant electrons in solids

without frustration（Heisenberg model）
Carleo and Troyer (2017), …

（Hubbard model)

Electron-phonon coupling

YN et al., (2017), …

（Holstein model)

with frustration

YN JPSJ (2020)（J1-J2 Heisenberg model）

Editorʼs Choice

YN and Imada (2021), …

Molecules（H2, LiH, …）Han et al., Choo and Carleo, …

Bose-Hubbard model（itinerant）
Saito (2017), …

Neural-network wave functions start to show its power in grand challenges in physics !
YN and M. Imada, Phys. Rev. X 11, 031034 (2021)

New direction : ﬁnite temperature
YN*, N. Yoshioka*, and F. Nori, Phys. Rev. Lett. 127, 060601 (2021)

15

Finite-temperature simulations using deep Boltzmann Machines
Puriﬁcation + DBM representation of puriﬁed states
Finite-T density matrix = pure state of extended system (+ trace out ancillae)

J1-J2 Heisenberg model（16 site)

Deep Boltzmann machine (DBM)

DBM (dots) vs Exact (solid curves)<latexit sha1_base64="5qe38tm1TIukrQEolRuHkxHfY4Y=">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</latexit>

h1

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0N←ancillae

h3 … hMh2<latexit sha1_base64="44BhsYRns2Ez3NOpJghtIjhSdmw=">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</latexit>…

Speciﬁc heat0
1

2

<latexit sha1_base64="xjZw70dzhBvlMJQHTJCdyj8n8zk=">AAACa3icSyrIySwuMTC4ycjEzMLKxs7BycXNw8vHLyAoFFacX1qUnBqanJ+TXxSRlFicmpOZlxpaklmSkxpRUJSamJuUkxqelO0Mkg8vSy0qzszPCympLEiNzU1Mz8tMy0xOLAEKRWTEV8coKvjWxgsoG+oZgIECboYyAxQE5AssZ4hhSGHIZ0hmKGXIZUhlyGMoAbJzGBIZioEwmsGQwYChACgWy1ANFCsCsjLB8qkMtQxcQL2lQFWpQBWJQNFsIJkO5EVDRfOAfJCZxWDdyUBbcoC4CKhTgUHV4KrBSoPPBicMVhu8NPiD06xqsBkgt1QC6SSI3tSCeP4uieDvBHXlAukShgyELrxuLmFIY7AAuzUT6PYCsAjIF8kQ/WVV0z8HWwWpVqsZLDJ4DXT/QoObBoeBPsgr+5K8NDA1aDYDF3EREGakZ2igZxhoouzgBI0KDgZpBiUGDWB4mzM4MHgwBDCEgsN5EsNshjmMb5mEmaSYZCBKmRiheoQZUACTKgBuIoxy</latexit>…

J2=0.0J2=0.5<latexit sha1_base64="VETmeNG9s1dGOXBA1h1Dku2oEkw=">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</latexit>N←system spin
Temperature

cf. Other ﬁnite-temperature methods：(path-integral) quantum Monte Carlo：
Thermal pure quantum states (TPQ)：
Tensor networks：

sign problem
limitation in system size
mostly focused on 1D for T > 0 (recently also 2D)・・・

For establishing powerful ﬁnite-temperature method（in particular for 2D systems）
YN*, N. Yoshioka*, and F. Nori, Phys. Rev. Lett. 127, 060601 (2021)

16