Reverse Engineering Hamiltonian from Spectrum Hiroyuki Fujita, - - PowerPoint PPT Presentation

• 1
• 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

2

• 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

3 something NOT deep Deep to be interesting?

Image compression

4

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.

5

Hubbard model

AFM Heisenberg model

half-filling, large U

Perturbation theory

2L

2L

O ✓ t2 U 2 ◆ 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

6

???????

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 chain trillion pixel image million pixel image We cannot simply use the (variant of) existing algorithms.

Totally unrealistic!

“PCA” for low-energy physics of Hubbard

7

Can we solve the “inverse problem” of diagonalization? A

e N × e N

A =

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 ≤ e N

Number counting

8

Exact Diagonalizaiton (ED)

e N × e N

A =

Number of independent params of is e

N × e N

A

Number of eigenvalues available is at most e

N

# of params in # of eigenvalues

A = O( e N × e N) A = O( e N)

∞

e N → ∞

Data Parameters

Too many parameters……

Physicists point of view

9 “Being physical” is a powerful constraint on the Hamiltonian Short-ranged interactions Few-body interactions Symmetries 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 → ∞ 0

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

10

Make the problem concrete

11 ED

E1, ......, 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 to

• 1. 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

12

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 − α ∂ ∂cj Cost(E, E0({cj}))

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

1 2N

N

X

i=1

(E0

i − Ei)2

cj

Cost

ED

E1, ......, EN

E0

1, ......, E0 N

diagonalization cost evaluation gradient descent

E1, ......, En

E0

1, ......, E0 n

H = X

i=1,...,M

ciHi

“Quantum computation” of the gradient

13

◆ ∂ ∂E0 Cost(E, E0({cj}))

Luckly, we know the perturbation theory of quantum mechanics:

H → X

i=1,...,M

ciHi + δcjHj

Ei ! Ei + δcj hΨi| Hj |Ψii

What we have to do is to

• 1. find a proper form of the ansatz
• 2. find the proper parameters for the fixed ansatz

∴ For a given ansatz, we can optimize its parameters ∂ ∂cj Cost(E, E0({cj})) = ✓ ∂ ∂cj E0 ◆

How? trivial H = X

i=1,...,M

ciHi

E0

1, ......, E0 N

Cost(E, E0) → Cost(E, E0) + λ 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)

Both and are nonzero contours of reg. term

Cost(E, E0)

contours of MES contours of reg. term

Demonstration: Hubbard chain at half-filling

15

Spin model ansatz

i

i + 2 i + 3

K1

K2 K3

i

i + 1 i + 2 i + 3

i + 1 i + 2 i + 3 (Si · Si+1)(Si+2 · Si+3)

i

i + 1 i + 2 i + 3 (Si · Si+1)(

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

• ptimized A with regul.
• ptimized 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

16

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)(

Simplified model

λ

Comparison with perturbation theory

17

Perturbation theory

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

Deviation from the perturb.

Difference betw. ML & Macdonald et. al.

e Ji = Ji − Jp

i

e Ki = Ki − Kp

i

∝ U −7

L=10, α = 0.1, Sz = 2, n=50

∝ U −14

Cross validation error Estimation is correct up to

From the viewpoint of image compression

18

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

trillion pixel image million pixel image

We demonstrated the following “image compression”

• riginal (30 MB)

5KB image cf)

something

Next application: entanglement

19 Quantum state (w.f.) ED entanglement spectrum

(ED)-1

Entanglement Hamiltonian Physical Hamiltonian ED DMRG etc. = spectrum of RDM

quantum state reduced density matrix entanglement spectrum Entanglement Hamiltonian

Thermodynamics from pure quantum state

20 B A A

Bath ・Gibbs ensemble ・Reduced density matrix ・Thermodynamic entropy ・Entanglement entropy

thermal fluctuation from the bath quantum fluctuation from B

e.g. entangled two-spins A B

Entanglement Hamiltonian

21 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

22 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 B

e.g.

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

Comparison with other method

23

Full diag. of RDM: exact but computationally hard

・need all the eigenvectors of the RDM → strong constraint: memory size < matrix dimension RDM → 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 20 1 10 100 1000 104 105

High-spin sector → small Hilb. space

chain

Hilbert space dimension

# 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” !

24

arXiv:1803.10856 arXiv:1712.03557

Outlook: Materials design for exotic quantum states

25

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 model e.g. Honeycomb iridates

with exotic quantum state e.g. Q. dimer model e.g. string-net condensation (lattice gauge)

Levin-Wen (2005)

Summary

26

We propose a scheme to construct Hamiltonians from a given spectrum

E1, E2, ......, EN

H

Energy spectrum Entanglement spectrum Low energy Hamiltonian Entanglement Hamiltonian

• Phys. Rev. B 97, 075114 (2018)

Parent Hamiltonian