2020/08/06

ボルツマンマシンを用いた量子多体波動関数表現：
深層ボルツマンマシンによる厳密な表現と制限ボルツマンマシンによる数値的近似表現

野村 悠祐
理研 CEMS

関連する発表：7/9 吉岡さん、9/3 斎藤先生

研究内容
t

量子多体論

U

DMFT, FLEX, …

新たな計算物質科学のフレームワーク

機械学習
RBM, neural network, …

第一原理計算DFT, DFPT, …

機能物性予測・設計へ

2

Machine Learning
Modeling unknown relationship between datax, y(x) : datay(x) = F[ (x), x]<latexit sha1_base64="qKZhg8U0H80fzxHjn5PKsZiAwTg=">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</latexit>

F ：functional (nonlinear)<latexit sha1_base64="jk55SSHw3gm0NIWxd+6hvlRNxnI=">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</latexit><latexit sha1_base64="1NabSqKGSCYygjGMZcve+1h8Gvs=">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</latexit><latexit

Example: Handwritten digit recognition<latexit sha1_base64="SqhNy17yX9iTNkSLOhMb/MnzaJU=">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</latexit><latexit

x : pixel data of images<latexit sha1_base64="SqhNy17yX9iTNkSLOhMb/MnzaJU=">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</latexit><latexit

xy(x)<latexit sha1_base64="ZWRX6ARps9h5jVhypRnBBlW1y34=">AAACbXichVHLSsNAFD2Nr1pfVREERYrF16bcSEFxJbhx6atVbEWSOK2haRKStFhLf8C14EIUFETEz3DjD7jwE8SFiwpuXHibBkRFvcPMnDlzz50zM6pt6K5H9BiSWlrb2jvCnZGu7p7evmj/QNq1So4mUpplWM6WqrjC0E2R8nTPEFu2I5SiaohNtbDU2N8sC8fVLXPDq9hip6jkTT2na4rH1HZluppVc7GD2sxuNE4J8iP2E8gBiCOIFSt6jSz2YEFDCUUImPAYG1DgcstABsFmbgdV5hxGur8vUEOEtSXOEpyhMFvgMc+rTMCavG7UdH21xqcY3B1WxjBBD3RDdbqnW3qi919rVf0aDS8VntWmVti7fUfD62//qoo8e9j/VP3p2UMO875Xnb3bPtO4hdbUlw9P6usLaxPVSbqkZ/Z/QY90xzcwy6/a1apYO0WEP0D+/tw/QXo2IVNCXk3GF5PBV4QxgnFM83vPYRHLWEGKzzVxjDOch16kIWlUGmumSqFAM4gvIU19AF+LjUo=</latexit>→ ８

y(x) : 0-9 digits<latexit sha1_base64="ZWRX6ARps9h5jVhypRnBBlW1y34=">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</latexit>→ ５
https://note.mu/sadaaki/n/n362a3f0fa5a9→ useful in extracting essential pattern from data

Application to quantum many-body systemsx：Basis of the Hamiltoniany(x)：Amplitude of quantum many-body wave function<latexit sha1_base64="SqhNy17yX9iTNkSLOhMb/MnzaJU=">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</latexit><latexit<latexit sha1_base64="ZWRX6ARps9h5jVhypRnBBlW1y34=">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</latexit>

Extract essential pattern of wave functions (data compression) andobtain accurate representations with a ﬁnite number of parameters3Quantum many body problems
H| i = E| i<latexit sha1_base64="6U6vIKPNQAk5aO44uIcfIDT5Gqc=">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</latexit>

Eigenstate

Hamiltonian

Example of many-body Hamiltonian：Heisenberg model (interacting S=1/2 quantum spins, eﬀective model for Mott insulator)

H=J<latexit sha1_base64="+aC91gqGwHDLDbCuko+2mvLkiWs=">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</latexit>

X

(i,j)

i·

j

Ground state：
Lowest-energy eigenstate（vector with exponentially large dimension）of Hamiltonian（matrix with exponentially large dimension）→ Most stable quantum states at zero temperatureH|<latexit sha1_base64="REhqCA9kAzRqB3g3lXwQD0wmHOk=">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</latexit>

GS i = E0 |GS i

Accurate ground-state calculations：Grand challenges in physics as well as in quantum chemistry| i=<latexit sha1_base64="yWA6aPgjNaoLO95+dsRYqgkbjBs=">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</latexit>X

model relation between x and Ψ(x)（machine learning）(x)|xi<latexit sha1_base64="8AKoe7VkU99plAe/YaOF4nNncEY=">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</latexit>x

sum over 2N conﬁgurations (Heisenberg)|xi = | 1z ,<latexit sha1_base64="KTxnYlDYLQ1g5qUeNrmQ5zN9HUI=">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</latexit>z
2, . . . ,

z
Ni

(x) ⇡

(x)

γ: parameter set (ﬁnite number)

4Restricted Boltzmann machine (RBM)
Paul Smolensky (1986)
G. E. Hinton, R. R. Salakhutdinov, Science. 313, 504 (2006)

Energy function

Mag. Field (bias term) bj

Interaction W ij

Boltzmann distribution

Marginal distribution
Mag. Field (bias term) ai

Single hidden layer + interlayer coupling only → restricted Boltzmann machine (RBM)
Marginal distribution

can represent any distribution over {0,1}N with inﬁnite M

K. Hornik, Neural Networks 4, 251 (1991); G. Cybenko, Mathematics of Control, Signals and Systems 2, 303 (1989); N. L. Roux and Y. Bengio, Neural Computation 20, 1631(2008).

5

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

Spin conﬁg.

wave function

z

( z)

<latexit sha1_base64="VYykaxCJEAhEg4XXUkOkJvCQ4WA=">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</latexit><latexit sha1_base64="Cg45e9Ms0cxZsdc31ouSyricbU8=">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</latexit>

RBM wave function
Interaction W ij
( z) =

X=ai iz +

z2, . . . ,zN: real space spin conﬁg.

hj = ±1: spin of hidden neuron

exp

i

{hj }
z

⇣X

z
1,

Mag. Field (bias term) ai

X
i,j

z
i Wij hj +Xb j hjj⌘

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

z

)=e

P

i ai

z
i

⇥

Y
j

⇣

2 cosh bj +Xi

Wij iz⌘6

Example: 1D Antiferromagnetic Heisenberg model (8site)
gauge transformation
σx,y → ­σx,y for one of sublattice

Optimization following variational principle : hHi =h |H| ih | i

E0

E0: ground state energy

optimization of nonlinear function using nonlinear cost function→ equivalent to machine learning task

wave function (real and positive for any x)

RBM interaction parameter
InitialOptimized

W

ψ(x)

Exact
Initial RBM
Optimized RBM

x

7

②

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 arXiv, …

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 physics① ※ 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)8

DBM (deep Boltzmann machine) wave function

d1

d2

d3

h1

h2

h3

z
1

z
2

z
3

deep layer

…

dM 0

0
Wjk

hidden layer…

hMvisible layer

z
N

Wij

( )=

X

d1

d2

d3

h1

h2

h3

z
1

z
2

z
3

h,d

…

dM 0

…

hM

…

z
N

…

P
P
P 0
X P
z P
z
0
a
+
W
h
+
b
h
+
h
W
d
+
ij j
jk k
i,j i
j j j
j,k j
k bk d k=
e i i ih,d9DBM representation of ground states
DBM compared with RBM
Prosmuch more ﬂexible representability
X. Gao and L.-M. Duan, Nat. Commun. 8, 662 (2017).

Cons
cannot trace out both h and d analytically(need to sample hidden spins to obtain wave function)

Key idea
reproduce imaginary-time evolution by dynamically modifying DBM network(no need to perform stochastic optimization of parameters! everything deterministic ! )⌧⌧| (⌧ )i = e H1 2 e H2 ⌧ . . . e H2 ⌧ e H1 2 |0i

Physical quantities are measured by MC sampling of classical visible and hidden spins

Novel class of quantum-to-classical mapping
G. Carleo, Y. Nomura, and M. Imada, Nat. Commun. 9, 5322 (2018) (see also N. Freitas et al., arXiv:1803.02118)

10

Example: Transverse-Field Ising model
G. Carleo, YN, and M. Imada, Nat. Commun, 9. 5322 (2018)

Hamiltonian:H = H1 + H2

Interaction (classical):Transverse-ﬁeld:

H1 =

X

z
Vlm lz m

l<m

X

H2 =

x
l l

l

How to express short time propagators by DBM ?
Interaction propagator:eTransverse-ﬁeld propagator:

e

⌧ Vlm

⌧

l

x
l

z
l

z
m

|DBMi

|DBMi

11

Example: Transverse-Field Ising model
G. Carleo, YN, and M. Imada, Nat. Commun, 9. 5322 (2018)

W̄lj = Wlj +Wlj = 012

DBM construction for Heisenberg model
G. Carleo, YN, and M. Imada, Nat. Commun, 9. 5322 (2018)a)Initialnetworkb)c)

Step 0Initial State
z
l

z
m

z
l

d[l]

d[lm]

z
m

z
l

z
m

d[m]

d[l]

d[lm]

<latexit sha1_base64="uBb4RTSONRM48nVS4HZU7/FqT0U=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtoQ9lsJu3SzSbsboQS+iO8eFDx6v/x5r9x2+ag1QcDj/dmmJkXZoJr47pfTmVldW19o7pZ29re2d2r7x886DRXDH2WilR1Q6pRcIm+4UZgN1NIk1BgJxzfzPzOIyrNU3lvJhkGCR1KHnNGjZU60aDoiWA6qDfcpjsH+Uu8kjSgRHtQ/+xHKcsTlIYJqnXPczMTFFQZzgROa/1cY0bZmA6xZ6mkCeqgmJ87JSdWiUicKlvSkLn6c6KgidaTJLSdCTUjvezNxP+8Xm7iy6DgMssNSrZYFOeCmJTMficRV8iMmFhCmeL2VsJGVFFmbEI1G4K3/PJf4p81r5ru3XmjdV2mUYUjOIZT8OACWnALbfCBwRie4AVencx5dt6c90VrxSlnDuEXnI9v3BOPcg==</latexit><latexit sha1_base64="1rHWp+jZUNKnTdkHtoE0FANu1w4=">AAAB7nicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsNu3S3U3c3Qgl9E948aDi1d/jzX/jts1BWx8MPN6bYWZelHGmjet+O5WV1bX1jepmbWt7Z3evvn/woNNcEeqTlKeqG2FNOZPUN8xw2s0UxSLitBONbqZ+54kqzVJ5b8YZDQUeSJYwgo2VunG/CLgIJ/16w226M6Bl4pWkASXa/fpXL05JLqg0hGOtA8/NTFhgZRjhdFLr5ZpmmIzwgAaWSiyoDovZvRN0YpUYJamyJQ2aqb8nCiy0HovIdgpshnrRm4r/eUFuksuwYDLLDZVkvijJOTIpmj6PYqYoMXxsCSaK2VsRGWKFibER1WwI3uLLy8Q/a1413bvzRuu6TKMKR3AMp+DBBbTgFtrgAwEOz/AKb86j8+K8Ox/z1opTzhzCHzifP6i3j+k=</latexit>

Step 1
New d and W 0

1d3h

Step 2Modify W

Step 3New h, W, W

h[lm1]

h[lm2]

2d6h
h[lm1]

0e⌧ J~ l ~ m|DBMi

Step 4
New h, W, W
(constraint)h[l]<latexit sha1_base64="AdYhSOzVr+yW8FFkwlyKLSKMXI8=">AAAB7XicbVBNS8NAEJ3Ur1q/qh69LBbBU0lEUG9FLx4rGFtIQ9lsN+3SzSbsToQS+iO8eFDx6v/x5r9x2+ag1QcDj/dmmJkXZVIYdN0vp7Kyura+Ud2sbW3v7O7V9w8eTJprxn2WylR3I2q4FIr7KFDybqY5TSLJO9H4ZuZ3Hrk2IlX3OMl4mNChErFgFK3UGfWLQIbTfr3hNt05yF/ilaQBJdr9+mdvkLI84QqZpMYEnpthWFCNgkk+rfVywzPKxnTIA0sVTbgJi/m5U3JilQGJU21LIZmrPycKmhgzSSLbmVAcmWVvJv7nBTnGl2EhVJYjV2yxKM4lwZTMficDoTlDObGEMi3srYSNqKYMbUI1G4K3/PJf4p81r5ru3XmjdV2mUYUjOIZT8OACWnALbfCBwRie4AVencx5dt6c90VrxSlnDuEXnI9v4jePdg==</latexit>h[m]<latexit sha1_base64="i8aGLRRgKHa/QbCpXExhzZjIRFM=">AAAB7XicbVBNSwMxEJ2tX7V+VT16CRbBU9kVQb0VvXis4NrCdinZNNuGJtmQZIWy9Ed48aDi1f/jzX9j2u5Bqw8GHu/NMDMvUZwZ6/tfXmVldW19o7pZ29re2d2r7x88mCzXhIYk45nuJthQziQNLbOcdpWmWCScdpLxzczvPFJtWCbv7UTRWOChZCkj2DqpM+oXkYin/XrDb/pzoL8kKEkDSrT79c/eICO5oNISjo2JAl/ZuMDaMsLptNbLDVWYjPGQRo5KLKiJi/m5U3TilAFKM+1KWjRXf04UWBgzEYnrFNiOzLI3E//zotyml3HBpMotlWSxKM05shma/Y4GTFNi+cQRTDRztyIywhoT6xKquRCC5Zf/kvCsedX0784bresyjSocwTGcQgAX0IJbaEMIBMbwBC/w6inv2Xvz3hetFa+cOYRf8D6+AeO8j3c=</latexit>h[lm3] h[lm4](real)

network after
time evolution

h[lm2]

2d4hh[lm3]
0h[lm5]

h[lm6]

h[lm1]

h[lm2]

13

Numerical result
G. Carleo, YN, and M. Imada, Nat. Commun, 9. 5322 (2018)(# hidden units)(system size) x (imaginary time)

1D Transverse-Field Ising

1D Antiferromagnetic Heisenberg

N = 20, Jδτ = 0.01

N = 80, Jδτ = 0.01
from empty networkfrom pre-optimized RBM

DBM reproduces exact time-evolution

better initial state => faster convergence14

Short Summary
Show deterministic construction of DBM to represent ground states
The number of hidden units grows linearly with system size and imaginary time, respectively

Additional hidden (deep) layer : “additional dimension” in statistical mechanics

DBM representation => New quantum-to-classical mapping

Unfortunately, there exist negative sign problem for e.g. frustrated spin systems

G. Carleo, Y. Nomura, and M. Imada, Nat. Commun. 9, 5322 (2018) (see also N. Freitas et al., arXiv:1803.02118)

15

②

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 arXiv, …

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※ 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)16

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 arXiv, …

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), …

17

Bosonic wave function vs Fermionic wave function
Boson<latexit sha1_base64="K0BowV/SOQvufAYUC9w7qYTblxY=">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</latexit>

Fermion(x1 , x2 ) = + (x2 , x1 )<latexit sha1_base64="8382yzwJMmpmyTMlgrUBhVIVQDs=">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</latexit>

(

)=

X

{hj }

exp

⇣X
i

(x2 , x1 )Mag. Field (bias term) bj

RBM wave functionz(x1 , x2 ) =ai iz +Xz
i Wij hj +

i,j

X
j

b j hj⌘Interaction W ij

→ Bosonic wave function
Mag. Field (bias term) ai

Application to Fermion systems
Mapping to interacting spin systems using Jordan-Wigner transformation
K. Choo et al, Nat. Commun. (2020)
Yoshioka et al.,18

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

RBM+PP

RBM

combine concepts frommachine learning (RBM) andphysics (pair-product(PP) state)

Pair-Product state (geminal wave function):

Boson wave function
no entanglement if hidden layer is absent
19

Fermion wave function
PP helps RBM to learn ground state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)

20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 arXiv, …

Editorʼs Choice

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

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

YN and M. Imada, arXiv:2005.14142
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, arXiv.2002.04613, …

21

2D square-lattice J1-J2 Heisenberg model

Materials

Unsettled phase diagram (J1=1)

Quantum spin liquid (QSL) around J2 = 0.5？
O. Mustonen et al, Nat. Commun., 9 1085 (2018)

Relation between QSL and superconductivity？

22

Candidates for intermediate phase(s)From Tao Xiangʼs lecture at ISSP

Valence Bond Solid (VBS)

23

Controversy
Quantum spin liquid (QSL)
Valence Bond Solid (VBS)QSL or VBS
Jiang et al., 2012 (DMRG)
Hu et al., 2012 (VMC+Lanczos)
Gong et al., 2014 (DMRG)
Morita et al., 2014 (mVMC)
Wang et al., 2016 (Tensor-product)
Wang and Sandvik, 2018 (DMRG)
Haghshenas and Sheng 2018 (iPEPS)→ Letʼs apply machine learning method !
24

RBM+PP wave function
restricted Boltzmann machine + pair-productYN and M. Imada, arXiv:2005.14142

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

Imposing quantum numbers (symmetry)
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

Ground state : S=0(even), K=0
Excited states : other quantum numbers

25

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]­0.49718(2)RBM+PP (present study)­0.497549(2) VMC+2nd order Lanczos [3]accurate[1] Szabó and Castelnovo, arXiv [2] Choo et al., PRB 2019[3] Hu, et al., PRB 2013 [4] Gong et al., PRL 2014 [5] Ferrari et al., PRB 2019

RBM+PP can accurately represent not only the ground state but also excited states→ enables excited-state level spectroscopy

26

Excited-state level spectroscopy
Suwa et al PRB (2016); Wang and Sandvik PRL (2018)

Excitation energy at 16x16 latticeVBS
QSL
AF-QSLQSL-VBSAF

cf. Analysis of correlation ratio 1

S(qpeak + q)/S(qpeak )(ground state property)

Two independent analyses agree
AF-QSL(one-to-one correspondence between
ground-state phase and excitation structure)
QSL-VBS27

Ground-state phase diagram

Comparison to previous results
Jiang et al., 2012 (DMRG)
Hu et al., 2012 (VMC+Lanczos)
Gong et al., 2014 (DMRG)
Morita et al., 2014 (mVMC)
Wang et al., 2016 (Tensor-product)
Wang and Sandvik, 2018 (DMRG)
Haghshenas and Sheng 2018 (iPEPS)

Quantum spin liquid (QSL)
Valence Bond Solid (VBS)

Present study (RBM+PP)QSL or VBS
Ferrari and Becca, arXiv (VMC)

28

Nodal quantum spin liquid

Excitation spectra (J2=0.5)

c.f. S(q,ω) in Heisenberg model

H. Shao et al., PRX 7, 041072 (2017)

Qualitatively consistent with Ferrari and Becca, PRB 98, 100405 (2018)

Accurate excited-states calculations clarify the nature of QSL

29

Summary

Boson system

Fermion system

Spin system (localized Mott insulator)
Itinerant electrons in solids

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

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

（Hubbard model)

Electron-phonon coupling

YN et al., (2017), …

（Holstein model)
YN JPSJ (2020)

Editorʼs Choice

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

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

Machine-learning method shows its power in grand challenges in physics !
YN and M. Imada, arXiv:2005.14142

30