Conventional

Inverse

⾃動微分を⽤いた欲しい性質をもつハミルトニアンの逆設計:
バンドトポロジーと量⼦エンタングルメントへの応⽤

Koji Inui, The University of Tokyo
2023/12/14 DLAP@上智⼤学

K. Inui, and Y. Motome, Commun. Phys. 6, 37 (2023).
https://github.com/koji-inui/automatic-hamiltonian-design

Table of contents
Introduction
Inverse problem in materials science
Framework
Inverse design of Hamiltonians with large anomalous Hall effectResults 1: Rediscovery of the Haldane model
Results 2: Discovery of a new Hamiltonian on a triangular lattice
Inverse design of quantum spin Hamiltonians with large quantum entanglement
Results 3: Automatic construction of the Kitaev model on the honeycomb lattice model
Summary and perspectives
Applicability to other problems

Inverse problem in materials science
Conventional theoretical approach

Conventional

1. Construct the Hamiltonian based on phenomenology or first principles
2. Tune the parameters in the Hamiltonian by calculating physical properties ofinterest
3

Inverse problem in materials science
Conventional theoretical approach

Conventional

Parameter space
Hopping terms 𝑡! , 𝑡" , 𝑡# …
Interaction terms 𝑈, 𝑉! , 𝑉" …
Temperature 𝑇
Magnetic field 𝐻(𝜔)Spin-orbit coupling 𝜆Lattice constant, aFilling 𝑁…

⊗Anisotropy
Site dependence
Layer dependence
Orbital dependence

Construct a simple model
based on intuition and feedback
Calculate the physical properties of interest

Laborious to explore !

4

Inverse problem in materials science
Inverse approach

Conventional

Inverse

1. Prepare the desired physical properties
2. Construct a Hamiltonian to realize them directly
Benefits

・It can bypass the laborious exploration
5

Inverse problem in materials science
Inverse approach

Conventional

Inverse

1. Prepare the desired physical properties
2. Construct a Hamiltonian to realize them directly
Benefits・It can bypass the laborious exploration
・It can reach the qualitatively new principles and materials

6Previous studies to estimate Hamiltonians
Bayesian
optimization

Generativemodels

R. Tamura and K. Hukushima, Phys. Rev. B 95 (2017) 164407.
K. Obinata, S. Katakami, Y. Yue, and M. Okada, J. Phys. Soc. Jpn. 88 (2019) 064802.

Machinelearning
T. Mertz and R. Valenti, Phys. Rev. Research 3 (2021) 013132.

Perturbation
theoryB. Sanchez-Lengeling and A. Aspuru-Guzik, Science 361 (2018) 360.

Geneticalgorithm

H. Fujita, Y. O. Nakagawa, S. Sugiura, and M. Oshikawa,Phys. Rev. B 95 (2017) 164407.

G. L. W. Hart et al., Nat. Mater. 4 (2005) 391.
A. Ajoy and P. Cappelaro, Phys. Rev. Lett. 110 (2013) 220503.

・Difficulty in learning because of the cost of collecting data and computation
・The validity of the results is not guaranteed
・Limitations on the models and physical quantities that can be applied

7

Research Purpose

Develop a versatile framework todiscover Hamiltonians with the desired properties
Proof of concept of the framework through application toanomalous Hall effect and quantum entanglement.
8

Framework

Automatic differentiation (Backpropagation)
Analytical derivatives of (almost) any function can be computedby applying the chain rule

Software libraries

Wikipedia,“automatic differentiation”

𝜕𝑓𝑤
!=𝜕𝑤

Some studies use overa trillion parameters

While it is used as backpropagation in deep learning,it can be applied to versatile applications.

W. Fedus et al., J. mach. Learn. Res. 23 (2022) 120.

(This framework does NOT
use neural networks) 10

Framework
Numerical method
Diagonalization

Hartree-Fock

Band calculation

Kohn-Sham eq.

DMFT…

Example)𝐿 𝜽 = −⟨𝑃 𝜽 ⟩𝐿 𝜽 = $ 𝑆"#$%&" 𝒌 − 𝑆 𝒌, 𝜽'𝒌Many parameters can be optimized
simultaneously
Possibility to discoverunprecedented Hamiltonians
beyond human intuition
11Framework
Neural networks

Our framework
Parametrized

Fixed

Hamiltonianℋ(𝜽)Parameters 𝜽Data

Parameters 𝜽ParametrizedDiagonalization Hartree FockBand calculation DMFT…
Perturbation

Neural network
model

ObjectivefunctionExample)

Example)𝐿 𝜽 =!
"##∑# 𝑦$%&'𝜽 − 𝑦()*+Fixed
Numerical
methodObjective physical𝐿 𝜽 = −⟨𝑃 𝜽 ⟩propertiesResult 1
Rediscovery of the Haldane modelTight binding model on honeycomb lattice

Twosublattices

Maximize anomalous Hall conductivity 𝜎)*

𝒌 = 100×100: 14 parameters
Objective function :!"!"

Set finite temperature to avoid !𝜽 becomes 014

Tight binding model on honeycomb lattice

Twosublattices

Increasing 𝜎)* automatically changes the Chern numbers to +1 and -1※ 𝜎*+ < 1 due tofinite temperature
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Tight binding model on honeycomb lattice

Twosublattices

Optimized parameters correspond to the center of the 𝐶 = 1 phasein the Haldane model
The phase diagram ofthe Haldane model

Different initial conditions also converge to the same state→ Our framework automatically rediscover the Haldane model

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Result 2

Discovery of a new Hamiltonian on a triangular latticeTight binding model on triangular lattice
Maximize anomalous Hall conductivity 𝜎)*

Foursublattices

𝒌 = 100×100: 38 parameters
Fixed to half-filling18

Tight binding model on triangular lattice

Foursublattices

Increasing 𝜎)* automatically changes the Chern numbers to [5,1,-3,-3]※ 𝜎$% < 6 due tofinite temperature

The total Chern number has a large value of 619

Tight binding model on triangular lattice

Foursublattices(approximately) three-fold rotational symmetry appears automaticallyTopview20

Tight binding model on triangular lattice

Foursublattices

Fictitious flux takes regular values
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𝜙4 , Φ4Φ'⊳ ≅ 0.9𝜋
Φ'⊲ ≅ 1.6𝜋𝑡/ , 𝑡0 = 10
𝜙12
∈ (0, 𝜋)7𝜋
Φ( ≅4𝑎)'𝜙&'(𝜙&𝑏()
𝜙&𝑐fictitious flux
,Φ+ = ∑𝜙+As far as we know, such a Hamiltonian has never been reported.
This proves that our framework can discover new Hamiltonians.

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Result 3

Application to quantum entanglement in many-body systems

Entanglement entropy (EE)
The measure of quantum entanglement in quantum many-body systems.

R Islam et.al., Nature528, 77 (2015)

𝑆5 = 0

𝑆5 > 0

Entanglement entropy
23

Entanglement entropy in various quantum phenomena
QuantumPhase transition

Topologicalorder

G.Vidal, et.al., PRL.
90, 227902 (2003)

A. Kitaev and J. Preskill,
PRL. 96, 110404 (2006)

Many-bodylocalization

Black holes

D. J. Luitz, et.al., PRB.
91, 081103(R) (2015)

T. Hartman and J. Maldacena,JHEP. 91, (R) (2006)

Quantum entanglement appears in various field such as condensed matterphysics, high-energy physics, and quantum information.

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Designing quantum entanglement
Conventional research has focused on investigating quantum entanglement
properties of specific quantum systems.
Hamiltonianℋ

Entanglement properties

Conventional

InverseMeanwhile, considering applications like quantum computation, we need methods todesign a system with desired quantum entanglement properties by solving inverseproblems.
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Objective function
Objective:
Designing Hamiltonians with large entanglement in the ground state
Is it OK to increase EE of the ground state? → No.

𝐿 = −𝑆!

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Objective function
Objective:
Designing Hamiltonians with large entanglement in the ground state
Is it OK to increase EE of the ground state? → No.
😥 The optimization becomes unstable when switching the ground states
Optimization process

1st Excited state

EE of the ground state
suddenly change

Ground state27

Objective function
Objective:
Designing Hamiltonians with large entanglement in the ground state
Is it OK to increase EE of the ground state? → No.
😥 There are several ways of partitioning into A and B
Group g=1,Pattern 𝜉 = 1

Group g=1,Pattern 𝜉 = 2Group g=2,
Pattern 𝜉 = 1…

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Objective function
Thermally ensemble EE (TEEE)Mean of TEEE for partitions𝑛: index of eigenstate(𝑔, 𝜉): partitioning pattern
𝐸- : Energy of state 𝑛𝑆-,.,/ : EE at state 𝑛, partitioning (𝑔, 𝜉)
𝛽: inverse temperature

Std of TEEE for partitions

😁 𝐿 does not change when switching the ground states

𝜆>0

Maximize TEEE

Make TEEE uniform for partitions
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Quantum spin systems on a honeycomb lattice

K. Inui, and Y. Motome, in preparation

18 parameters
Bipartition patterns

30Quantum spin systems on a honeycomb lattice

K. Inui, and Y. Motome, in preparation

By increasing the quantum entanglement, the Kitaev model automatically appears

31Various Lattices

K. Inui, and Y. Motome, in preparation

We run calculation with 100 different initial conditions.
Kitaev model

Kitaev model

Inhomogeneous

Inhomogeneous32

Maple-leaf lattice

K. Inui, and Y. Motome, in preparation

𝐿 ≅ −1.708
We found a new Kitaev-like modelwith large quantum entanglement.
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Summary and perspective

K. Inui, and Y. Motome, Commun. Phys. 6, 37 (2023).
https://github.com/koji-inui/automatic-hamiltonian-design

We have developed and demonstrated a framework to automatically construct
a Hamiltonian with the desired properties using automatic differentiation.
We apply this framework to maximizing the Hall conductivity
1. Rediscovering the Haldane model on honeycomb lattices
2. Discovery of a new Hamiltonian with large Hall coefficient and topologically nontrivial properties ona triangular lattice
We apply it to maximizing the entanglement entropy
1. The method is found to generate the Kitaev model on the honeycomb lattice.
2. It generate a new model with large enganglement on the maple-leaf lattice.

Perspective: the method has a wide range of applicability beyond entanglement
Numerical methodsl Tensor networksl DMRGl DFT

Physical propertiesl Topological entanglement entropyl Superconductivity
l Reproduction of experiments
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