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Phys. Rev. B 97, 075114 (2018)

Reverse Engineering Hamiltonian

from Spectrum

Hiroyuki Fujita, Insitute for Solid State Physics

Deep Learning and Physics 2018, Osaka

Collaborators

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

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

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

I am going to talk about

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something NOT deep

Deep to be interesting?

Image compression

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512×512 256×256

Simple but not efficient → principal component analysis (PCA)

(1) (2)

Optimization maximizing the data variance

e.g. (1) is better in the variance of the projected data

PCA is to find a subspace onto which

e.g. Simple average of nearby pixels

the data are projected.

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Hubbard model

AFM Heisenberg model

half-filling, large U

Perturbation theory

2L

2L

O

✓t2

U2

◆4L

4L

Low-energy model = “compressed image”

What is “PCA” for this problem?

“Image compression” in cond. mat. ?

4 states/site 2 states/site

Straightforward(?) approach

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???????

from sklearn.decomposition import PCA

pca = PCA(n_components = ncomp) pca.fit(Hubbard_H)

pca_res = pca.transform(Hubbard_H) Spin_H = pca.inverse_transform(pca_res)

???????

e.g. L = 10 Hubbard chaintrillion pixel image million pixel image

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

Totally unrealistic!

“PCA” for low-energy physics of Hubbard

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Can we solve the “inverse problem” of diagonalization?

A

eN ⇥ eNA =

Reconstruct the matrix

Hermitian matrix

based on the low-energy properties of the Hubbard model

The “PCA” has to construct the effective model

= low-energy spectrum

N eN

Number counting

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Exact Diagonalizaiton (ED)eN ⇥ eNA =

Number of independent params of is eN ⇥ eNA

Number of eigenvalues available is at most eN

# of params in

# of eigenvalues

A = O( eN ⇥ eN)

A = O( eN)1

eN ! 1Data

Parameters

Too many parameters……

Physicists point of view

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“Being physical” is a powerful constraint on the Hamiltonian

Short-ranged interactions Few-body interactionsSymmetries such as U(1), SU(2), …

# of params in

# of eigenvalues

H = O(Ln)

Dimension of Hilbert space is exponential in the system size

L ! 10

Data

Parameters

(up to n-body)

Solvable because it’s physics!

Original Hamiltonian ED Low-energy spectrum

(ED)-1

Low-energy Hamiltonian

e.g. Hubbard model @ half-filling

e.g. AFM Heisenberg model

Quantum state (w.f.)ED

entanglement spectrum

(ED)-1

Entanglement Hamiltonian

What can we do with (Exact Diagonalization)-1

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Make the problem concrete

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EDE1, ......, EN (Small number of eigenvalues)

H =X

i=1,...,M

ciHi

Heisenberg, four-spins interactions, …

ED-1

Spin model with parameters to be optimized

What we have to do is to1. find a proper form of the ansatz

2. find the proper parameters for the fixed ansatz

Correlated electrons in 1D

Formulate as a supervised learning problem

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H =X

i=1,...,M

ciHi

= data = model= prediction

(1) Define cost function

(2) Calculate the gradient in terms of the parameters

(3) Update the parameters as

Gradient descent algorithm

cj = cj � ↵@

@cjCost(E,E0({cj}))

Cost(E,E0({cj})) =1

2N

NX

i=1

(E0i � Ei)

2

cj

Cost

EDE1, ......, EN E01, ......, E

0N

diagonalization

cost evaluation

gradient descent

E1, ......, En

E01, ......, E

0n

H =X

i=1,...,M

ciHi

“Quantum computation” of the gradient

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@

@cjCost(E,E0({cj})) =

✓@

@cjE0

◆@

@E0Cost(E,E0({cj}))

Luckly, we know the perturbation theory of quantum mechanics:

H !X

i=1,...,M

ciHi + �cjHj Ei ! Ei + �cj h i|Hj | ii

What we have to do is to1. find a proper form of the ansatz2. find the proper parameters for the fixed ansatz

∴ For a given ansatz, we can optimize its parameters

@

@cjCost(E,E0({cj})) =

✓@

@cjE0

◆@

@E0Cost(E,E0({cj}))

How?

trivial

H =X

i=1,...,M

ciHi E01, ......, E

0N

Cost(E,E0) ! Cost(E,E0) + �X

j=1,...,M

|cj |Prefer small parameters

Model selection by regularization

�14“Sparse” nature of the estimation → Model selection

L1 norm regularization

contours of MES

Cost(E,E0) ! Cost(E,E0) + �X

j=1,...,M

|cj |

Both and are nonzero

contours of reg. term

Cost(E,E0) ! Cost(E,E0) + �X

j=1,...,M

|cj |

contours of MEScontours of reg. term

Demonstration: Hubbard chain at half-filling

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Spin model ansatz

i i+ 1 i+ 2 i+ 3K1

K2

K3

i i+ 1 i+ 2 i+ 3

i i+ 1 i+ 2 i+ 3

(Si · Si+1)(Si+2 · Si+3)

i i+ 1 i+ 2 i+ 3

i i+ 1 i+ 2 i+ 3

i i+ 1 i+ 2 i+ 3

(Si · Si+1)(Si+2 · Si+3)

i i+ 1 i+ 2 i+ 3

i i+ 1 i+ 2 i+ 3

i i+ 1 i+ 2 i+ 3

i i+ 1 i+ 2 i+ 3

i i+ 1 i+ 2 i+ 3

i i+ 1 i+ 2 i+ 3

optimized A with regul.

optimized A w/o regul.

Effective model of

?

Model selection based on the insensitivity to the regularizationλ

w/o regul. trapped by local minima

“Important” terms should survive under strong λ

Hierarchical structure

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Hierarchical structure

～ order of importance?

insensitivity to the regul.

→ Estimate parameters again WITHOUT regul.

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

K2

K3

i i+ 1 i+ 2 i+ 3

i i+ 1 i+ 2 i+ 3

i i+ 1 i+ 2 i+ 3

i i+ 1 i+ 2 i+ 3

i i+ 1 i+ 2 i+ 3

(Si · Si+1)(Si+2 · Si+3)

Simplified model

λ

Comparison with perturbation theory

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Perturbation theoryA. H. MacDonald, S. M. Girvin, and D. Yoshioka, PRB (1988). A. Rej, D, Serban, and M. Staudacher, JHEP (2006)

Deviation from

the pe

rturb.

Difference betw. ML & Macdonald et. al.

eJi = Ji � Jpi

eKi = Ki �Kpi

/ U�7

L=10, α = 0.1, Sz = 2, n=50

/ U�14Cross validation error

Estimation is correct up to

From the viewpoint of image compression

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

trillion pixel image million pixel image

We demonstrated the following “image compression”

original (30 MB) 5KB image

cf) something

Next application: entanglement

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Quantum state (w.f.) ED entanglement spectrum

(ED)-1

Entanglement HamiltonianPhysical Hamiltonian

EDDMRGetc.

= spectrum of RDM

quantum state

reduced density matrix

entanglement spectrum

Entanglement Hamiltonian

Thermodynamics from pure quantum state

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BA A Bath

・Gibbs ensemble・Reduced density matrix

・Thermodynamic entropy・Entanglement entropy

thermal fluctuation from the bathquantum fluctuation from B

e.g. entangled two-spins

A B

Entanglement Hamiltonian

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Pushing forward the analogy

Entanglement Hamiltonian

e.g. entangled two-spins

A B

Physical Hamiltonian

free spin

spectrum of (ED)-1

“log (matrix)”

Construction of Entanglement Hamiltonian

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Short-ranged interactions Few-body interactions

Symmetries such as U(1), SU(2), … True for ?

A

Entanglement-edge correspondence

Similar to physical edge Hamiltonian EH is local & few-body?

A B

As long as the ent-edge corresp. holds, we can estimate EH

A

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

A Be.g.

defined for “virtual” cut defined for “physical” cut

Comparison with other method

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Full diag. of RDM: exact but computationally hard

・need all the eigenvectors of the RDM

→ strong constraint: memory size < matrix dimensionRDM → EH

Computational cost

Our method: approximate but computationally cheap

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

・use of symmetry

e.g. magnetization conservation

5 10 15 201

10

100

1000

104105

High-spin sector → small Hilb. space

chain

Hilbert s

pace

dim

ension

# of evals. > # of params.

We can reduce dim. as long as

# of up spins

RDM → ES

ES → EH w/ transl. symm.

w/o transl. symm.

Not just a “Hello, World” !

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arXiv:1803.10856

arXiv:1712.03557

Outlook: Materials design for exotic quantum states

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Models written with emergent d.o.f (e.g. Majoranas, dimers, lattice gauge fields)

Model of emergent d.o.f

Parent Hamiltonian written with

physical d.o.f (electrons, spins)

Energy spectrum= low energy spectrum of unknown parent Hamiltonian

(ED)-1

Materials design

ED

e.g. Majorana fermion

e.g. Kitaev modele.g. Honeycomb iridates

with exotic quantum state

e.g. Q. dimer model

e.g. string-net condensation (lattice gauge)

Levin-Wen (2005)

Summary

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We propose

a scheme to construct Hamiltonians from a given spectrum

E1, E2, ......, EN ! HE1, E2, ......, EN ! H

Energy spectrum

Entanglement spectrum

Low energy Hamiltonian

Entanglement Hamiltonian

Phys. Rev. B 97, 075114 (2018)

Parent Hamiltonian