Reverse Engineering Hamiltonian
from Spectrum
Deep Learning and Physics 2018, Osaka

Hiroyuki Fujita, Insitute for Solid State Physics
Phys. Rev. B 97, 075114 (2018)1

Collaborators
Yuya. O. Nakagawa(PhD @ ISSP -> Fintech company)

S. Sugiura(PD @ ISSP -> PD @ Harvard)

M. Oshikawa(Prof. @ ISSP, now @ Wien)2

I am going to talk aboutsomething NOT deep

Deep to be interesting?

3

Image compression
e.g. Simple average of nearby pixels

256 256512 512

Simple but not eﬃcient → principal component analysis (PCA)
(1)

(2)

PCA is to ﬁnd a subspace onto which
the data are projected.
Optimization maximizing the data variancee.g. (1) is better in the variance of the projected data4

“Image compression” in cond. mat. ?
Hubbard model
half-ﬁlling,large UAFM Heisenberg model

4LL

4

✓ 2◆
t
O
U2

L

2
2L

Perturbation theory

4 states/site

2 states/site

Low-energy model = “compressed image”
What is “PCA” for this problem?

5

Straightforward(?) approach???????

from sklearn.decomposition import PCA
pca = PCA(n̲components = ncomp)???????

pca.ﬁt(Hubbard̲H)
pca̲res = pca.transform(Hubbard̲H)
Spin̲H = pca.inverse̲transform(pca̲res)

Totally unrealistic!
e.g. L = 10 Hubbard chain
trillion pixel image

million pixel image

We cannot simply use the (variant of) existing algorithms.

6

“PCA” for low-energy physics of Hubbard
The “PCA” has to construct the eﬀective model
based on the low-energy properties of the Hubbard model= low-energy spectrum

A=e ⇥NeN

Hermitian matrix

Reconstruct
the matrix AeN N

Can we solve the “inverse problem” of diagonalization?
7Number counting
Exact Diagonalizaiton (ED)A=e ⇥NeN

Number of independent params ofA is Ne ⇥ Nee
Number of eigenvalues available is at most NParameters
Datae ⇥Ne)
# of params in A = O(Ne)# of eigenvalues A = O(Ne !1
NToo many parameters……

18

Physicists point of view
“Being physical” is a powerful constraint on the Hamiltonian
Short-ranged interactions

Few-body interactions

Symmetries such as U(1), SU(2), …
Dimension of Hilbert space is exponentialin the system size
(up to n-body)Parameters

Datan# of params in H = O(L )# of eigenvalues

L!10

Solvable because itʼs physics!
9

What can we do with (Exact Diagonalization)
Original Hamiltonian

EDe.g. Hubbard model @ half-ﬁlling-1

Low-energy spectrum(ED)-1
Low-energy Hamiltoniane.g. AFM Heisenberg model

Quantum state (w.f.)

EDentanglement spectrum(ED)-1
Entanglement Hamiltonian
10Make the problem concrete

Correlated electrons in 1DED
ED-1

E1 , ......, EN (Small number of eigenvalues)
H=Xc i Hi

i=1,...,M

Spin model with parametersto be optimized

Heisenberg, four-spins interactions, …

What we have to do is to
1. ﬁnd a proper form of the ansatz
2. ﬁnd the proper parameters for the ﬁxed ansatz
11Formulate as a supervised learning problem
E1 , ......, EN

H=X

c i HiED0
E10 , ......, EN

i=1,...,M= data= prediction= modelGradient descent algorithm
NX
10(Ei0Cost(E, E ({cj })) = 2N
i=1(1) Define cost function

Ei ) 2(2) Calculate the gradient in terms of the parameters(3) Update the parameters as

E1 , ......, En

H=Xi=1,...,M

cj = cj

cost evaluation

E10 , ......, En0

gradient descent@Cost(E, E 0 ({cj }))
↵@cj
Cost

diagonalizationc i Hicj

12

“Quantum computation” of the gradient✓✓◆◆
trivial@@@@ 0 0 @@0 0
0 0
Cost(E,E({c}))=E
Cost(E,E
Cost(E, E ({cECost(E, E({c({cj j })) =j }))
j }))
0
0
@c@c
@c@c
@E
@E
j j
j j
How?

Luckly, we know the perturbation theory of quantum mechanics:

H!

Xc i Hi + c j Hj

i=1,...,MH=X

E i ! Ei + c j hc i Hii | Hj |ii

0E10 , ......, EN

i=1,...,MFor a given ansatz, we can optimize its parameters

What we have to do is to
1. ﬁnd a proper form of the ansatz
2. ﬁnd the proper parameters for the ﬁxed ansatz
13Model selection by regularization
00

Cost(E, E ) ! Cost(E, E ) +X

j=1,...,M|cj |L1 norm regularization

Prefer small parameters

contours of MES
contours of reg. term

contours of MES
contours of reg. term

Cost(E, E 0 ) ! Cost(E, E 0 ) +BothandX
0|c|Cost(E,j E )!C

j=1,...,M

are nonzero“Sparse” nature of the estimation → Model selection

14

Demonstration: Hubbard chain at half-fillingEﬀective model of(Si · Si+1 )(Si+2 · Si+3 )?
Spin model ansatz

ii+1i+2i+3i

i+1i+2i+3ii+1i+2i+3ii+1i+2i+3ii+1i+2i+3ii+1i+2i+3ii+1i+2i+3ii+1i+2i+3ii+1i+2i+3(Si · Si+1 )(Si+2 · Si+3 )K1ii+1i+2i+3K2ii+1i+2i+3K3ii+1i+2i+3Model selection based on the insensitivity to the regularizationλoptimized A w/o regul.

w/o regul. trapped by local minima“Important” terms should survive under strong λoptimized A with regul.

15

Hierarchical structure
insensitivity to the regul.

Hierarchical structure
L=10, total Sz = 2, U = 8, n = 50, α = 0.01

λ〜 order of importance?

(Si · Si+1 )(Si+2 · Si+3 )

ii+1i+2ii+1i+2i+3ii+1i+2i+3ii+1i+2i+3i+3Simpliﬁed modelK2K3ii+1i+2i+3→ Estimate parameters again WITHOUT regul.

16Comparison with perturbation theory
Perturbation theory

Deviation from the perturb.

A. H. MacDonald, S. M. Girvin, and D. Yoshioka, PRB (1988).

A. Rej, D, Serban, and M. Staudacher, JHEP (2006)

Diﬀerence betw. ML & Macdonald et. al.

Jei = Jie i = Ki
Kp
Jip
Ki

Cross validation error

/U7/U

14

Estimation is correct up to
L=10, α = 0.1, Sz = 2, n=50

17From the viewpoint of image compression
We demonstrated the following “image compression”

trillion pixel image

million pixel image

preserving the 99.9999% of the essential information (for U=10).

cf)

somethingoriginal (30 MB)

5KB image18

Next application: entanglement
Quantum state (w.f.)
EDDMRGED

entanglement spectrum= spectrum of RDM(ED)-1etc.
Physical Hamiltonian

Entanglement Hamiltonian
reduced density matrix

quantum state

entanglement spectrum
Entanglement Hamiltonian19

Thermodynamics from pure quantum state
ABA

Bath・Reduced density matrix

・Gibbs ensemble

・Entanglement entropy・Thermodynamic entropy

quantum ﬂuctuation from B

thermal ﬂuctuation from the bathe.g. entangled two-spins

AB20

Entanglement Hamiltonian
Pushing forward the analogy

Physical Hamiltonian

Entanglement Hamiltoniane.g. entangled two-spins

Afree spinB

spectrum of(ED)-1“log (matrix)”21

Construction of Entanglement Hamiltonian
Short-ranged interactions

Few-body interactions

True forSymmetries such as U(1), SU(2), …

?

Entanglement-edge correspondenceAB

deﬁned for “virtual” cut

e.g.

AAB

A

deﬁned for “physical” cut

Similar to physical edge Hamiltonian

EH is local & few-body?

As long as the ent-edge corresp. holds, we can estimate EH
see Phys. Rev. B 97, 075114 (2018)22

Comparison with other method
Full diag. of RDM: exact but computationally hard・need all the eigenvectors of the RDM→ strong constraint: memory size < matrix dimension

Our method: approximate but computationally cheap

Computational cost
RDM → EH

・only a small number of eigenvalues → memory eﬃcient algos. (e.g. Lanczos)
・use of symmetry

Hilbert space dimension

e.g. magnetization conservation

High-spin sector → small Hilb. space
We can reduce dim. as long as# of evals. > # of params.

105
104
1000
100chain

10

RDM → ESw/o transl. symm.

1
510

15

20

# of up spins

ES → EH

w/ transl. symm.

23

Not just a “Hello, World” !
arXiv:1712.03557

arXiv:1803.10856

24

Outlook: Materials design for exotic quantum states
Models written with emergent d.o.f (e.g. Majoranas, dimers, lattice gauge ﬁelds)
e.g. Q. dimer modele.g. string-net condensation (lattice gauge)

Levin-Wen (2005)

Model of emergent d.o.f
with exotic quantum statee.g. Majorana fermion

Energy spectrum

ED= low energy spectrum of
unknown parent Hamiltonian-1(ED)

Parent Hamiltonian written withMaterials design
physical d.o.f (electrons, spins)
e.g. Honeycomb iridates

e.g. Kitaev model

25

Summary
We propose
a scheme to construct Hamiltonians from a given spectrum

E1E, E12,,E......,E!
H,......,EN2
N !HLow energy Hamiltonian

Energy spectrum
Parent Hamiltonian
Entanglement spectrum

Entanglement Hamiltonian

Phys. Rev. B 97, 075114 (2018)26