13HPCI magazine Vol.

Boosting Novel
Materials Discovery−mVMC Software Accurately
Simulates Strongly CorrelatedElectron Systems
In May 2023, the first HPCI Software Award was announced. In the development
category, the software named mVMC was selected for the Grand Prize, and thedevelopment team received recognition. mVMC is capable of efficiently and accurately
calculating structures of interacting electrons that strongly influence each other.
“Utilizing this feature, it becomes possible to explore novel materials,” said Dr. Kota Ido,
one of the key members involved in developing mVMC. He hopes people aiming toevaluate and develop new materials will make use of it. We recently sat down with Dr.
Ido to discuss the software’s development process and its distinctive features. Dr.
Takahiro Misawa, Project Associate Professor at the Institute for Solid State Physics(ISSP), The University of Tokyo, who has been a close collaborator in Dr. Ido’s researchand has contributed to making mVMC open-source, joined the conversation.

Kota Ido

Research Associate
The Institute for Solid State Physics (ISSP), The University of Tokyo01

13

HPCI magazine Vol.

Capable of Simulating Strongly CorrelatedElectron Systems with Up to 1,000 Particles“mVMC, in a nutshell, is a ‘simulator for accurately predicting thestructures of electrons in materials’,” explains Dr. Ido. Many properties ofsubstances are closely related to the structures of electrons within them. Asan everyday example, electric current flows through metals due to thepresence of freely moving electrons, and the direction a magnet points isdetermined by the alignment of the electron’s spin. Therefore, in the field ofquantum condensed matter physics, which investigates the macroscopic
properties of materials based on the behavior of microscopic particles,understanding the movement of electrons is of paramount importance.

mVMC is an application implementing the many-variable variational Monte
Carlo method, which also serves as the inspiration for its name. It has
been under development for a long time. Recently, with the contributionsof Dr. Ido, Dr. Misawa and others, the software has been extended toefficiently represent structures of around 1,000 electrons. Additionally,user-friendly features such as a refined user interface and result
visualization have been incorporated, transforming it into an open-sourcetool accessible to everyone. Looking ahead, mVMC is expected to be
widely used by researchers aiming to discover and develop novel
materials like high-temperature superconductors and quantum materials.
Recognizing its contribution to quantum condensed matter physics,
mVMC was awarded the 2023 HPCI Software Award (Photo 1).

Particularly in “strongly correlated electron systems,” whereelectron-electron interactions are much stronger compared to typicalmaterials, unique phenomena such as superconductivity and quantum
entanglement arise. Given the ongoing development of materials likesuperconductors with minimal energy loss for efficient wiring or materials
indicating strong quantum entanglement crucial for realizing quantum
computers, the significance of simulating the underlying strongly
correlated electron systems has increased.
However, when calculating electronic structure in strongly correlated
systems, the computational complexity increases exponentially with thenumber of electrons. As a result, the exact treatment of a larger numberof electrons has been limited to around 20 electrons, requiring the use ofapproximations to handle more electrons. While approximations can yieldreasonably accurate results when electron interactions are weak,
introducing approximations to calculate the electronic structure in strongly
correlated systems significantly compromises accuracy. Addressing thischallenge and enabling accurate treatment of structures involving around
1,000 electrons is the achievement of the mVMC software.

02

Photo 1 :Dr. Ido holds the certificate for the HPCI Software Award (Development Category, GrandPrize), accompanied by Dr. Misawa holding the trophy.

Boosting Novel Materials Discovery−mVMC Software Accurately Simulates Strongly Correlated Electron Systems

Software Blossoming on the Foundation ofPrevious Generations’ Efforts
First of all, how do we go about calculating the structure of electrons? The
structure of electrons is represented by something called a wave function(Ψ) that spreads out like a wave. This wave function is determined by solving
the Schrödinger equation (Fig. 1). In this equation, the environment inwhich the electrons are situated (for example, how atoms are arranged ina solid) is represented by an operator called the Hamiltonian ( ). Solving
this equation also yields the energy ( ) of the wave function.
However, it should be noted that solving the Schrödinger equation exactlyis limited to special cases, such as the hydrogen atom with a single
electron. As a result, in fields like condensed matter physics that deal witha large number of electrons in materials, researchers have resorted toapproximations to solve this equation. However, as mentioned earlier, inthe case of strongly correlated electron systems, applying approximations
while dealing with a large number of electrons severely compromises
accuracy, making it challenging to obtain calculation results that could be
useful for designing new materials.
One of the solutions to this challenge is mVMC, but the journey to Dr. Ido
and his team’s recognition was quite difficult.

Fig. 1 : Schrödinger equation

The micro world, like electrons, follows the rules of quantum mechanics. TheSchrödinger equation is an equation based on quantum mechanics used to calculate
the structures of micro particles such as electrons.

The HPCI Software Award
The HPCI Software Award was established in 2023 by the High Performance
Computing Infrastructure (HPCI) Consortium, a general incorporated
association. The award is intended to recognize and honor organizations,particularly those led by young developers, that have made significant
contributions to the advancement of computational science and havedeveloped software that is deemed especiallybeneficial. It also acknowledges entities that havecontributed to the dissemination of such software. The
award is divided into two categories: the Development
Award and the Dissemination Award. Among these,
our consortium honored mVMC development teamwith the top prize in the Development Award category.

Hirofumi Tomita,

President of the HPCI Consortium

A method that has been used for a long time to represent the structures ofmany electrons is the “variational Monte Carlo method,” which combines
the concepts of a “trial wave function” and the “Monte Carlo method.” In
this approach, one starts by assuming an electron’s structure andconstructs a trial wave function to represent it. Then, through repeatedcalculations, the energy is determined while adjusting the parameters of
the trial wave function to optimize it, aiming to find the wave function thatyields the lowest energy. However, depending on the shape of the trial
wave function, it’s sometimes impossible to calculate energy exactly.
Therefore, the Monte Carlo method is employed to predict the expectedvalues of quantities like energy. With advancements in computing, it has
become possible to compute the structures of around 1,000 electrons.
However, the accuracy of representing the structures of many electrons
depends on the assumed trial wave function. Increasing the number of03

13

HPCI magazine Vol.

parameters to improve accuracy also increases the computational effort,
and the number of parameters that can be introduced in the trial wave
function has been limited to maintain tractability.
Dr. Ido explains: “In 2001, Dr. Sandro Sorella from Italy proposed a good
optimization method that allowed the introduction of more parameters.
Furthermore, this approach was further developed by Mr. Daisuke Taharaand Dr. Masatoshi Imada*1 from The University of Tokyo in 2008, who
introduced the ‘many-variable variational Monte Carlo method.’ This
allowed the incorporation of many parameters, providing flexibility to therepresentation capability of the wave function and increasing accuracy.
When the number of parameters was limited, we were restricted to trial
wave functions that ‘forced’ certain aspects, like the alignment of spins.
However, since 2008, it became possible to use trial wave functions thatcan represent various states using a single form, enabling therepresentation of more complex electronic structure flexibly.”“However, at that time, it felt like an internally used computational code. In
2016, Dr. Misawa received support from the ISSP’s project for theadvancement of software usability in materials science, and he shaped it
into a form that could be publicly released as software. Thus, mVMC’sdevelopment involved the contributions of many individuals. Additionally, Iaimed to enhance the accuracy of the trial wave function and functionalityof the software. Without years of accumulated effort, we wouldn’t behere,” Dr. Ido modestly says. However, Dr. Misawa highlights, “Thecomputational complexity to compute electronic structures among 1000
interacting electrons requires at least 21000. Even if we started running thesupercomputer Fugaku from the moment the universe began, we still
wouldn’t have completed such calculations. Dr. Ido devised a way tocalculate this in a realistic timeframe without losing the essence ofstrongly correlated electron systems.” He freely praises Dr. Ido’sachievements (Photo 2).

04

Photo 2 : Reflecting on the development of mVMC -- Drs. Ido and Misawa

Achieving Significant Results Using mVMC
In April 2022, Dr. Ido and his team issued a press release announcing
their achievement of elucidating the unique properties of a quantum spin
liquid that arises in organic solids using mVMC. Organic solids arematerials in which organic compounds are regularly arranged. If theconstituent organic compounds are complex molecules, the resultingorganic solid will also have intricate structures. For instance, the organic
solid Pd(dmit)2 salt, which was the focus of this study, is composed ofalternating layers of a metal complex layer (Pd(dmit) 2) and a cation layer(Fig. 2a). By altering the compounds in the cation layer, a transformation
has been observed experimentally from a magnetic phase with orderedelectron spin arrangements to a quantum spin liquid phase where the spinorientation remains undetermined.

Boosting Novel Materials Discovery−mVMC Software Accurately Simulates Strongly Correlated Electron Systems(a)(b)
Cation layer

Energy difference between the quantum spin liquid phaseand the magnetic phase[meV]
Results of this study □Experimental results ○0.8
0.6
0.40.2

Me4PMe4As
Me4Sb0Metal complex layer

etureraptem
10

on N[K]siti
20na
Tr30-0.2-0.4 0.4040

EtMe3 Sb-0.3

Et2 Me2 As-0.2-0.10

Strength of frustration( t c - t b )/t aCation layer
Magnetic phase

Quantum spin liquid phaseFig. 2 : Expanded crystal structure of organic solid EtMe3Sb[Pd(dmit)2]2 (a) and phase diagram of organic solid Pd(dmit)2 salts (b)(a) Crystal structure of the organic solid EtMe3Sb[Pd(dmit)2]2, which is one of the Pd(dmit)2 salts. The structure consists of alternating layers of the cation layer of EtMe3Sb and the metal
complex layer [Pd(dmit)2]2. Here, Et stands for ethyl, Me for methyl, Sb for antimony, Pd for palladium, and dmit for 1,3-dithiole-2-thione-4,5-dithiolate. In the metal complex layer, two dmit
molecules surround Pd (gray spheres).
(b) The vertical panel illustrates the dependence of the energy difference between the quantum spin liquid phase and the magnetic phase (vertical axis), calculated in this study, on thestrength of frustration among electron spins (horizontal axis). When the value on the vertical axis is negative, the quantum spin liquid phase is realized. The lower panel plots the relationshipbetween the temperature (transition temperature) observed in experiments for the onset of the magnetic phase and the strength of frustration. The blue and orange shaded regions indicate
the experimentally observed regions where the magnetic phase and the quantum spin liquid phase appear, respectively. The color coding in the vertical panel is an extension of the color
coding in the lower panel. Me4P, Me4As, Me4Sb, Et2Me2Sb and EtMe3Sb are cation molecules present in each of the organic solid Pd(dmit)2 salts. Changing the cation molecules results invariations in the plotted data. Only EtMe3Sb[Pd(dmit)2]2 highlighted in red exhibits a quantum spin liquid, successfully reproducing experimental resultsSource: Ido, K., Yoshimi, K., Misawa, T. and Imada, M. Unconventional dual 1D–2D quantum spin liquid revealed by ab initio studies on organic solids family.
(2022). https://doi.org/10.1038/s41535-022-00452-8. 7, 4805

13HPCI magazine Vol.

Quantum spin liquids, known for exhibiting strong quantum entanglement*2,
are currently drawing attention due to potential applications in quantumcomputing. In the case of Pd(dmit)2 salt, it was speculated that changing thetype of cation layer would alter the frustration*3 between electron spins anddestabilize the electron spins responsible for the magnetic properties ofthe metal complex layer, leading to the emergence of a quantum spin
liquid. However, the precise conditions and characteristics of thisphenomenon remained largely unknown.
To address this, Dr. Ido’s team selected five different types of Pd(dmit)2
salts and conducted calculations using mVMC. By reproducing thestructures of electrons within the substance on a computer, they clarified
the conditions under which a quantum spin liquid appears. Their findingsaligned with existing experimental results (Fig. 2b). Furthermore, a closer
examination of the properties of the quantum spin liquid provided novel
insights that could contribute to the development of quantum devices.

Aiming for Further Advancement of the Software“In various fields, mVMC has been selected for the award, so I want tofurther develop mVMC in a way that lives up to the honor of the award,”says Dr. Ido. He is already advancing the sophistication of mVMC.
In fact, in the aforementioned press release study, artificial neuralnetworks were employed during the optimization of trial wave functions.
As a result, computational accuracy improved, leading to plans for
integrating artificial neural network capabilities as a standard feature in
the publicly available mVMC.
Furthermore, including the research described in the press release, they
have previously focused on solving only the lowest-energy ground stateas the wave function. However, experimental methods for understanding
the electronic structure often involve irradiating materials with light or06

neutrons to gain information about excited states (states with higher energy
than the ground state). To elucidate electronic structures through comparison
between experiments and simulations, it’s crucial to be able to computeexcited structures of electrons. Therefore, enabling the representation of
excited structures of electrons is a major future goal for Dr. Ido, and he’s
working on this challenge within a project (hp230213) of the Program for
Promoting Research on the Supercomputer Fugaku for fiscal 2023.
In simulations of quantum condensed matter physics, researchers
represent their target materials using Hamiltonians and solve theSchrödinger equation. Given that mVMC can flexibly represent electronicstructures, it’s applicable not only to the quantum spin liquid highlighted in
the press release, but also to various strongly correlated electronsystems. Additionally, since it’s easy to input information about the target
material, several research groups, both domestically and internationally,
have already started using mVMC. Moreover, as Dr. Misawa points out,
“You don’t need a supercomputer like Fugaku to run this software. Whatmatters is how you write a Hamiltonian to employ mVMC. Even with asmall number of electrons, there’s ample potential for revealing
interesting phenomena.”Drs. Ido and Misawa have built upon the foundation laid by theirpredecessors, nurturing mVMC into an application that’s freely available
to many researchers. It’s poised to continue blossoming with further
improvements in performance and being employed on numerous
computing platforms, not just within HPCI, contributing significantly to thefield of quantum condensed matter physics.
*1 Current affiliation: Waseda Research Institute for Science and Technology, Waseda
University/ Faculty of Science and Technology, Sophia University*2 Strong coupling phenomenon between two micro-particles. When the state of one isdetermined, the state of the other is also determined.
*3 Phenomenon in which the arrangement of electron spins becomes unstable due to thecrystal structure.

13HPCI magazine Vol.

Boosting Novel Materials Discovery−mVMC Software Accurately Simulates Strongly Correlated Electron Systems

About theResearcher“Even my hobbies are a form of research,” Dr. Ido says. He constantly juggles multiple
research projects, finding diversion in some of them as a way to take a break. “Ifsomething seems interesting, even if there’s a high barrier, it becomes a way to unwind.
When things go well, there’s that feeling of ‘Yes!’” During his student days, he neverintended to become a researcher. It was with a light-hearted approach that he got
involved in challenging simulation techniques to learn some technical skills before
entering the workforce, and he got completely hooked. That passion from those days stillburns bright, continuously driving him, and it’s what led to the development andachievements we’ve discussed here.

Associated Research Projects:・AI numerical spectroscopy for analyzing emergent structures of quantum entanglement in correlated quantum materials (hp230213)
Principal Investigator: Youhei Yamaji, National Institute for Materials Science・Basic Science for Emergence and Functionality in Quantum Matter - Innovative Strongly-Correlated Electron Science by Integration of Fugaku and Frontier Experiments (hp200132/ hp210163/ hp220166)
Principal Investigator: Masatoshi Imada, School of Engineering, Waseda Research Institute for Science and Technology, Waseda University

Registered Institution for Facilities Use Promotion /The Representative for HPCI Operation

Research Organization
for Information Science and Technology

1-5-2, Minatojimaminami-machi, Chuo-ku, Kobe, Hyogo, 650-0047 JAPAN
Tel: +81-78-599-9511

HPCI Portal site

https://www.hpci-office.jp/en

HPCI PR site

https://fugaku100kei.jp/e/Issued: September 2023
© 2023 Research Organization for Information Science and Technology07