Interface tool from Wannier90 to RESPACK: wan2respack

TL;DR

The paper introduces 'wan2respack', an interface tool that converts Wannier functions from Wannier90 into a format compatible with RESPACK, enabling accurate derivation of low-energy Hamiltonians for correlated materials.

Contribution

We developed and validated 'wan2respack', a tool that bridges Wannier90 and RESPACK, facilitating the calculation of low-energy Hamiltonians with consistent Wannier functions.

Findings

'wan2respack' accurately converts Wannier functions between formats.

The low-energy Hamiltonians from both Wannier90 and RESPACK formats agree.

The tool simplifies workflows for correlated material simulations.

Abstract

We develop the interface tool $\verb|wan2respack|$, which connects $\verb|RESPACK|$ (software that derives the low-energy effective Hamiltonians of solids) with $\verb|Wannier90|$ (software that constructs Wannier functions). $\verb|wan2respack|$ converts the Wannier functions obtained by $\verb|Wannier90|$ into those used in $\verb|RESPACK|$, which is then used to derive the low-energy effective Hamiltonians of solids. In this paper, we explain the basic usage of $\verb|wan2respack|$ and show its application to standard compounds of correlated materials, namely, the correlated metal SrVO$_3$ and the high-$T_{c}$ superconductor La$_2$CuO$_4$. Furthermore, we compare the low-energy effective Hamiltonians of these compounds using Wannier functions obtained by $\verb|Wannier90|$ and those obtained by $\verb|RESPACK|$. We confirm that both types of Wannier functions give the same Hamiltonians. This benchmark comparison demonstrates that $\verb|wan2respack|$ correctly converts Wannier functions in the $\verb|Wannier90|$ format into those in the $\verb|RESPACK|$ format.

Equations

$$w_{i\mathbf{R}}(\mathbf{r})=\frac{1}{\sqrt{N_{\mathbf{k}}}}\sum _{\mathbf{k}}\sum _m{U}_{im}(\mathbf{k}){\psi }_{m\mathbf{k}}^{\mathrm{opt}}(\mathbf{r})e^{-i\mathbf{k}\cdot \mathbf{R}},$$

$${\psi }_{m\mathbf{k}}^{\mathrm{opt}}(\mathbf{r})=\sum _n{U}_{mn}^{\mathrm{opt}}(\mathbf{k}){\psi }_{n\mathbf{k}}(\mathbf{r}),$$

$${\psi }_{n\mathbf{k}}(\mathbf{r})=\frac{1}{\sqrt{N_k}}\frac{1}{\sqrt{\Omega }}\sum _{\mathbf{G}}{C}_{\mathbf{G}n}(\mathbf{k})e^{i(\mathbf{k}+\mathbf{G})\cdot \mathbf{r}},$$

$$w_{i0}(\mathbf{r})=\frac{1}{N_k}\sum _{\mathbf{k}}\sum _{\mathbf{G}}{\tilde{C}}_{\mathbf{G}i}(\mathbf{k})\frac{1}{\sqrt{\Omega }}{e}^{i(\mathbf{k}+\mathbf{G})\cdot \mathbf{r}}.$$

$${\tilde{C}}_{\mathbf{G}i}(\mathbf{k})=\sum _{m,n}{U}_{im}(\mathbf{k})U_{mn}^{\mathrm{opt}}(\mathbf{k})C_{\mathbf{G}n}(\mathbf{k}).$$

$$H=\sum _{\sigma ,\mathbf{R}{\mathbf{R}}^{\prime },\alpha \beta }{t}_{\alpha \mathbf{R},\beta {\mathbf{R}}^{\prime }}{c}_{\alpha \mathbf{R}\sigma }^{†}{c}_{\beta {\mathbf{R}}^{\prime }\sigma }$$

$$\displaystyle+\frac{1}{2}\sum_{\sigma\rho,\bm{R}\bm{R}^{\prime},\alpha\beta}\Big{[}U_{\alpha\bm{R},\beta\bm{R}^{\prime}}c_{\alpha\bm{R}\sigma}^{\dagger}c_{\beta\bm{R}^{\prime}\rho}^{\dagger}c_{\beta\bm{R}^{\prime}\rho}c_{\alpha\bm{R}\sigma}$$

$$+J_{\alpha \mathbf{R},\beta {\mathbf{R}}^{\prime }}(c_{\alpha \mathbf{R}\sigma }^{†}{c}_{\beta {\mathbf{R}}^{\prime }\rho }^{†}{c}_{\beta \mathbf{R}\rho }{c}_{\alpha {\mathbf{R}}^{\prime }\sigma }$$

$$\displaystyle+c_{\alpha\bm{R}\sigma}^{\dagger}c_{\alpha\bm{R}\rho}^{\dagger}c_{\beta\bm{R}^{\prime}\rho}c_{\beta\bm{R}^{\prime}\sigma})\Big{]},$$

$$t_{\alpha \mathbf{R},\beta {\mathbf{R}}^{\prime }}$$

$$U_{\alpha \mathbf{R},\beta {\mathbf{R}}^{\prime }}$$

$$\times W(\mathbf{r},{\mathbf{r}}^{\prime })w_{\beta {\mathbf{R}}^{\prime }}^{\ast }({\mathbf{r}}^{\prime })w_{\beta {\mathbf{R}}^{\prime }}({\mathbf{r}}^{\prime }),$$

$$J_{\alpha \mathbf{R},\beta {\mathbf{R}}^{\prime }}$$

$$\times W(\mathbf{r},{\mathbf{r}}^{\prime })w_{\beta {\mathbf{R}}^{\prime }}^{\ast }({\mathbf{r}}^{\prime })w_{\alpha \mathbf{R}}({\mathbf{r}}^{\prime }).$$

Full text

††journal: Computer Physics Communications

PROGRAM SUMMARY

Program title: wan2respack

Licensing provisions: GNU General Public License version 3

Programming language: Fortran and python3

Computer: PC, cluster machine

Operating system: Unix-like system, tested on Linux and macOS

Keywords: Wannier functions, $ab$ $initio$ downfolding, constrained random phase approximation, strongly correlated electron systems.

External routines/libraries: Quantum ESPRESSO (version 6.6), Wannier90 (version 3.0.0), RESPACK (version 20200113), tomli.

Nature of problem: Using RESPACK, one can derive low-energy effective Hamiltonians of solids from maximally localized Wannier functions. However, due to the differences in the representation of Wannier functions, the Wannier functions obtained by Wannier90 cannot be directly used in RESPACK.

Solution method: wan2respack converts the Wannier functions in the Wannier90 format into those in the RESPACK format. Using the converted Wannier functions, one can derive the low-energy effective Hamiltonians using RESPACK.

1 Introduction

Wannier functions, constructed through unitary transformations of Bloch functions, offer simple, convenient representations of the electronic structures of solids [1]. After the development of an efficient method of obtaining unique sets of maximally localized Wannier functions (MLWFs) [2, 3], MLWFs have been widely used to analyze the electronic structures of solids. For example, MLWFs play an essential role in calculating electronic polarizations [4, 5] and orbital magnetizations [6, 7]. MLWFs are also used to construct tight-binding models [8], which reproduce low-energy bands near the Fermi level.

MLWFs are also vital for deriving the $ab$ $initio$ low-energy effective Hamiltonians of solids  [9, 10]. Based on MLWFs, the screened interactions in low-energy effective Hamiltonians are evaluated through constrained random phase approximation (cRPA). The derivation of low-energy effective Hamiltonians is often called $ab$ $initio$ downfolding. Correlation effects beyond the conventional density functional theory (DFT) [11, 12] can be considered by solving low-energy effective Hamiltonians using accurate solvers. In the last decade, this method has been applied to a wide range of strongly correlated compounds [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38].

The open-source software package Wannier90 [39, 40, 41, 42] was developed to construct MLWFs from the results of $ab$ $initio$ band calculations. The MLWFs obtained by Wannier90 can be used to calculate many important properties of solids, such as Berry phases and anomalous Hall conductivity [43], electrical conductivity [44], and spin Hall conductivity [45]. Because of this versatility, many $ab$ $initio$ calculation software packages, such as Quantum ESPRESSO [46, 47], VASP[48, 49], WIEN2k[50], ABINIT[51], and OpenMX[52, 53, 54], have an interface to Wannier90.

Nakamura et al. recently developed the open-source software package RESPACK [55, 56], which implements $ab$ $initio$ downfolding. After its release, RESPACK has been applied to a wide range of correlated materials [57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71]. RESPACK also provides functions (RESPACK-Wannier) that construct MLWFs using the $ab$ $initio$ band calculation results obtained by Quantum ESPRESSO or xTAPP [72, 73]. The RESPACK-Wannier code was developed independently of Wannier90. Therefore, their representation of Wannier functions has technical differences, such as the choice of $k$-point mesh (e.g., based on irreducible or reducible representation), although the constructed Wannier functions themselves are equivalent. Due to these differences, RESPACK cannot be directly connected with Wannier90.

In this paper, we introduce the interface tool wan2respack [74], which converts Wannier functions in the Wannier90 format into those in the RESPACK format. After the conversion of Wannier functions, the screened Coulomb and exchange interactions can be evaluated using RESPACK. The rest of this paper is organized as follows. In Section 2, we give an overview of the formats of the MLWFs implemented in Wannier90 and RESPACK and explain how to convert them using wan2respack. We also explain how to derive low-energy effective Hamiltonians based on the converted MLWFs. In Sections 3 and 4, we explain the installation and usage, respectively, of wan2respack. In Section 5, we show the application of wan2respack to the correlated materials SrVO3 and La2CuO4. Section 6 summarizes this paper.

2 Overview

This section provides an overview of the formats of the Wannier functions implemented in Wannier90 and RESPACK. We also explain how wan2respack converts Wannier functions in the Wannier90 format into those in the RESPACK format. Finally, we discuss how to derive the low-energy effective Hamiltonians of solids based on the converted MLWFs.

We obtain MLWFs through unitary transformations of Bloch functions. The Wannierization procedure consists of the following optimization of two quantities:

Projection [$U({\bm{k}})^{\rm opt}_{mn}$ in Eq. (1)] of the Bloch functions within the given energy window to minimize the gage-invariant Wannier spread 2. 2.

Unitary transformation [$U_{im}({\bm{k}})$ in Eq. (2)] of the optimal Bloch wave functions ($\psi^{\rm opt}({\bm{k}})$ in Eq. (2)) obtained by the first step

According to the Wannierization procedure, the $i$th Wannier function around site $\bm{R}$ ($\bm{R}$ is often called the Wannier center), $w_{i\bm{R}}(\bm{r})=w_{i\bm{0}}(\bm{r}-\bm{R})$, is defined as

[TABLE]

where $N_{\bm{k}}$ is the number of $\bm{k}$-point meshes and $U_{im}({\bm{k}})$ is the unitary matrix, which transforms the $m$th optimized Bloch wave function $\psi_{m\bm{k}}^{\rm opt}({\bm{r}})$ into the $i$th Wannier function. Here, $\psi_{m\bm{k}}^{\rm opt}({\bm{r}})$ is given by

[TABLE]

where $U_{mn}^{\rm opt}({\bm{k}})$ is the projection matrix. The Bloch wave function $\psi_{n\bm{k}}({\bm{r}})$ is defined as

[TABLE]

where $\Omega$ is the volume of the calculation unit cell and $C_{\bm{G}n}(\bm{k})$ is the expansion coefficient of the plane wave.

Here, we explain the difference between the Wannier90 and RESPACK formats of Wannier functions. By substituting the definition of the Bloch functions into Eq. (1) for $\bm{R}=\bm{0}$, we obtain

[TABLE]

The relation between $\tilde{C}_{\bm{G}i}(\bm{k})$ and $C_{\bm{G}n}$ is

[TABLE]

The Wannier functions in RESPACK are defined in Eq. (4) and used in the calculations of the screened Coulomb interactions. RESPACK reads wave functions in the irreducible Brillouin zone (IBZ), $C_{\bm{G}n}(\bm{k})$, where $\bm{k}\in{\rm IBZ}$, to reduce the computational cost, particularly that of the dielectric function. Since crystal symmetry is not taken into account in the standard Wannierization procedure[75], $U_{im}(\bm{k})$, $U_{mn}^{\rm opt}(\bm{k})$, and $\tilde{C}_{\bm{G}i}(\bm{k})$ are calculated on the full $\bm{k}$-point mesh by expanding $C_{\bm{G}n}(\bm{k})$ in the IBZ to the full $\bm{k}$-point mesh. When using wan2respack, we first calculate wave functions in the IBZ, $C_{\bm{G}n}(\bm{k})$, to compute the dielectric function. Then, we separately calculate wave functions on the full $\bm{k}$-point mesh and generate Wannier functions using Quantum ESPRESSO and Wannier90. wan2respack computes $\tilde{C}_{\bm{G}i}(\bm{k})$ based on these outputs. Here, we have to use the same full $\bm{k}$-point mesh used in RESPACK.

After the Wannier functions are obtained, the following $ab$ $initio$ Hamiltonians are derived:

[TABLE]

where $c_{\alpha\bm{R}\sigma}^{\dagger}$ ($c_{\alpha\bm{R}\sigma}$) is a creation (annihilation) operator of the $\alpha$th Wannier orbital on $\bm{R}$ with the spin $\sigma$. The transfer integral and the screened Coulomb and exchange interactions are defined as

[TABLE]

Here, $H_{0}(\bm{r})$ and $W(\bm{r},\bm{r}^{\prime})$ represent the one-body part of the Hamiltonian (Kohn–Sham Hamiltonian) and the static screened Coulomb interaction obtained through cRPA, respectively. The integrals are obtained over the crystal volume $V$. RESPACK calculates these quantities based on the MLWFs.

3 Installation of wan2respack

wan2respack can be downloaded from the following GitHub repository:

https://github.com/respack-dev/wan2respack

Users can compile wan2respack using CMake. The typical installation procedure of wan2respack is as follows:

$cd$PATH_to_wan2respack $mkdir build$ cd build $cmake ../ -DCONFIG=$Type_of_Configure -DCMAKE_INSTALL_PREFIX=$PATH_to_Install$ make $ make install

$PATH_to_wan2respack is the path to the wan2respack directory and$PATH_to_Install is the path to the installation directory. By replacing $Type_of_Configure with the name of CMake configuration files, users can specify their desired compilers.

All binary files and Python scripts are installed to $PATH_to_Install/bin. The roles of the scripts in bin are as follows:

init.py: Common functions are defined in this Python module.

• 2.

qe2respack.py: This Python script is used to generate the input files of RESPACK from the output files obtained by Quantum ESPRESSO band calculations.\cprotect111This script is originally distributed under GNU GPL version 3 by the open-source software RESPACK— version 20200113.

• 3.

wan2respack.py: This is the main Python script. The configuration file name should be written in the toml format [76]. This script calls wan2respack_pre.py for preprocessing and wan2respack_core.py for core processing. These scripts are described as follows:

• (a)

wan2respack_pre.py: This Python script is used to save the Quantum ESPRESSO results and export ${\bm{k}}$ points with gen_mk.x and qe2respack.py.

• i.

gen_mk.x: This execution file, written in Fortran90, for calculating the ${\bm{k}}$-point mesh for Wannier90.

• (b)

wan2respack_core.py: This Python script is used to prepare files about Wannier functions in the RESPACK format using gen_wan.x and qe2respack.py.

• i.

gen_wan.x: This execution file, written in Fortran90, converts the Wannier function information in the Wannier90 format into that in the RESPACK format.

4 Usage of wan2respack

The calculation flow of wan2respack is as follows:

$Ab$ $initio$ DFT calculations are performed using Quantum ESPRESSO. 2. 2.

${\bm{k}}$ points are generated for Wannier90 using wan2respack. 3. 3.

Wannier functions are constructed using Wannier90. 4. 4.

The Wannier functions are converted using wan2respack. 5. 5.

The screened interactions are calculated using RESPACK.

Through these procedures, we obtain the screened Coulomb and exchange interactions in the low-energy effective Hamiltonians of solids. This flow is summarized in Fig. 1.

In the following, we explain the usage of wan2respack. The input files for performing these calculations are in samples/seedname.lattice.kmesh, where seedname is the name of the target compound, lattice is the abbreviation of the Bravais lattice, and kmesh is the number of $k$ points. This directory consists of the following directories:

PP: This includes the pseudopotentials. 2. 2.

inputs: This includes the input files for self-consistent field (scf) calculations (seedname.scf.in), non self-consistent field (nscf) calculations (seedname.nscf.in), Wannierization (seedname.pw2wan.in and seedname.win.ref), and the downfolding by RESPACK (respack.in). An input file for wan2respack is also included (conf.toml). The basic calculation flow is written in submit.sh. 3. 3.

inputs_selfk: This includes the input files used by the user when setting ${\bm{k}}$ points for Wannier90. The input files for Wannier90 (seedname.nscf_wannier.in and seedname.win) are prepared. 4. 4.

reference: This includes the reference input files for generating the Wannier functions using calc_wannier in RESPACK (RESPACK-Wannier).

4.1 DFT calculations for irreducible k points

DFT calculations of scf and nscf are performed using Quantum ESPRESSO by executing the following commands:

$QE/bin/pw.x < seedname.scf.in > seedname.scf.out$QE/bin/pw.x < seedname.nscf.in > seedname.nscf.out

Here, $QE indicates a path to the installation directory of Quantum ESPRESSO. The calculated wave functions are used to compute the dielectric function using RESPACK, so$\bm{k}$ points should be irreducible to reduce the computational cost.

4.2 Export of k points to be calculated by Wannier90

Next, for preprocessing, we generate a full $\bm{k}$-point list used by RESPACK-Wannier. This ${\bm{k}}$-point list is exported to the input files of the nscf calculation and Wannier90 by executing the following commands:

$python$PATH_to_Install/bin/wan2respack.py -pp conf.toml

The contents of conf.toml are as follows:

[base] QE_output_dir = "./work/seedname.save" seedname = "seedname" [pre.ref] nscf = "seedname.nscf.in" win = "seedname.win.ref" [pre.output] nscf = "seedname.nscf_wannier.in" win = "seedname.win"

conf.toml should be written in the toml format [76]. In the [base] section, the output directory of Quantum ESPRESSO and the seed name are specified using QE_output_dir and seedname, respectively. In the [pre.ref] section, the reference files for generating the input files are specified using nscf and win, respectively. In the [pre.output] section, the names of the output files generated by wan2respack are indicated by nscf and win. After each calculation, the dir-wfn directory and seedname.nscf_wannier.in and seedname.win files are generated (${\bm{k}}$ points are added to seedname.nscf_wannier.in and seedname.win).

4.3 Generation of Wannier functions

Wannier functions are generated using Quantum ESPRESSO, Wannier90, seedname.nscf_wannier.in, and seedname.win by executing the following commands:

$QE/bin/pw.x < seedname.nscf_wannier.in > seedname.nscf_wannier.out$Wannier90/wannier90.x -pp seedname $QE/bin/pw2wannier90.x < seedname.pw2wan.in > seedname.pw2wan.out$Wannier90/wannier90.x seedname

Here, $Wannier90 indicates a path to the installation directory of Wannier90.

4.4 Conversion of Wannier functions into RESPACK format

The Wannier functions obtained by Wannier90 are converted into the RESPACK format by executing the following command:

$python$PATH_to_Install/bin/wan2respack.py conf.toml

After these calculations, the following files are generated in the dir-wan directory:

dat.wan: $\tilde{C}_{\bm{G}i}(\bm{k})$

• 2.

dat.ns-nb: $N_{s},N_{b}$

• 3.

dat.umat: $\sum_{m}U({\bm{k}})_{im}U^{\rm opt}({\bm{k}})_{mn}$

• 4.

dat.wan-center: $\langle w_{i0}|{\bm{r}}|w_{i0}\rangle$

4.5 Calculation of screened interactions using RESPACK

The input file for RESPACK is respack.in. Using this file, we can calculate the screened Coulomb and exchange interactions through cRPA in RESPACK. The execution command is as follows:

$RESPACK/bin/calc_chiqw < respack.in > LOG.chiqw$RESPACK/bin/calc_w3d < respack.in > LOG.W3d $RESPACK/bin/calc_j3d < respack.in > LOG.J3d

For comparison, we provide an input file for RESPACK-Wannier in reference/respack.in. As shown later, the obtained Wannier functions and screened interactions by Wannier90 and RESPACK quantitatively agree.

5 Application Examples

In this section, we show the application of wan2respack to typical compounds for correlated electron systems, namely, SrVO3 and La2CuO4, whose crystal structures are shown in Fig. 2. The inputs are uploaded to the GitHub repository as samples. Some outputs in this section are also available elsewhere[78]. Note that we use only the disentanglement scheme when we construct the Wannier functions by Wannier90. Nevertheless, as we show later, the Wannier functions constructed from Wannier90 and RESPACK show good agreement with each other.

5.1 SrVO3

SrVO3 is a typical correlated metal compound[79, 80]. Three $t_{2g}$-like orbitals are located around the Fermi energy and isolated from the other bands[81]. As seen in Fig. 3(a), the Wannier bands of the $t_{2g}$ orbitals obtained using RESPACK and Wannier90 reproduce the original DFT band structure well. Figure 3(b) and (c) show Wannier functions with an orbital index $\alpha=1$. The sum of the spreads of all the Wannier orbitals are 5.5161 $[{\rm\AA^{2}}]$ for Wannier90 and 5.5156 ${[\rm\AA^{2}}]$ for RESPACK\cprotect222As the unit of the spread, the square of the Bohr radius $a_{0}^{2}$ is employed in the outputs of RESPACK—.. These results suggest that both software can generate the same Wannier orbitals quantitatively.

Since RESPACK and Wannier90 obtain almost the same Wannier functions, the effective interactions obtained using either Wannier function should be the same if wan2respack works correctly. To confirm this statement, we plot an effective Coulomb interaction written in dat.WvsR.001 in Fig. 4. The two results show excellent agreement, which indicates that wan2respack works well.

5.2 La2CuO4

We show the application of wan2respack to La2CuO4, which is a parent compound of high-$T_{\rm c}$ cuprate superconductors[82, 83]. Although this compound is a Mott insulator of an antiferromagnetic order, most of the DFT results suggest that this compound is a paramagnetic metal in its ground state[84, 85]. In contrast to the DFT calculations, it is shown that the Mott insulating state can be reproduced by solving the effective Hamiltonians derived from $ab$ $initio$ downfolding [36, 86].

Compared with SrVO3, La2CuO4 has a more complicated band structure[84, 85, 36]. The $d_{x^{2}-y^{2}}$ orbital of copper hybridized with the $p$ orbitals of oxygen crosses the Fermi energy. Contrary to that of SrVO3, this band is not well isolated: the other bands, such as the $d_{3z^{2}-r^{2}}$ orbital of copper and the $p$ orbitals of oxygen, exist near the Fermi energy. Such complexity of band structures sometimes leads to difficulties in constructing Wannier orbitals. Nevertheless, as shown in Fig. 5, the band structures near the Fermi level generated by RESPACK and Wannier90 are consistent with each other. We also confirm that the Wannier spreads are also consistent with each other (4.0231 [${\rm\AA^{2}}$] for Wannire90 and 4.0235 [${\rm\AA^{2}}$] for RESPACK). Moreover, we confirm that the effective interactions calculated via wan2respack agree well with the original RESPACK results, as shown in Fig. 6.

6 Summary

We developed the interface tool wan2respack, which converts Wannier functions in the Wannier90 format into those in the RESPACK format. Using wan2respack, one can perform RESPACK calculations using the Wannier functions obtained by Wannier90. For example, one can obtain the $ab$ $initio$ low-energy effective Hamiltonians of solids using RESPACK. To demonstrate the use of wan2respack, we derive low-energy effective Hamiltonians for the correlated metal SrVO3 and the high-$T_{c}$ superconductor La2CuO4 using Wannier functions obtained by Wannier90 and those obtained by RESPACK-Wannier. From these applications, we confirm that the low-energy effective Hamiltonians derived through both implementations of Wannier functions are the same. Since Wannier90 is a standard tool for constructing Wannier functions, the connection between Wannier90 and RESPACK via wan2respack further enhances the usability of RESPACK.

Acknowledgments

We are indebted to Kazuma Nakamura for valuable discussions and for providing us with several codes. We acknowledge Tetsuya Shoji, Noritsugu Sakuma, and Tetsuya Fukushima for fruitful discussions and important suggestions. A part of this work is financially supported by TOYOTA MOTOR CORPORATION. KY and TM were supported by Building of Consortia for the Development of Human Resources in Science and Technology from the MEXT of Japan. We thank Oakbridge-CX in the Information Technology Center, the University of Tokyo and the Supercomputer Center, the Institute for Solid State Physics, the University of Tokyo for their facilities. This work has also been supported by JST-Mirai Program (JPMJMI20A1), Grant-in-Aid for Scientific Research (Nos. 21H01003, 21H01041, 21H04437, 22K03447, and 22K18954) from Ministry of Education, Culture, Sports, Science and Technology, Japan, and the National Natural Science Foundation of China (Grant No. 12150610462).