Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

3.3

Strongly Correlated
Quantum Systems

171

Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Tensor-Network Renormalization-Group Study of Critical
Phenomena
Naoki KAWASHIMA
Institute for Solid State Physics,
The University of Tokyo, Kashiwa-no-ha, Kashiwa, Chiba 277-8581
We developed efficient methods, algorithms,

operator and the scaling dimensions. Ourparallelized programs, and sometimes new

benchmark calculation on the 2D Ising model

concepts, based on novel numerical techniques

established validity of this basic idea. The newincluding the tensor network (TN) method andscheme may provide a direct method for

applied them to relevant physical problems. To

estimating the scaling dimensions in higher

list subjects of our research in 2021, (1)dimensions.
In [2], we proposed a method for obtaining

development of new numerical methods basedon the TN representation [1,2], (2) study of TN

theground states such as Kitaev spin liquid [3], (3)calculations. In this method, we consider anstudy of spin liquid in frustrated spin systems

imprurity tensor representing a locally excited[4],state, and construct an eigenvalue equation in(4)

developmentof

open-source

excitation

spectrum

based

on

TN

the space of such excited states, which can beapplications [5].
In [1], we re-considered the method forblock-diagonalized by Frourier transformation.

extracting the critical information from the

By diagonalizing each block we can obtain

RSRG calculation with TNs. The most common

excitation spectra of various excitations. We

method for obtaining the scaling dimension isappliedto form a transfer matrix from the scale-quantum spin chain and successfully computed

invariant tensor. Then, we can obtain theits excited states and the dynamical structure

scaling dimensions from its eigenvalues using

factor. We further investigated entanglement

the relationship derived from the conformal

properties of excited states.

thisschemetoone-dimensional

field theory. However, this method has not been

In [3], we considered in the Kitaev model

generalized to three or higher dimensions. In

on the honeycomb lattice and its extention to

our scheme, we construct a super-operator that

S=1. The models with arbitrary spin quantum

represents the RSRG tarnsformation. Without

numbers are not exactly solvable in contrast to

explicitly relying on the CFT, we derive thethe well-known Kitaev spin liquid solution ofrelashion between the eigenvalues of the super-

the spin-1/2 system. To unveil the important172

Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

difference between these states, we use theextereme dilution to clarify the validity of thetensor-network wave functions, which proved,

analytical predictions for the universal behavior

in our previous work, to be essential inof the model in this region. We confirmed thecapturing the nature of the Kitaev spin liquid.

universal behaviors. In addition, we estimated

We found that the distinct sign structures of themodel-dependent parameters such as the s-tensor-network wave function for the integer-

wave scattering length in a good agreement

and half-integer-spin quantum spin liquids arewith the prediction as well as the non-universal

responsible for completely different groundcorrection term of the third order in the particle

states in the spatially anisotropic limit.

density, which has never been estimated before.

In [4], we investigated S=1/2 XXZ laddersystem with four-spin interaction. The model

References

can be viewed as a one-dimensional variant of[1] X. Lyu, R. G. Xu and N. Kawashima, Phys.

the J-Q model on a square lattice. By means ofRev. Research 3, 023048 (2021).

variational

state[2] W.-L. Tu, H.-K. Wu, N. Schuch, N.

(VUMPS) calculations and an effective field

Kawashima and J.-Y. Chen, Phys. Rev. B 103,

theory, we determine the phase diagram of the205155 (2021).

model and present evidence that the Néel-VBS[3] H.-Y. Lee, T. Suzuki, Y. B. Kim and N.

transition is continuous and belongs to the

Kawashima, Phys. Rev. B 104, 024417 (2021).

Gaussian universality class with the central[4] T. Ogino, S. Furukawa, R. Kaneko, S.

charge c=1. The full phase diagram consists ofMorita and N. Kawashima, Phys. Rev. B 104,8075135 (2021).

distinct

uniform

phasesmatrixincluding

product

magneticallyordered phases, Haldane-like phases, and VBS[5] Y. Motoyama, K. Yoshimi, A. Masaki-Kato,phases.

T. Kato and N. Kawashima, Comput. Phys.

In [5], we developed DSQSS, an open-

Commun. 264, 107944 (2021).

source software for quantum Monte Carlo[6] A. Masaki-Kato, Y. Motoyama and N.

simulation of lattice system. In [6], we applied

Kawashima, J. Phys. Soc. Jpn. 90, 034711

it to the Bose-Hubbard model in the case of(2022).

173

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Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Numerical studies of Mott transitionsin quantum spin liquid candidates
Youhei YAMAJI
Center for Green Research on Energy and Environmental Materials,
National Institute for Materials Science
Namiki, Tsukuba-shi, Ibaraki 305-0044

The Kitaev model is a unique solvable twodimensional quantum system [1]. As shown byKitaev, the ground state of the model hamiltonian is a quantum spin liquid. Later, Jackeli
and Khaliullin proposed that the effective lowenergy hamiltonian of Mott insulating phasesof honeycomb iridium oxides such as Na2 IrO3
could be the Kitaev model [2]. Although theground state of Na2 IrO3 was found to show
a magnetic long-range order, the dominant
terms in the ab initio effective spin hamiltonian of Na2 IrO3 are given by the Kitaev-typeexchange couplings [3].
Even though there emerge new candidate
materials represented by α-RuCl3 on the frontof the search for the spin-liquid ground states,
an important question has not been solved yet:
It has not been clarified yet whether the small
Mott gaps of these candidate materials affectthe properties of them or not. To elucidate an
impact of the itinerant nature of electrons on
the properties of a spin-liquid candidate, we
studied the original ab initio itinerant hamiltonian consisting of iridium t2g orbitals [3], instead of the spin hamiltonian.
In the present research, we studied the abinitio t2g hamiltonian of Na2 IrO3 [3]. The t2g
hamiltonian consists of the kinetic energy andthe intra-atomic and inter-atomic Coulomb repulsions as,plings, ĤU denotes the intra-atomic Coulombinteractions, and ĤV is the inter-atomic
Coulomb repulsion.
Naı̈vely, ĤV is expected to be relevant tocharge channels but irrelevant to spin degrees
of freedom. However, it is not true. Wheninter-atomic exchange couplings are estimatedby the second-order perturbation theory, theintra-atomic and intra-orbital Coulomb repulsion U in the energy denominator is replaced
by U −V , where V is the inter-atomic Coulomb
repulsion. Therefore, when the inter-atomic
Coulomb repulsions are controlled as parameters, not only the charge but also the spin
degrees of freedom are affected.
We controlled the nearest-neighbor, second nearest-neighbor, and 1/r long-range
Coulomb interactions to examine the competition among the magnetic and charge orders
and suppression of the magnetic orders, by using an open-source code mVMC [4].

References[1] A. Kitaev: Ann. Phys. (Amsterdam) 321(2006) 2.
[2] G. Jackeli and G. Khaliullin: Phys. Rev.
Lett. 102 (2009) 017205.

(1)[3] Y. Yamaji, et al.: Phys. Rev. Lett. 113(2014) 107201.

where ĤK represents the kinetic terms including trigonal distortion and spin-orbit cou-

[4] T. Misawa, et al.: Comput. Phys. Commun. 235 (2019) 447.

Ĥt2g = ĤK + ĤU + ĤV ,177

Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Theoretical study of symmetry breaking and quantum
transport phenomena in charge-spin-orbital coupled systems
Yukitoshi MOTOME
Department of Applied Physics,
The University of Tokyo, Bunkyo, Tokyo 113-8656

We have theoretically studied a variety ofthe effect of randomness on the spin dynamics

intriguing phenomena in correlated electronin the Kitaev quantum spin liquid [13]. We alsosystems

elucidatedwithcharge-spin-orbital

coupling

linearandnonlinearthermal(project numbers: 2021-Ca-0050 and 2021-Cb-transport in the presence of both uniform and0014). During the last fiscal year, we havestaggered magnetic fields [14]. We have madeachieved substantial progress on the following

collaborations with experimental groups on the

topics. We summarize the main results below.

topological nature in an external magnetic field(i)

Regarding[15,16]. In addition, we showed how to create

topological spin crystals, such as skyrmion andand control fractional excitations by analyzing

hedgehog lattices, as moiré interferences

the effect of local lattice distortions [17]. We

between spin helices, we clarified how theiralso proposed that iridium ilmenites can bemagnetic and topological properties can begood candidates for Kitaev magnets [18].

modulated by changing twist angles between(iii) Multipole physics: We collaborated withthe directions of the constituent spin density

experimental groups on a chiral magnet

waves [1] and shifting relative phases [2,3,4].

Pb(TiO)Cu4(PO4)4 to clarify the control of

We also elucidated charge modulation on suchmagnetic domains [19] and nonreciprocal

topological spin crystals [5]. In addition, we

directional dichroism [20,21].

clarified the ground state phase diagrams [6](iv) Development of new theoretical methods:

and spin excitations [7] for effective spin

We developed a many-body numerical methodmodels with long-range interactions. We also

for quantum spin systems based on theclarified linear and nonlinear optical responses

Lefschetz thimbles technique [22]. We alsoin quasi-one-dimensional chiral magnets [8,9].

developed new methods based on machine

We have collaborated with experimental groups

learning techniques: determinant-free fermionic[10,11]. We wrote a review article on the

wave function approach [23] and inverse design

itinerant frustration mechanism [12].

of(ii) Kitaev quantum spin liquids: We clarified

differentiation [24].

Topologicalspin

crystals:

178

Hamiltonians

by

using

automatic

Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

References[14] K. Nakazawa, Y. Kato, and Y. Motome,[1] K. Shimizu, S. Okumura, Y. Kato, and Y.

preprint (arXiv:2201.04274), accepted for Phys.

Motome, Phys. Rev. B 103, 184421 (2021).

Rev. B.

[2] S. Hayami, T. Okubo, and Y. Motome, Nat.

[15] T. Yokoi, S. Ma, Y. Kasahara, S. Kasahara,

Commun. 12, 6927 (2021).

T. Shibauchi, N. Kurita, H. Tanaka, J. Nasu, Y.

[3] K. Shimizu, S. Okumura, Y. Kato, and Y.

Motome, C. Hickey, S. Trebst, and Y. Matsuda,

Motome, preprint (arXiv:2201.03290).

Science 373, 568 (2021).

[4] K. Shimizu, S. Okumura, Y. Kato, and Y.

[16] S. Suetsugu, Y. Ukai, M. Shimomura, M.

Motome, J. Phys.: Conf. Ser. 2164 012069

Kamimura, T. Asaba, Y. Kasahara, N. Kurita, H.

(2022).

Tanaka, T. Shibauchi, J. Nasu, Y. Motome, and[5] S. Hayami and Y. Motome, Phys. Rev.

Y. Matsuda, preprint (arXiv:2203.00275).

B 104, 144404 (2021).

[17] S.-H. Jang, Y. Kato, and Y. Motome, Phys.

[6]

Y.

Kato

and

Y.

Motome,preprint

Rev. B 104, 085142 (2021)(arXiv:2202.04407).

[18] S.-H. Jang and Y. Motome, Phys. Rev.

[7] Y. Kato, S. Hayami, and Y. Motome, Phys.

Materials 5, 104409 (2021).

Rev. B 104, 224405 (2021).

[19] K. Kimura, Y. Kato, S. Kimura, Y.

[8] S. Okumura, T. Morimoto, Y. Kato, and Y.

Motome, and T. Kimura, npj Quantum Mater. 6,

Motome, Phys. Rev. B 104, L180407 (2021).

54 (2021).

[9] S. Okumura, T. Morimoto, Y. Kato, and Y.

[20] T. Katsuyoshi, K. Kimura, Z. Yang, Y.

Motome, J. Phys.: Conf. Ser. 2164 012068

Kato, S. Kimura, Y. Motome, Y. Kohama, and T.

(2022).

Kimura, J. Phys. Soc. Jpn. 90, 123701 (2021).

[10] K. Fujiwara, Y. Kato, T. Seki, K. Nomura,[21] M. Akaki, K. Kimura, Y. Kato, Y. Sawada,

K. Takanashi, Y. Motome, and A. Tsukazaki,

Y. Narumi, H. Ohta, T. Kimura, Y. Motome,

Commun. Mater. 2, 113 (2021).

and M. Hagiwara, Phys. Rev. Research 3,[11] N. D. Khanh, T. Nakajima, S. Hayami, S.

L042043 (2021).

Gao, Y. Yamasaki, H. Sagayama, H. Nakao, R.

[22] P. A. Mishchenko, Y. Kato, and Y. Motome,

Takagi, Y. Motome, Y. Tokura, T. Arima, and S.

Phys. Rev. D 104, 074517 (2021).

Seki, Adv. Sci. 2105452 (2022).

[23] K. Inui, Y. Kato, and Y. Motome, Phys.

[12] S. Hayami and Y. Motome, J. Phys.:

Rev. Research 3, 043126 (2021).

Condens. Matter 33, 443001 (2021).

[24][13] J. Nasu and Y. Motome, Phys. Rev. B 104,

(arXiv:2203.07157).

045116 (2021).

179

K. Inui and Y. Motome, preprint

Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Low-energy excitations of skyrmion crystals in acentrosymmetric spin-charge coupled magnet
Masahito MOCHIZUKI
Department of Applied Physics, Waseda University
Okubo, Shinjuku-ku, Tokyo 169-8555

In this project, we have theoretically studied the low-energy dynamics of skyrmion crystals in a centrosymmetric Kondo-lattice magnet with conduction-electron-mediated magnetic interactions (project ID: 2021-D-0002).
We computed the dynamical spin- andcharge-structure factors of skyrmion crystals ina centrosymmetric spin-charge coupled magnet by taking the Kondo-lattice model witha classical treatment of localized spins. Ourlarge-scale adiabatic molecular dynamics simulations revealed four Goldstone modes in theabsence of an external magnetic field [Figure1]. Three of the four Goldstone modes are associated with the spontaneous breaking of theSO(3) symmetry in spin space, and thus linearly dispersing. The remaining one is the socalled phonon mode associated with the displacement of skyrmions, and thus quadratically dispersing. Surprisingly, we observed
clear segregation of spin and charge excitations: the three linearly dispersive Goldstone
modes have full contribution from the spin sector and the quadratically dispersive Goldstone
mode has full contribution from the charge sector [1]. Note that the dynamical structure factors obtained with the adiabatic molecular dynamics simulations perfectly coincide with thedispersion relations obtained with the linearspin-wave calculation.
In our adiabatic molecular dynamics simulations, instead of repeated full diagonalizations of the quadratic form Hamiltonian,
we utilized the kernel polynomial method in

combination with the automatic differentiation technique to evaluate the local physicalquantities, for example, expectation values ofitinerant electron spins and local charge occupations. The Chebyshev moments are evaluated using the combination of processing andthreaded parallelization techniques on 18,432
cores (144 nodes) of System B Ohtaka.
(a) S(q,ω)(b) N(q,ω)

Figure 1: Dynamical (a) spin- and (b) chargestructure factors of the skyrmion crystal in theabsence of an external magnetic field.

References[1] R. Eto, R. Pohle, and M. Mochizuki,
preprint [arXiv:2203.01496].

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Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Theoretical study on the photodriven spin andcharge dynamics of topological materials
Masahito MOCHIZUKI
Department of Applied Physics, Waseda University
Okubo, Shinjuku-ku, Tokyo 169-8555

We have theoretically studied the dynamics
of topological magnetism and topological electron states under periodic driving (project ID:
2021-Ba-0015, 2021-Cb-0009). The main results of these projects are listed below.
(i) Dynamical magnetic topology switching ofskyrmion crystals: We revealed novel dynamics of skyrmion crystals in a centrosymmetric spin-charge coupled system under the microwave magnetic field [1]. By taking the
Kondo-lattice model with a classical treatmentof localized spins, we clarified the dynamic
topological phase transitions of the skyrmion
crystals among their skyrmion number Nsk =2, 1, and 0 through the large-scale spin dynamics simulations in combination with the kernel polynomial method and the automatic differentiation technique. The topological phasetransitions are induced by the effective magnetic field perpendicular to the rotating planeof the circularly polarized microwave magnetic
field. In our numerical simulations with thekernel polynomial method and the automaticdifferentiation technique, we utilized our own
hybrid parallelization code using both the processing parallelization with the message passing interface and the threaded parallelization
with OpenMP. Moreover, in the automatic differentiation process, we reduced the computational cost for the evaluation of Chebyshev
moments by using the relation 2Tm (x)Tn (x) =Tm+n (x) + T|m−n| (x) where Tm (x) denotes them-th order first-kind Chebyshev polynomial.
(ii) Spin-wave eigenmodes of the quadruple-Q

magnetic hedgehog lattices in chiral magnets:
We studied the spin-wave eigenmodes of thequadruple-Q hedgehog lattices which appear
in the Kondo-lattice model on the cubic lattice by the spin-dynamics simulations in combination with the kernel polynomial method[2]. As a result, under the [001] magnetic field,
we clarified two eigenmodes with counterclockwise fashion for the ac magnetic field perpendicular to [001] and four eigenmodes for theac magnetic field parallel to [001]. Three ofthe four eigenmodes parallel to [001] are localized around point defects of magnetic textures,and the remaining one is closely related to thephase degrees of freedom.
(iii) Photodriven dynamical phase transitionsin the α-type organic salt: We clarified the
dynamical phase transitions in the charge ordered organic salt α-(BEDT-TTF)2 I3 under
the irradiation of laser by the time-dependentmean-field calculation and the Floquet theory[3]. The suﬀicient intensity of laser melts the
charge order and induces the semimetal phase.
A much stronger laser induces the Chern insulator phase with nonzero Hall conductivity.

References[1] R. Eto and M. Mochizuki, Phys. Rev. B
104, 104425 (2021).
[2] R. Eto and M. Mochizuki, in preparation.
[3] Y. Tanaka and M. Mochizuki, preprint[arXiv:2203.04542].

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Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Topological phases, excitation modes, effects of exceptional
points and transport phenomena in strongly correlated
systems
Norio KAWAKAMI
Department of Physics,
Kyoto University, Kitashirakawa, Sakyo-ku, Kyoto 606-8502
and hold the possibility of being applied indiodes or high-frequency optical devices. Whilenonlinear responses in noninteracting models
have been explored widely, the effect of strong
correlations on the nonlinear response is still
poorly understood. In [2], we first derive aformula for calculating the nonlinear response
at finite temperatures that is entirely based on
single-particle
Green’sfunctions.
Wefurthermore analytically show that nonlinear
responses are significantly enhanced by therenormalization of the band structure and nonHermiticity in the Green’s function. Finally, we
confirm our analytical prediction by performing
numerical calculations for several strongly
correlated models.
In [3], we use this Green’s function method
to study the nonlinear Hall effect in Ce3Bi4Pd3.
Experiments measured a huge nonlinear Hall
effect in this material. From symmetry analysis,
we develop a model describing Ce3Bi4Pd3 as a
Weyl-Kondo semimetal. Performing dynamical
mean-field calculations for this model, we findthat below the Kondo temperature, Weyl points
emerge close to the Fermi energy in the band
structure; at high temperatures, this material isa metal. Due to strong correlations, the band
structure is strongly renormalized. We thendirectly calculate the nonlinear longitudinal
conductivity andthenonlinear Hall
conductivity and find that the nonlinear Hall
resistivity is strongly enhanced compared to anoninteracting system. We find good agreement
between our numerical calculations and theCorrelated materials have appeared as anarena to study non-Hermitian phenomena, suchas exceptional points emerging in the singleparticle Green’s function. Exceptional points
are degeneracies of eigenvalues due to the nonHermiticity of a matrix. Contrary to Hermitiandegeneracies, the eigenvalues and theeigenvectors coalesce at an exceptional point,
giving rise to novel phenomena.
In [1], we show that topological Kondo
insulators are an ideal platform for studying
these phenomena due to a combination ofstrong correlations and surface states. Using
numerical simulations, we demonstrate theemergence of exceptional points in the singleparticle Green’s function on the surface of thematerial while the bulk is still insulating. We
believe that this situation is perfect forexperimental observation of exceptional points.
In particular, we reveal how quasiparticle stateswith long lifetimes are created on the surface
by non-Hermitian effects while the Dirac cones
are smeared. This effect is related to the surface
Kondo breakdown at which heavy Dirac cones
disappear from the single-particle spectrum andare replaced by light states.
Furthermore, we study nonlinear responses
in strongly correlated systems, specificallyanalyzing the effect of renormalization andnon-Hermiticity on the second-order nonlinear
response.
Nonlinear responses in condensed matterare recently intensively studied because they
provide rich information about the materials

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Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

experimental observations.
In [4], we analyze the effect of strong
correlations on fragile topological insulators.
Fragile topological insulators (FTIs) offer
obstruction to the formation of exponentiallylocalized Wannier functions, but they can betrivialized by adding certain trivial degrees of
freedom. For the same reason, FTIs do not hostsymmetry-protected flow of edge states
between bulk bands in cylindrical boundary
conditions. Still, they are expected to have aspectral flow between the fragile bands andother bands under certain twisted boundary
conditions. We analyze commonly observed
effects of strong correlations, such as the Mott
insulator transition and magnetism, on a known
model hosting fragile topology. We show thatfragile topology, along with the twisted
boundary states, is stable with interactions
below a critical interaction strength in thenonmagnetic case. Above this interaction
strength, a transition to the Mott insulatingphase occurs, and the twisted boundary states
disappear. Furthermore, the fragile topology is
destroyed by a homogeneous magnetic field.
However, we show that a magnetic field can
induce a topological phase transition whichconverts a fragile topological insulator to a
Chern
insulator.
Finally,we
study
ferromagnetic solutions of the fragile
topological model.
In [5], we investigate magnetic instabilities
in charge-neutral twisted bilayer graphene close
to so-called “magic angles” using a
combination of real-space Hartree-Fock anddynamical mean-field theories. We find that
localized magnetic states emerge for values of
the Coulomb interaction U that are significantly
smaller than what would be required to render
an isolated layer antiferromagnetic. Ourfindings shed new light on perspectives forexperimental realization of magnetic states incharge-neutral twisted bilayer graphene.

In [6], we analyze the impact of criticalnematic fluctuations on the low energy
properties of phonons. We propose how to
identify the three-state Potts nematicfluctuations by ultrasound attenuation. The
Landau damping term becomes isotropic due tofluctuations of the C3-breaking bond-order, andthe nemato-elastic coupling is also isotropic.
The transverse sound attenuation coefficient
diverges isotropically, in contrast to the case of
the C4-breaking bond-order that shows strong
anisotropy. The shift of the first order transitionline at low temperatures toward the secondorder line renders the transition a weak firstorder in a wide range of parameters.

References[1] R. Peters, K. Kimura, Y. Michishita, T.
Yoshida, and N. Kawakami: “Surface
exceptional points in a topological Kondo
insulator” Phys. Rev. B 104 (2021) 235153.
[2] Y. Michishita and R. Peters: “Effects ofrenormalization and non-Hermiticity on
nonlinear responses in strongly-correlated
electron systems” Phys. Rev. B 103 (2021)
195133.
[3] A. Kofuji, Y. Michishita, and R. Peters:“Effects of strong correlations on the nonlinear
response in Weyl-Kondo semimetals” Phys.
Rev. B 104 (2021) 085151.
[4] A. Joshi and R. Peters: “Mott transition andmagnetism in a fragile topological insulator”
Phys. Rev. B 103 (2021) 165130.
[5] J. Vahedi, R. Peters, A. Missaoui, A.
Honecker, and G. T. de Laissardière:“Magnetism of magic-angle twisted bilayer
graphene” SciPost Phys. 11 (2021) 083.
[6] K. Kimura, M. Sigrist, and N. Kawakami:“Probing three-state Potts nematic fluctuations
by ultrasound attenuation” Phys. Rev. B 105(2022) 035130.

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Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Finite-temperature methods for quantum many-body systems
using Boltzmann machines
Yusuke NOMURA
RIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wako, 351-0198

The

thermalbehaviorofnumericallymany-body

optimizemachinesstatistical mechanics provides a theoretical

Hamiltonian evolution as accurately as possible

framework for describing finite-temperature

within the representability of deep Boltzmann

properties. However, in quantum many-bodymachines. We apply the methods to thesystems, due to the "curse of dimensionality" oftransverse-field Ising model and the two-

Hilbert space, it is very difficult to capture thedimensional J1-J2 Heisenberg model. The

complex competition between thermal andnumerical benchmarks show that our scalable

quantum

variousapproach can successfully simulate the finite-

numerical methods have been developed, suchtemperature properties of the quantum spinas quantum Monte Carlo, diagrammatic, andproblems.

Althoughreproduce

Boltzmannsystems is a fundamental problem. Celebrated

fluctuations.

to

deep

imaginarytimetensor network methods, it is challenging toestablish a reliable methodology that can beextended to the most exotic and interestingregions in quantum many-body physics.
Here, we take a novel approach using
artificial neural networks. In particular, we
employ deep Boltzmann machines that haveflexible representability. We propose twodifferent approaches, offering paths to encode
many-body

thermalphysics

into

deep

Fig. 1: Structure of deep Boltzmann machine

Boltzmann machines. In the first method, we
findaconstructcompletely
deep

deterministic

Boltzmannwaymachines

employed in the present study.

toReferences

that

reproduce the exact purified expression of finite[1] Y. Nomura, N. Yoshioka, and F. Nori, Phys.

temperature states. In the second method, we

Rev. Lett. 127, 060601 (2021)

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Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Exotic phenomena induced by strong correlations andnon-Hermitian topology
Tsuneya Yoshida
Department of Physics,
1-1-1 Tennodai, Tsukuba, Ibaraki, JAPAN 305-8571
In this year, we have mainly studied non-interacting level.

Hermitian topological phenomena by

Furthermore, we have explored thefocusing on exceptional points [1-4]. In

topological structure of the classicalparticular, we have analyzed exceptional

systems[5-7]. In particular, we havepoints at surfaces of topological Kondo

observed Hermitian and non-Hermitian

insulators[1].

havetopological physics in systems of the headapplied the dynamical mean-filed theory

conduction and the evolutionary gamecombinedtheory. The topological properties ofSpecifically,withrenormalization

the

group

we

numerical

methodtoasuperconductors are also discussed [8,9].

variant of the periodic Anderson model
which describes the topological Kondo
insulator

SmB6.

Our

numerical

simulation has elucidated that finite
lifetimeofquasi-particles

induces

exceptional points at surfaces of the
topologicalKondo

insulator.

These

exceptional points induce additional low
energy excitations and change the spin
structure of surface states (see Fig. 1).
Fig.1 Spectral weight for a surface of a

We also have analyzed correlations

topological Kondo insulator. The emergence ofeffects on the non-Hermitian topology inzero

dimension[4],

where

we

have

-

exceptional

pointsinduceslowenergyexcitations and affects the spin structure of theobserved that the classification of the***point-gap topology is reduced to Z2 while***

surface states (The figure is taken from Ref. [1])the classification is Z at the non-

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Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

References[6] S. Makino, T. Fukui, T. Yoshida, and Y.

[1] R. Peters, K. Kimura, Y. Michishita, T.

Hatsugai, Phys. Rev. E 105, 024137

Yoshida, and N. Kawakami, Phys. Rev. B(2021)

104 235153 (2021).

[7] T. Yoshida, T. Mizoguchi, and Y.

[2] T. Yoshida, R. Okugawa, and Y.

Hatsugai, Sci. Rep. 12, 1-7 (2021)

Hatsugai, Phys. Rev. B 105, 085109[8] T. Yoshida, T. Mizoguchi, Y. Kuno, and[3] H. Wakao, T. Yoshida, and Y. Hatsugai

Y. Hatsugai, Phys. Rev. B 103, 235130

arXiv:2203.05773(2021)[4] T. Yoshida and Y. Hatsugai, Phys. Rev.

[9] Y. Yanase, A. Daido, K. Takasan, andB 104, 075106 (2021).

T. Yoshida, Physica E, 140, 115143 (2022)[5] T. Yoshida, T. Mizoguchi, and Y.
Hatsugai, Phys. Rev. E 104, 025003

186

Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Extreme response of isotropic spin ﬂuctuations
emerging from anisotropic antiferromagnets undermoderate spin dilution
Hidemaro SUWA
Department of Physics, University of Tokyo, Tokyo 113-0033

The stability and response of magnetic
materials to external stimuli are determined by spin ﬂuctuations dictated by magnetic anisotropy. The two-dimensional (2D)
isotropic antiferromagnet is one of the ideal
cases to pursue an extreme magnetic response. However, magnetic anisotropy is signiﬁcant in most materials, reducing the magnetic response. As a perturbation, magnetic
anisotropy introduces a characteristic lengthscale ξani . The signiﬁcance of anisotropy follows from the fact that the correlation length
increases rapidly with decreasing temperatureand easily reaches ξani . In contrast, the Zeeman energy from a typical achievable magnetic
ﬁeld is weaker, and the length scale ξext introduced by the external ﬁeld is much larger, i.e.,ξext > ξani . Moreover, the Zeeman ﬁeld in antiferromagnets must be a staggered ﬁeld thatis linearly coupled to the staggered magnetization, which is diﬃcult to realize.
Our primitive 2D antiferromagnet is realized on a pseudo spin-1/2 square lattice embedded in a [(SrIrO3 )1 /(SrTiO3 )2 ] superlattice, with the adjacent IrO2 planes well separated by non-magnetic spacers. Strong spinorbit coupling of iridium stabilizes the Jeﬀ =1/2 moment antiferromagnetically coupled toeach other via the Heisenberg super-exchangeinteraction. An essential feature of such a
Jeﬀ = 1/2 square lattice is that the large
Dzyaloshinskii–Moriya interaction arising fromthe staggered rotation of the IrO6 octahedronaround the c axis is canceled throughout thelattice, and the pseudospins tend to be almost
completely isotropic, which is called a hidden
SU(2) symmetry. This phenomenon allows thein-plane uniform magnetic ﬁeld to act as aneﬀective staggered ﬁeld. Nevertheless, higherorder super-exchange paths due to Hund’s coupling cause small easy-plane anisotropy, eﬀectively reducing the spin symmetry from SU(2)to U(1).
We have theoretically and experimentallydemonstrated that we can switch the orderof ξext and ξani utilizing magnetic dilution.
Using ISSP System B in class C projects(ID: 2021-Ca-0117) and the ALPS/looper(http://github.com/wistaria/alps-looper)
code, we simulated large-scale 2D AFM
systems of more than one million quantum spins.
We calculated the Berezinskii–Kosterlitz–Thouless transition temperatures for several dilution fractions. Using
the crossover argument with multiple perturbations, we have shown that the response
to magnetic ﬁelds changes from the U(1)model to the SU(2) model above the thresholddilution fraction because of the length-scale
switch (ξext < ξani ). The theoretical calculation successfully accounts for the experimental
observations, conﬁrming the extreme response of isotropic ﬂuctuations emerging fromanisotropic antiferromagnets under moderate
spin dilution.

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Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Research of Multi-Channel Kondo Eﬀect
Emerging from Heavy Rare-Earth Ions
Takashi HOTTA
Department of Physics, Tokyo Metropolitan University
1-1 Minami-Osawa, Hachioji, Tokyo 192-0397

We have discovered the three-channel Kondo
eﬀect in the cubic Ho compound [1]. We haveanalyzed a seven-orbital impurity Andersonmodel hybridized with Γ8 and Γ7 conduction
electrons for a case with ten local f electrons
corresponding to a Ho3+ ion by using a numerical renormalization group (NRG) method.
Then, we have found a residual entropy of log ϕ√with the golden ratio ϕ = (1 + 5)/2 as a signal of the three-channel Kondo eﬀect for the
local Γ5 triplet ground state.
For the construction of the local f -electron
Hamiltonian Hloc , we introduce one f -electronstate, deﬁned by the eigenstate of spin-orbit
and crystalline electric ﬁeld (CEF) potential
terms. Under the cubic CEF potential, we obtain Γ7 doublet and Γ8 quartet from j = 5/2
sextet, whereas we ﬁnd Γ6 doublet, Γ7 doublet,and Γ8 quartet from j = 7/2 octet. By using
those one-electron states as bases, we obtain
Hloc asHloc =∑†(λj + Bj,µ )fjµτ
fjµτ + nEf

j,µ,τ

+

∑ ∑ ∑

j1∼j4 µ1∼µ4 τ1∼τ4

Iµj11jτ21,jµ32jτ42 ,µ3 τ3 µ4 τ4 (1)× fj†1 µ1 τ1fj†2 µ2 τ2fj3 µ3 τ3fj4 µ4 τ4 ,where fjµτ is the annihilation operator of a localized f electron in the bases of (j, µ, τ ), j
is the total angular momentum, j = 5/2 and7/2 are denoted by “a” and “b”, respectively,µ distinguishes the cubic irreducible representation, Γ8 states are distinguished by µ = α
and β, while the Γ7 and Γ6 states are labeledby µ = γ and δ, respectively, τ denotes thepseudo-spin, which distinguishes the degeneracy concerning the time-reversal symmetry, n
is the local f -electron number at an impurity
site, and Ef is the f -electron level to control
n. Here we set h̄ = kB = 1 and the energy unit
as eV.
Concerning the spin-orbit term, we obtain
λa = −2λ and λb = (3/2)λ, where λ isthe spin-orbit coupling of f electron. In thisstudy, we set λ = 0.265 for the Ho ions.
Concerning the CEF potential term for j =5/2, we obtain Ba,α = Ba,β = 1320B40 /7
and Ba,γ = −2640B40 /7, where B40 denotes
the fourth-order CEF parameter for the angular momentum ℓ = 3. For j = 7/2, we
obtain Bb,α = Bb,β = 360B40 /7 + 2880B60 ,Bb,γ = −3240B40 /7 − 2160B60 , and Bb,δ =360B40 − 3600B60 /7. In the present calculations, we treat B40 and B60 as parameters.
For the Coulomb interaction terms, we do notshow the explicit forms of I here, but they
are expressed by the four Slater-Condon parameters, F 0 , F 2 , F 4 , and F 6 . These values should be determined from experimental
results, but here we simply set the ratio as
F 0 /10 = F 2 /5 = F 4 /3 = F 6 = U , where U indicates the Hund’s rule interaction among thef orbitals and the magnitude is set as unity inthis study.
In Fig. 1, we show the local CEF groundstate phase diagram for n = 10. The groundstate multiplet for B40 = B60 = 0 is characterized by the total angular momentum J = 8.
Under the cubic CEF potentials, the sept-188

Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

0.005
B60 [meV]

1.5

n=10
U=1
l=0.265

G5 triplet
W=10-3G1 singlet

0.0

ublet
G 3 dot-0.005
iple
G tr
4

-0.01
-0.1

-0.05

W=-10-3G3 doublet
0.0
B40 [meV]

0.05

f-electron entropy

0.01

0.1

1

Ho3+ (f10)x=0.0, W=10-3eV
V=0.805eV
U=1eV
l=0.265eV

0.5

log φ
0 -12 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1
10 10 10 10 10 10 10 10 10 10 10 10

Temperature [eV]
Figure 1: Local CEF ground-state phase diagram on the (B40 , B60 ) plane for n = 10 with
U = 1 and λ = 0.265.
dectet of J = 8 is split into four groups as one
Γ1 singlet, two Γ3 doublets, two Γ4 triplets,
and two Γ5 triplets. Then, we obtain fourkinds of local ground states for n = 10, as
shown in Fig. 1. The dashed rhombus denotes the trajectory of B40 = W x/15 and
B60 = W (1 − |x|)/180 for −1 ≤ x ≤ 1 with
W = ±10−3 . Roughly speaking, the Γ5 triplet
appears widely for B60 > 0, whereas the Γ3
doublet is found for B60 < 0. In the region of
B60 ≈ 0 and B40 > 0, the Γ1 singlet was stabilized. For B40 < 0, we ﬁnd the Γ3 doublet andΓ4 triplet in a narrow region between the Γ5
triplet and Γ3 doublet.
Now we include the Γ7 and Γ8 conduction
electron bands. Here we consider only the hybridization between the conduction and j =7/2 electrons. The Hamiltonian is given byH =∑εkc†kµτckµτ +k,µ,τ

+ Hloc ,∑Vµ(c†kµτ fbγτ +h.c.)

k,µ,τ

(2)

where εk is the dispersion of conduction electron with wave vector k, ckγτ is an annihilation operator of conduction electrons, and Vµdenotes the hybridization between f electronin the µ orbital and conduction electron of theµ band. Here we set Vα = Vβ = Vγ = V .
In this research, we analyze the model by
employing the NRG method [2]. We introduce

Figure 2: Entropy vs. temperature for n = 10with x = 0.0 and W = 10−3 , corresponding tothe Γ5 triplet ground state.
a cut-oﬀ Λ for the logarithmic discretizationof the conduction band. Due to the limitationof computer resources, we keep M low-energy
states. Here we use Λ = 8 and M = 5, 000.
In the NRG calculation, the temperature T isdeﬁned as T = DΛ−(N −1)/2 , where D is half
the conduction band width, which was set as
1 eV, and N is the number of renormalization
steps.
Here we brieﬂy discuss the typical result forn = 10 [1]. In Fig. 2, we show entropy vs. temperature for the Γ5 triplet ground state. We
observe a residual entropy of log ϕ at low tem√
peratures with the golden ratio ϕ = (1+ 5)/2,
which is characteristic of the three-channel
Kondo eﬀect. We conﬁrm the residual entropy of log ϕ in the wide region of the local Γ5
triplet ground state. Thus, we conclude thatthe three-channel Kondo eﬀect occurs in theΓ5 triplet ground state of Ho3+ ions.

References[1] T. Hotta, J. Phys. Soc. Jpn. 90, 113701(2021).
[2] H. R. Krishna-murthy, J. W. Wilkins, andK. G. Wilson, Phys. Rev. B 21, 1003
(1980).

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Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Numerical study of topological thermal transport in
Kitaev-related systems
Joji NASU
Department of Physics, Tohoku University
Sendai 980-8578

The Kitaev quantum spin model has been
intensively investigated since the proposal by
A. Kitaev as a canonical model exhibiting
quantum spin liquids. This model is considered to describe the magnetism of iridium oxides and α-RuCl3 where the strong spin-orbit
coupling and electron correlations are present.
However, these materials show a magnetic order at low temperatures due to magnetic interactions additional to the Kitaev term. As additional terms, eﬀects of the Heisenberg interaction and oﬀ-diagonal symmetric term (Γ term)
have been examined intensively. Moreover, recent experiments in α-RuCl3 clariﬁed that an
applied magnetic ﬁeld induces nonzero thermal
Hall conductivity, whose sign depends on themagnetic ﬁeld direction. However, the ﬁeldangle dependence of the Kitaev-Heisenberg
and Kitaev-Heisenberg-Γ models remains unclear in the presence of magnetic orders.
In this project, we systematically examine
the ﬁeld-angle dependence of magnetic orders
and excitation spectra in these models. To address the case with arbitrary spin directions ina magnetic order, we have newly developed thecalculation technique by extending the conventional spin-wave theory [1, 2]. In this method,
we can implement large-sublattice calculations
straightforwardly. Our spin-wave theory needs
the Cholesky decomposition. We use the LAPACK library to perform it. First, we examine the ground-state spin conﬁguration bychanging the magnetic ﬁeld direction and obtain the phase diagram. Based on the ob-tained ground-state magnetic orders, we calculate the energy and Berry curvature of themagnon bands. Furthermore, we calculate thetemperature dependence of the thermal Hall
conductivity. To compute the thermal Hall coeﬃcient, we need to sum up the Berry curvature for the wave-number vector in the Brillouin zone. We parallelize the k-summation in
the ISSP supercomputer to perform this procedure eﬃciently. The thermal Hall conductivity is enhanced in magnetically ordered stateswith noncoplanar spin conﬁgurations. Moreover, we also ﬁnd that the thermal Hall eﬀect
originating from the topological nature of themagnon dispersion is absent when the ﬁeld isapplied along with one of the spin axes in the
Kitaev-Heisenberg model. We also ﬁnd that
this feature disappears by introducing the oﬀdiagonal symmetric Γ interaction. In addition,
we clarify that the Γ interaction suppresses thethermal Hall conductivity because of enhancing the low-energy magnon gap.

References[1] S. Koyama and J. Nasu: B 104 (2021)
075121.
[2] S. Koyama and J. Nasu: J. Phys.: Conf.
Ser. 2164 (2022) 012071.

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Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Searching for meron crystal in itinerant magnets
Satoru Hayami
Department of Applied Physics, The University of Tokyo, Tokyo 113-8656

Topological spin textures, such as a magnetic skyrmion, meron, and hedgehog, have attracted considerable attention, since they giverise to fascinating physical phenomena, suchas the topological Hall eﬀect. The fundamental understanding of the stabilization mechanisms for these nontrivial spin textures would
be useful for potential applications to nextgeneration electronic and spintronic devices.
In the project with number 2021-Ca-0042, wehave theoretically investigated the possibilityof such topological spin textures based on anitinerant electron model and its eﬀective spinmodel. We have presented the main results
this year below.
(i) Skyrmion crystals in trigonal systems:
We have theoretically investigated a new stabilization mechanism of the skyrmion crystals
in itinerant trigonal systems [1]. By focusing on the anisotropic exchange interactionthat originates from the lack of horizontal mirror symmetry and twofold rotational symmetry along the threefold axis, we obtained two
types of skyrmion crystals with the skyrmion
number of one and two. We have shown that
two SkXs are transformed with each otherby changing the magnetic ﬁeld. Our ﬁndings
suggest itinerant trigonal systems as a newskyrmion-hosting system in addition to conventional hexagonal and tetragonal systems.
(ii) Mechanism of skyrmion crystals in EuPtSi: We have theoretically studied the stabilization mechanism of the skyrmion crystals
in the f -electron compound EuPtSi based onan eﬀective spin model [2]. We found two
important ingredients to induce the skyrmion
crystal: (1) the multiple-q superpositions of

the spirals with low-symmetric ordering vectors and (2) the synergy between spin-charge
and spin-orbit couplings in itinerant magnets.
We have clariﬁed that the model to satisfy two
conditions well describes the skyrmion physics
in EuPtSi by performing unbiased simulated
annealing.
(iii) Meron-antimeron crystal on a triangular lattice: A meron-antimeron crystal consisting of the periodic array of merons andantimerons with a half skyrmion number has
attracted growing attention since its observation in Co8 Zn9 Mn3 and Gd2 PdSi3 . Nevertheless, its origin has not been fully identiﬁed owing to the lack of a stabilization mechanism
and an appropriate model. To understandthe origin of the meron-antimeron crystals on
the basis of metallic magnets, we theoretically
performed simulated annealing for an eﬀective spin model derived from the Kondo lattice model on a noncentrosymmetric triangular lattice [3]. As a result, we showed that several types of meron-antimeron crystals are stabilized by the synergy among the biquadraticinteraction, the single-ion anisotropy, and the
Dzyaloshinskii-Moriya interaction. The emergent meron-antimeron crystals are owing to thenature of itinerant electrons, which have notbeen obtained in localized spin models. We
also found other topological spin textures including the skyrmion crystal can be realized by
the same eﬀective model at zero ﬁelds.
(iv) Charge density waves in multiple-Q spinstates: We elucidated the charge density wave
formation in the presence of the spin density
wave [4]. Based on the perturbation in termsof the spin-charge coupling, we theoretically191

Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

derived a useful formula for the relationshipbetween charge density waves and spin density waves, which can be applied to any typeof magnetic texture and lattice structure. We
have tested the obtained formula for several
single-Q and double-Q magnetic orderings including the skyrmion crystal on a square lattice. We have also shown the eﬀects of large
spin-charge coupling beyond the perturbative
regime and the spin-orbit coupling to comprehensively understand the charge density wave
formation in itinerant magnets.
(v) Phase shift in skyrmion crystals: We
have theoretically studied the possibility ofcontrolling the skyrmion crystals by using thephase degree of freedom among the spin density waves [5]. We have shown that a phaseshift can occur in the skyrmion crystals to stabilize a new type of vortex crystals with a different pattern of an emergent magnetoelectricﬁeld through local scalar spin chirality, which
gives rise to nonreciprocal transport phenomena without the spin-orbit coupling. Analyzing the Kondo lattice model and its eﬀective
spin model, we found three mechanisms tocause the phase shift: exchange interactions
between the localized spins, thermal ﬂuctuations, and long-range chirality interactions.
(vi) Skyrmion crystal in centrosymmetric bilayer magnets: We proposed a stabilization
mechanism to realize the skyrmion crystal incentrosymmetric bilayer magnets [6]. By performing the simulated annealing on the bilayer
triangular-lattice spin model, we have shownthat the interplay between the layer-dependent
Dzyaloshinskii-Moriya interaction and interlayer exchange interaction gives rise to theskyrmion crystal even without the frustrated
exchange interaction and multiple-spin interactions. In addition, we have obtained two
triple-Q states with a uniform scalar chirality
while increasing and decreasing the magnetic
ﬁeld in the skyrmion crystal phase that has notbeen obtained in the single-layer system.

onal magnets: We theoretically investigatedthe possibility of realizing multiple skyrmion
crystals against an external magnetic ﬁeld in
tetragonal itinerant magnets [7]. By focusing
on multiple peaks at bare susceptibility of itinerant electrons, we found that three types ofskyrmion crystals with the skyrmion numbersof one and two, which are characterized by different superpositions of helices, are stabilized
in the ground state. Moreover, we found thatthey are transformed with each other by an
external magnetic ﬁeld.

References[1] R. Yambe and S. Hayami, Sci. Rep. 11,11184 (2021)[2] S. Hayami and R. Yambe, J. Phys. Soc.
Jpn. 90, 073705 (2021).
[3] S. Hayami and R. Yambe, Phys. Rev. B
104, 094425 (2021).
[4] S. Hayami and Y. Motome, Phys. Rev. B
104, 144404 (2021).
[5] S. Hayami, T. Okubo, and Y. Motome,
Nat. Commun. 12, 6927 (2021).
[6] S. Hayami, Phys. Rev. B 105, 014408(2022).
[7] S. Hayami, J. Phys. Soc. Jpn. 91, 023705(2022).

(vii) Multiple skyrmion crystals in tetrag-

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Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Analysis of string excitations in the opticalconductivity of doped Mott insulators
Takami TOHYAMA
Department of Applied Physics, Tokyo University of Science, Tokyo 125-8585

The optical conductivity has provided important information on the nature of the holedoped Mott insulators. The most striking
feature is dynamical spectral-weight transfer
upon chemical doping and photodoping appearing at mid-infrared (MIR) region. The
weights at MIR are sensitive to dimensionality and are found in two dimensions but not in
one dimension. MIR weights are expected tocontain essential information on the dynamical properties of holes in the two-dimensional
Mott insulators. A hole moving in antiferromagnetic (AFM) spin background creates misarranged spins with respect to their sublatticeand consequently induces string-type excitation of spins. MIR weights may have a relation to string structures. However, it is unclear how MIR weights actually are related tostring excitations that are the result of a complex process due to spin-charge coupling. If ahole moves in an AFM state, it leaves tracesof spin mismatches with sublattice magnetization called Sz strings. The Sz strings are
inadequate to describe a hole surrounded by
AFM spins since the hole picks up a nontrivial
U (1) phase when it hops in an AFM spin background. This phase generates another type ofstrings, called phase strings, caused by transverse spin component.
In this project, we demonstrate howmuch Sz and phase strings contribute to
MIR weights.
We calculate the optical conductivity of the Hubbard model inthe strong-coupling regime and the t-J
model by using time-dependent Lanczos andtime-dependent density-matrix renormalization group (tDMRG) methods. We focus on
doped Mott insulators with a single hole andconsider two-leg ladders up to a 20×2 sites [1]and square lattice up to 6×6 sites [2] underopen boundary conditions. Turning on and oﬀ
the eﬀect of phase strings, we examine how
they contribute to MIR weights. We ﬁnd thatphase strings play an essential role in MIR
weights. MIR weights are crucially suppressed
for both the Hubbard and t-J models if we remove phase strings. Although Sz strings also
contribute to MIR weights, their contribution
is smaller than phase strings. We consider that
this is because Sz strings can be self-healed
via quantum spin ﬂips, while phase strings are
not reparable. Our ﬁndings suggest that a mutual Chern-Simons gauge ﬁeld, which is an elementary force between spin and charge in thephase-string theory, is signiﬁcant for obtaining
MIR weights. This indicates that a hole does
not pick up a U (1) phase when moving in AFM
spin background if we remove this gauge ﬁeld.
As a result, we can characterize a doped holesurrounded by AFM spins via a spin-polaron
quasiparticle, which has a cosinelike energy
dispersion in the single-particle spectral function.

References[1] K. Shinjo, S. Sota, and T. Tohyama:
Phys. Rev. B 103 (2021) 035141.
[2] K. Shinjo, S. Sota, and T. Tohyama: inpreparation.

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Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Theoretical Calculation on Coulomb Repulsionof EDXT Oligomer Charge Transfer Salts
Tomoko FUJINO
Institute for Solid State Physics,
The University of Tokyo, Kashiwa-no-ha, Kashiwa, Chiba 277-8581
typical low molecular weight 1D conductors

We have synthesized discrete oligomer
modelsfor

dopedpoly(3,4-(TTF•TCNQ: 4tTTF ≈ 0.2–0.4 eV, 4tTCNQ ≈ 0.5

ethylenedioxythiophene) (PEDOT) [1,2]. DopedeV), revealing the origin of the high conductivity

PEDOT is a commonly employed conductiveof doped PEDOT. Interestingly, the elongation

polymer in organic electronic devices owing toof characteristic oligomer length from dimer totheir flexibility, lightness, transparency, andtrimer exhibited decreased W (≈ 0.90 eV), while

concisetheir

it improved the conductivity with 70%

complicated packing structures based on a non-activation energy (0.20 eV) of that for dimer.

uniform molecular weight distribution severely

These data indicated that extending thelimit structural data acquisition, especiallyconjugation lengths of the oligomers markedly

electronic structural data, requisite for further

lowered effective Coulomb repulsion (Ueff),

improving the physical properties.

thereby decreasing the ratio of Ueff/W andsynthetic

routes.

However,This study calculated the band structures

improving the conductivity [2]. The details on

based on the single-crystal structures of charge

Ueff by theoretical calculations will be discussed

transfer salts of the dimer as the shortest models

in near future.

for doped PEDOT [1]. The band calculations
functional theory (DFT)-based by OpenMX

References

software revealed their one-dimensional (1D)

[1]. R. Kameyama, T. Fujino, S. Dekura, M.

metallic

Kawamura, T. Ozaki, and H. Mori: Chem. Eur.

band

structures

with

a

strongintracolumnar orbital interaction (calculated

J. 27 (2021) 6696.

band width W ≈ 1 eV). This interaction was[2] R. Kameyama, T. Fujino, S. Dekura, and H.

significantly stronger than that observed for

Mori: Phys. Chem. Chem. Phys. 24 (2022), 9130.

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Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Studies of the superconductivity and magnetic statesin the strongly correlated electron systems based on
Hubbard models.
Atsushi Yamada
Department of Physics, Chiba University
Chiba 263-8522, Japana, Chiba 277-8581

A spin liquid state, which is a purely
nonmagnetic Mott insulator without spontaneously broken spatial or spin symmetry, has
attracted a lot of interest. This state is realized in geometrically frustrated systems likethe charge organic transfer salts κ-(BEDTTTF)2 X[1] and Cs2 CuCl4 .[2]study larger systems, we need to revise ourprogram using MPI technique. We are now
improving our programs, even though we encounter a lot of diﬃculties.

A spin liquid could arise also in the intermediate coupling region of strongly correlated
systems between a semi-metal and orderedstate, because in this case a correlation-driven
insulating gap might open before the system
becomes ordered. This possibility might be realized in the half-ﬁlled Hubbard model on thehoneycomb lattice, where a semi-metal is realized at U = 0, and in fact is studied, for example, using Quantum Monte Carlo (QMC) simulations. Meng et al[3] found that a spin liquidstate is realized in the Honeycomb lattice Hubbard model. But suqsequent QMC studies[4]with larger lattice size ruled out this possibility.

[1] Y. Shimizu, K. Miyagawa, K. Kanoda,
M. Maesato, and G. Saito, Phys. Rev.
Lett. 91, 107001 (2003); Y. Kurosaki, Y.
Shimizu, K. Miyagawa, K. Kanoda, and
G. Saito, Phys. Rev. Lett. 95, 177001(2005).

We have studied the magnetic and metal-toinsulator transitions by variational cluster approximation using 10-site and 16-site clusters
as a reference system. Parts of numerical calculations were done using the computer facilities of the ISSP. We found that UAF = 2.7 and
UMI = 3.0 for 10-site cluster, and UAF = 2.7
and UMI = 3.2 for 16-site cluster.[5] This result
also rules out the existence of the spin liquid inthis model. Our results agree with recent large
scale Quantum Monte Carlo simulations.[4] To[4] S. Sorella, Y. Otsuka, and S. Yunoki, Sci.
Rep. 2, 992 (2012); F. F. Assaad andI. F. Herbut, Phys. Rev. X 3, 031010(2013); F. Parisen Toldin, M. Hohenadler,
F. F. Assaad, and I. F. Herbut, Phys. Rev.
B 91, 165108 (2015).

References[2] R. Coldea, D.A. Tennant, A.M. Tsvelik,
and Z. Tylczynski, Phys. Rev. Lett. 86,
1335 (2001); R. Coldea, D.A. Tennant,and Z. Tylczynski, Phys. Rev. Lett. 68,
134424 (2003).
[3] Z. Y. Meng et al., Nature 464, 08942(2010).

[5] A. Yamada, Int. J. Mod. Phys. B 30,
1650158 (2016).

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Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Non-Abelian anyons trapped in vacancies of theKitaev model and non-local spin correlations
Masahiro O. TAKAHASHI and Satoshi FUJIMOTO
Department of Materials Engineering Science, Osaka University
Machikaneyama 1-3, Toyonaka, Osaka 560-8531

We have clariﬁed, using massive parallel
computer simulations, that a non-local spinspin correlation exists between speciﬁc spatially well-separated sites in the Kitaev spin
liquid with some point vacancies, which reveals the realization of Majorana zero modes(MZMs) strongly trapped in vacanceis.
The Kitaev model [1] is a spin-1/2 model deﬁned with bond-dependent Ising interactions
on the honeycomb lattice and it is exactly solvable with the use of an itinerant Majorana
fermion and a Z2 ﬂux deﬁned on each plaquette. In a chiral spin liquid phase MZMs
can appear exhibiting a non-local correlation
and Z2 ﬂuxes binding them obey non-Abelian
statistics, in principle. It is expected to exploit
the composite particles for topological quantum computation. It is diﬃcult, however, tointroduce and control the composites in the
ground state of the pure Kitaev model since
they are thermal excitations. To avoid the difﬁculty, we focus on point vacancies in the system. It is well known that each vacancy binds
Z2 ﬂux in the ground state, or the boundﬂux sector [2], while MZMs strongly trappedin them are not yet reported.
In this study, we reveal the importance of
localized Majorana fermions caused by vacancies, which have not been properly taken into
account in previous studies. In the Kitaev’s
notation, a spin operator at j-site, Sjγ (γ =x, y, z), is represented as Sjγ = ibγj cj /2, using an itinerant Majorana fermion operator
cj and a localized Majorana fermion opera-

tor bγj (Fig. 1). The former behaves as afree Majorana fermion described as a tightbinding model while the latter forms a Z2 gageﬁeld with the other localized Majorana fermion
from the nearest site on each bond. Then, if avacancy exists, the localized Majorana fermion
from the nearest site of the vacancy site loses
the partner and it can couple with itinerant
Majorana fermions, as shown in Fig. 1, whenan external magnetic ﬁeld is applied. This nontrivial Majorana hopping term, or b-c hopping
term, contributes to realizing MZMs bound inZ2 ﬂuxes that should be naturally found in thebound-ﬂux sector.

Figure 1: Majorana fermions for each spin andnontrivial Majorana hoppings between bγj andck around a vacancy vA, represented as arrows.
The eﬀective Majorana Hamiltonian Heﬀ forthe Kitaev model with vacancies, which are
spatially well-separated, works well to host
MZMs. We assume two vacancies, vA and vB,for simplicity and the direction of a magnetic
ﬁeld parallel to the in-plane a-axis. At nonzero

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Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

magnetic ﬁelds, a pair of MZMs appears in thebulk gap and is stable against an increase ina magnetic ﬁeld strength unless the boundﬂux ground state is valid. This property is
not conﬁrmed without b-c Majorana hopping
terms: the Majorana gap grows as increasing
the strength whereas pseudo-MZMs [3] disappear rapidly to form the bulk gap. Moreover,
it is discovered that more than 99% of the realspace amplitudes of MZMs are on the localized
Majorana fermions, which implies that they
are strongly trapped in Z2 ﬂuxes in the groundstate.
To approach a pair of MZMs by a physicalobservable using a non-local property, we consider a non-local spin correlation between thenearest sites of each vacancy, say jA and jBsite. Our interest is written as below at inverse temperature β,⟨SjzA (t) SjzB (0)⟩ ≡[]Tr PF SjzA (t) SjzB (0) e−βHeﬀ
Tr [PF e−βHeﬀ ](1)

where PF is a projection operator to
the physical subspace.
Especially theas shown in Fig. 2. Indeed, the structure factor in the zero-ﬂux sector diﬀers from the one
in the bound-ﬂux ground state in a point thatthe signal comes from b-Majorana excitations
shift to a higher ω region, very close to the
Majorana bulk gap, which implies the absenceof MZMs.
In summary, non-Abelian anyons composedof MZMs and Z2 ﬂuxes, both of which aretrapped in vacancies, can stably exist in the
ground state of the vacancy-induced Kitaev
model with a magnetic ﬁeld. They exhibit the
non-local spin-spin correlation in a chiral spin
liquid phase, and this system may become thenew platform for topological quantum computation.

References[1] A. Kitaev, Ann. Phys. 321 (2006) 2.
,[2] A. J. Willans et al., Phys. Rev. Lett. 104(2010) 237203.
[3] W.-H. Kao and N. B. Perkins, Ann. Phys.
435 (2021) 168506.

Figure 2: The structure factor of a non-local
spin-spin correlation and the Majorana spectrum structure.
Fourier-transformed dynamical structure fac∫ ∞ i(ω+iδ)t z
tor Ψzz⟨SjA (t) SjzB (0)⟩jA jB (ω) ≡0 eclearly reveals the contribution from MZMs
and b-Majorana excitations inside the bulk gap

197

Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Discrete time crystals in solids
Tatsuhiko Ikeda
Institute for Solid State Physics, University of Tokyo
Kashiwa-no-ha, Kashiwa, Chiba 277-8581

We have studied a discrete time crystal(DTC) in solid-state materials using Supercomputer System B at the ISSP. This year, apaper on this project was published [1]. We
will report this project below.
A DTC is an exotic nonequilibrium phasein periodically driven dissipative quantum systems that breaks the discrete time translation
symmetry. The DTC had been well studied
and experimentally demonstrated in artificial
quantum systems without dissipation, such as
trapped ions, NV centers in diamond, andquantum computers. However, it had not been
explored in natural materials, like solids, with
dissipation. Nonetheless, if we create the DTC
in natural materials, it could oﬀer a new possibility of time crystals in material science.
In this study, we have investigated the DTC
in solid-state materials. To uncover the mechanism and properties of the DTC in solids, wehave considered a periodically driven quantum
Ising model with dissipation using the FloquetRedfield equation that describes a periodicallydriven quantum system subject to dissipation.
The Floquet-Redfield equation is much harder
to solve than the usual Schrödinger equation
because we need to treat a D × D density matrix rather than a D-dimensional wave function in the Floquet-Redfield equation (D is theHilbert space dimension). Therefore, in thisstudy, we have employed the mean-field theory
and numerically solved the DTC order parameter self-consistently.

the phase diagram of the DTC in solids (seeFig. 1). This phase diagram highlights a nontrivial behavior that the separate two DTC regions appear. Also, we have shown that this
DTC is robust against any perturbation. We
believe that these results pave the way for a
new research field of time crystals in materialscience.

DTCτNormalT
Figure 1: Phase diagram of the DTC in solids.
The horizontal (vertical) axis denotes the temperature of the bath (the period of the Hamiltonian).

References[1] K Chinzei and TN Ikeda: Phys. Rev. Research 4, 023025 (2022).

Performing a massive numerical calculation
with the supercomputer, we have obtained198

Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Interaction-induced quantum spin Hall state in the
Dirac electron system α-(BEDT-TSeF)2I3
Akito KOBAYASHI
Department of Physics, Nagoya University
Nagoya 464-8602

We have constracted extended-Hubbard
model using first-principles calculations todeal with spin-orbit and non-local repulsive interactions and have shown that aninteraction-induced quantum spin Hall (QSH)
state is stabilized, focusing on the recentlydiscovered candidate topological insulator
α-(BEDT-TSeF)2 I3 having two-dimensionalcharge-neutral Dirac cones.[1, 2, 3, 4] In the absence of repulsive interactions, a topological insulator characterized by a nonzero spin Chern
number appears by the spin-orbit coupling. By
considering up to next-nearest neighbor repulsions at Hartree-Fock level, the intrinsic spinorbit gap is found to grow and a QSH state
appears that has both a nonzero spin Chern
number and order parameter as shown in Fig.
1. The loops ϕi shown in Fig. 1 represents
the spin current flowing between BEDT-TSeF
molecules. The spin currents between nextnearest neighboring molecules also exist, butthey are omitted in Fig. 1 for simplicity. These
spin currents increase at low temperatures below 50 K and widen the gap. Transport coeﬀcients and spin susceptibility have been calculated and found to be consistent with most of
the experimental findings, including the metalto-insulator crossover occurring at ∼ 50 K as
well as the Berry phase change from 0 to pi under hydrostatic pressure. We argue that such a
QSH state does not necessitate a sizeable spinorbit interaction to generate a large insulating gap, which is highly advantageous for the
search of novel topological phases in genericmaterials having small spin-orbit coupling.

References[1] S. Kitou, et al., Phys. Rev. B 103, 035135(2021).
[2] T. Tsumuraya et al., Eur. Phys. J. B, 94,
17 (2021).
[3] Y. Suzumura et al., JPSJ 90, 124707(2021).
[4] D. Ohki, K. Yoshimi, and A. Kobayashi,
arXiv:2110.05010

Figure 1: Interaction-induced quantum spin
Hall state in α-(BEDT-TSeF)2 I3 .

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Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Study of cuprate high-temperature superconductors
using a four-band d-p model
Hiroshi WATANABE
Research Organization of Science and Technology, Ritsumeikan University
1-1-1 Noji-Higashi, Kusatsu-shi, Shiga 525-8577

The discovery of superconductivity incuprates [1] has brought about the significant
progress in strongly-correlated electron systems. However, we have not yet reached aunified understanding of their properties, including their material dependence of the superconducting transition temperature Tc . Recent
development of theories and experiments sheds
light on the importance of the orbital degree of
freedom in cuprates [2, 3]. To obtain the unified description of cuprate high-temperature
supercondctors, we study the four-band d-p
model that properly includes the orbital degreeof freedom [4]. We construct the tight-binding
model for La2 CuO4 and HgBa2 CuO4 systems
based on the first-principles calculation and examine the effect of Coulomb interactions withthe variational Monte Carlo (VMC) method.
The Gutzwiller-Jastrow type wave function isused for the VMC trial wave function. The system size for the calculation is N =24×24=576
unit cells (and thus 576×4=2304 orbitals in total), which is large enough to avoid finite size
effects.
The superconducting correlation function
vs hole doping rate x for the La2 CuO4
system is shown in Fig. 1. At x = 0, thesystem is insulating and the superconductivity is completely suppressed: P dd = 0. As x
increases, mobile carriers are introduced into
the system and the mobility of the Cooper pairincreases. On the other hand, the strengthof the d-d pairing itself is reduced by doping because the electron correlation is also re-duced. The balance between these two factors results in the dome-shaped behavior of
P dd . The Ud /t1 dependence of P dd can bealso understood with this picture. We consider that Ud /t1 ∼ 8 is a realistic value for the
La2 CuO4 system. For the HgBa2 CuO4 system, while the dome-shaped behavior is observed for Ud /t1 = 7 and 8, P dd monotonicallydecreases with x for Ud /t1 = 6. This is because the system is metallic for Ud /t1 = 6 andthe picture of “doped Mott insulator” is nolonger valid. It results from the smaller ∆dp
in the HgBa2 CuO4 system that leads to theweaker electron correlation compared with the
La2 CuO4 system. The difference between two
systems are well described in our four-band d-p
model.

P dd

Figure 1: Superconducting correlation function P dd vs hole doping rate x for the La2 CuO4
system.

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Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

References[1] J. G. Bednorz and K. A. Muller, Z. Phys.
B 64, 189 (1986).
[2] H. Sakakibara et al., Phys. Rev. Lett. 105,
057003 (2010).
[3] C. E. Matt et al., Nat. Commun. 9, 972(2018).
[4] H. Watanabe et al., Phys. Rev. Research
3, 033157 (2021).

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Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Superconductivity in correlated electron systems
Takashi YANAGISAWA
National Institute of Advanced Industrial Science and Technology (AIST)
1-1-1 Umezono, Tsukuba 305-8568

1

2Introduction

Our computations are mainly based on the
optimized variational Monte Carlo method[1].
We also employ the Gutzwiller-Jastrow wave
function where the doublon-holon correlation
is taken into account. We have investigated
the ground-state phase diagram of the twodimensional Hubbard model and the twodimensional d-p model. We performed parallel computation in Monte Carlo calculations.
In order to reduce statistical errors, we carried out ∼ 500 parallel calculations. Parallel
computing is very important to reduce Monte
Carlo statistical errors.
The many-body wave function is written in
the form ψ (1) = exp(−λK)PG ψ0 , where K denotes the kinetic energy part (non-interactingpart) of the Hamiltonian and PG = PG (g) is
the Gutzwiller operator to control the double occupancy with the variational parameterg. ψ0 indicates a trial wave function which
is usually taken as the Fermi sea, the BCS
wave function or the state with some magnetic(or charge) orders. We can improve the wave
function systematically by multiplying by operators PG and e−λK repeatedly. We can consider ψ (2) = exp(−λ′ K)PG (g ′ )ψ (1) for different
variational parameters λ′ and g ′ . This wave
function is a very good many-body wave function because the ground-state energy is lowered greatly and the ground-state energy islower than those that are evaluated by anyother wave functions. We also employ the
Jastrow-type wave function which is written
as ψJ = PG PQ PJ ψ0 where PJ indicates anearest-neighbor number correlation operator
and PQ controls the nearest-ngighbor doublonholon correlation.

202

Why does the pairing state
become stable?

We have shown that the superconducting
phase exists in the strongly correlated region of electronic models[2].
We investigated kinetic-energy effect in superconductivity in the strongly correlated phase of the twodimensional Hubbard model. Let us examine why the superconducting state becomesstable in the strongly correlated region. The
Gutzwiller-BCS state is stabilized as a potential energy driven superconductivity because the Coulomb interaction energy is lowered while the kinetic energy increases in thisstate. In contrast, we showed that in the ψλ BCS wave function ψλ−BCS = e−λK PG ψBCS ,the Coulomb energy increases and instead the
kinetic energy is lowered in the strongly correlated phase where the Coulomb repulsive interaction U is large. The correlated superconducting state is realized as a kinetic energy
driven pairing state and this indicates the enhancement of superconductivity due to kineticenergy effect.
We estimate the kinetic energy in the superconducting state ψλ−BCS [3]. We define theSC condensation energy ∆Esc as a sum of twocontributions ∆Ekin−sc and ∆EU −sc :∆Esc = E(∆ = 0) − E(∆ = ∆opt ),∆Ekin−sc = Ekin (∆ = 0) − Ekin (∆ = ∆opt ),∆EU −sc = EU (∆ = 0) − EU (∆ = ∆opt ),where ∆ = ∆sc is the SC order parameter and∆opt is the optimized value which gives the energy minimum. We have
∆Esc = ∆Ekin−sc + ∆EU −sc .

(1)

The kinetic energy in ψλ−BCS is lower than
the kinetic energy in the normal state ψλ . The

Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

0.5

Coulomb energy expectation value increases as∆sc increases. The results show

3

Kinetic energy enhancement of superconductivity

We define the difference of the kinetic energyas∆Ekin = Ekin (ψG ) − Ekin (ψλ ),
(3)

sc

∆E

kin-sc

/N

0.012

0.2

0.008

0.1

0.004

0

sc

0.3

∆E /N

for ψλ−BCS with U = 18t and the hole density
x = 0.12.

∆E /N
U

(2)

0.016

U

kin

∆EU −sc < 0,

kin

/N

E /N

0.4

∆E /N, E /N

∆Ekin−sc > 0,

0.02

∆E

0

-0.1

-0.004
0

5

10

15

20

U/t

where Ekin (ψG ) and Ekin (ψλ ) indicate the kinetic energy for ψG and ψλ , respectively. We
can write ∆Ekin = Ekin (λ = 0) − Ekin (λ) for
the optimized value of λ. ∆Ekin has the close
relation with the SC condensation energy ∆Escand its kinetic part ∆Ekin−sc .
We show ∆Ekin /N in Fig. 1 for x = 0.12
where x is the hole doping rate. The Coulomb
energy EU /N and the superconducting condensation energy ∆Esc /N are also shown in
Fig. 1. ∆Ekin begins to increase after the
Coulomb energy EU reaches the peak whenU ≈ 8t. The y axis on the right shows∆Ekin−sc /N in Fig. 1. ∆Ekin−sc shows a similar behavior to ∆Ekin . ∆Ekin−sc may change
sign as a function of U , which is consistent withthe analysis for Bi2 Sr2 CaCu2 O8+δ [4]. Thisshows the kinetic energy enhancement of superconductivity.

Figure 1: Kinetic-energy difference ∆Ekin /Nand the kinetic-energy gain ∆Ekin−sc /N in the
superconducting state ψλ−BCS as a functionof U on a 10 × 10 lattice where Ne = 88 andt′ = 0. The Coulomb energy EU /N for ψλ andthe condensation energy ∆ESC are also shown.
We use the periodic boundary condition in one
direction and antiperiodic one in the other direction. The y axis on the right shows the superconducting condensation energy ∆Esc andthe kinetic condensation energy ∆Ekin−sc /Nfor ψλ .

4[2] T. Yanagisawa, J. Phys. Soc. Jpn. 85,
114707 (2016).
T. Yanagisawa, J. Phys. Soc. Jpn. 88,
054702 (2019).
T. Yanagisawa, Condensed Matter 4, 57(2019).

Summary

We examined the kinetic energy effect in SC
states. The Gutzwiller-BCS state is the potential energy driven SC state, because the SC
condensation energy comes from the Coulombinteraction energy. This is the same as theoriginal BCS state where the superconductivity appears due to the attractive interaction.
We evaluated the kinetic energy in the improved SC state (ψλ−BCS ) to find that the kinetic energy gain stabilizes SC state while theexpectation value of Coulomb interaction energy increases. This indicates that superconductivity is enhanced due to the kinetic energy
effect.

203

References[1] T. Yanagisawa et al., J. Phys. Soc. Jpn.
67, 3867 (1998).
T. Yanagisawa, Phys. Rev. B75, 224503(2007) (arXiv: 0707.1929).

[3] T. Yanagisawa, Phys. Lett. A403, 127382(2021).
T. Yanagisawa et al., EPL 134, 27004(2021).
[4] G. Deutscher, A. F. Santander-Syro, N.
Bontemps: Phys. Rev. B 72, 092504(2005).

Activity Report 2021 / Supercomputer Center, Institute for Solid State Physics, The University of Tokyo

Photoinduced nonequilibrium dynamics in correlated
electron systems
Atsushi ONO
Department of Physics, Tohoku University,
Sendai 980-8578

Antiferromagnets are promising for spintronics applications owing to their robustness
against a perturbative ﬁeld and to a higher
energy scale compared with that of ferromagnets. Recently, a current-induced staggered
torque, called Néel spin-orbit torque (NSOT),
has been recognized as an eﬃcient mechanismthat rotates the Néel vector in antiferromagnets. However, since most of the previous
studies considered the electrical switching of
the antiferromagnets, it is not fully understoodwhether and how the NSOT is generated by
optical pulses.
We have studied the real-time dynamics induced by a static electric ﬁeld and a terahertz electric-ﬁeld pulse [1]. We considered aminimal tight-binding model of antiferromagnetic Dirac semimetals, where a nonsymmorphic symmetry associated with the direction of
the Néel vector protects the crossing of doubly
degenerated energy bands, i.e., Dirac points.
The time evolution is governed by the Landau–
Lifshitz–Gilbert equation for the localized moments and the von Neumann equation for theitinerant electrons. We solved these coupledequations by using the ISSP supercomputer
system.
Figure 1 shows the real-time dynamics induced by a mono-cycle electric-ﬁeld pulse. We
found that the Néel vector rotates in a picosecond timescale, depending on the amplitude and polarization of light, and the Néelvector rotation accompanies the change in theenergy gap at the Dirac points. Our analy-

sis revealed that the Néel vector is canted by
optical NSOT and then rotated by a magneticanisotropy torque. We also showed that in thissystem a magneto-optical eﬀect can be used toobserve the direction of the Néel vector in realtime.

Figure 1: Time evolution of the Néel vector n, uniform magnetization m, excited electron density, and bandgap [1]. A mono-cycle
electric-ﬁeld pulse F (t) is applied.

References[1] A. Ono and S. Ishihara, npj Comput.
Mater. 7, 171 (2021).

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