Topological Mott insulator phase and metal-insulator crossover in organic Diracelectron system α-(BEDT-TSeF)2I3

Daigo Ohki1,∗ Kazuyoshi Yoshimi2, and Akito Kobayashi11Department of Physics, Nagoya University, Furo-cho,

Chikusa-ku, Nagoya, 464-8602 Japan2Institute for Solid State Physics,

University of Tokyo, Chiba 277-8581, Japan

(Dated: October 12, 2021)

To elucidate the low-temperature (T ) insulating state of α-(BEDT-TSeF)2I3 (α-(BETS)2I3), weconstruct a two-dimensional extended Hubbard model with transfer integrals including spin-orbit-coupling (SOC) and Coulomb interaction, based on first-principles calculation. We investigate thelow-T electronic state using Hartree Fock approximation, and found that the topological Mottinsulator (TMI) state is a candidate for the insulating state. In this state, energy gap in the high-Tweak topological insulator (TI) state is strongly enhanced by next-nearest-neighbor interactions V ′.With the increase in T , the TMI state continuously changes to a high-T weak TI state. Calculationsusing random-phase-approximation also indicate that the ferrimagnetic spin fluctuation appears inthe high-T TI state, when weak onsite interaction is considered.

I. INTRODUCTION

Dirac electrons in solids such as graphene [1, 2], bis-muth [3, 4], and several organic conductors [5–12] exhibitvarious physical properties, including quantum conduc-tion [13], large diamagnetism [14], and anomalous elec-tron correlation effects. Therefore, Dirac electron sys-tems have attracted attention in both theoretical andexperimental research. In organic Dirac electron systemssuch as α-(BEDT-TTF)2I3 and α-(BEDT-TSeF)2I3 (α-(ET)2I3 and α-(BETS)2I3), which are the main focus ofthis study, the Coulomb interaction between Dirac elec-trons is relatively large because of the small values of thetransfer integrals, resulting in low group velocity.

The two-dimensional (2D) massless Dirac electron sys-tem in α-(ET)2I3 transitions to the insulator phase withstripe charge order [15–17] owing to nearest-neighbor(N.N.) Coulomb interaction. Mass generation becauseof the stripe charge order leads to unique temperature(T ) and pressure (P ) dependencies of the transport coef-ficients and spin gap [18–22]. In the massless Dirac elec-tron phase, ferrimagnetic spin fluctuation, velocity renor-malization, and exotic spin fluctuation owing to electroncorrelation effects have been reported by theoretical stud-ies as well as nuclear magnetic resonance (NMR) exper-iments under an in-plane magnetic field [23–28].

Organic conductor α-(BETS)2I3 is similar to α-(ET)2I3. Both have almost the same crystalline struc-ture and 2D Dirac electron systems, obtained by first-principles calculation using high-accuracy X-ray diffrac-tion data [29]. The transfer integrals and spin-orbit cou-pling (SOC) of α-(BETS)2I3 are greater than those of α-(ET)2I3 [30]. The slightly stronger SOC of α-(BETS)2I3

compared to that of α-(ET)2I3 is derived from the sub-stitution of sulfur (S) atoms by selenium (Se) atoms, and

∗ dohki@s.phys.nagoya-u.ac.jp

its value is estimated to be approximately 2 meV usinggeneralized gradient approximation (GGA) [31, 32]. Thisresult suggests that α-(BETS)2I3 becomes a topologicalinsulator (TI) with a finite SOC gap at the Dirac cone.The energy band structure of α-(BETS)2I3 has been ex-tensively studied [33–35], and the appearance of a weakTI due to SOC has been reported [31, 32].

Interestingly, although α-(BETS)2I3 has a TI state,a clearly different insulating state appears with the de-crease in T . Quantum conduction has been observed bydirect current (DC) resistivity measurements at T > 50K (high-T ) and P > 5 kbar [36–38], and the DC resis-tivity increases sharply at T < 50 K (low-T ) and P < 5kbar. Shubnikov–de Haas oscillation measurements for acarrier-doped sample (∼ 10 K) indicate that the Onsagerphase factor switches from 1/2 to zero at P ' 5 kbar atwhich the insulating state vanishes [37, 38]. The results ofa synchrotron X-ray diffraction experiment suggest thatthe insulating state of α-(BETS)2I3 does not involve spa-tial inversion symmetry breaking and change in the bondlength between N.N. BETS molecules [29]. Furthermore,NMR experiments show that the spin susceptibility and1/T1T are proportional to T and T 2, respectively, nearthe emergence of the insulating state, and no signs of aspin order are observed [39, 40]. Therefore, an insulatingstate unrelated to the charge, bond, and spin order isexpected to appear at low-T in α-(BETS)2I3.

In our recent study [41], we had investigated the insu-lating state of α-(BETS)2I3 using a 2D Hubbard modelwith Hartree approximation; a spin ordered massiveDirac electron (SMD) phase, which involves only time-reversal symmetry breaking, was obtained as the stablesolution. However, signs of a spin order have not beenobserved in recent NMR experiments [39, 40]. We needto consider not only the effect of onsite Coulomb inter-action with the electronic state but also the contributionof long-range components because these interactions re-main finite in the Dirac electron system, even when the

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screening effect is considered [42–44]. In this study, weinvestigate the possibility of a topological Mott insula-tor (TMI) state [45–50] as a candidate for the insulatingstate of α-(BETS)2I3. It is known that the TMI state iscaused by the phase modulation in the transfer integrals[45], the contribution of SOC [46], and the next-nearest-neighbor (N.N.N.) Coulomb interaction [47].

The remainder of this paper is organized as follows.In Section II, we perform first-principles calculation us-ing X-ray data at 30 K under ambient pressure for con-structing a 2D extended Hubbard model with SOC. TheCoulomb interactions, including the long-range compo-nents, are calculated using constrained random phase ap-proximation (cRPA). In Section III, the electronic stateof α-(BETS)2I3 at low-T is firstly calculated applyingHartree Fock (HF) approximation. Next, the stability ofthe TMI state due to the presence of N.N. and N.N.N.Coulomb interactions is investigated, and the phase mod-ulation of the transfer integrals due to the emergence ofTMI is calculated. Moreover, physical quantities, such asthe Onsager phase factor and DC conductivity, are cal-culated using T -matrix approximation [13, 51–54], andcompared with a recent experiment [37]. Finally, we in-vestigate the effect of spin fluctuation on high-T elec-tronic state of α-(BETS)2I3 using random phase approx-imation (RPA) [7, 23–26]. In Sec. IV, the study is sum-marized and the relationship with preceding studies isdiscussed, and the scope for future research is presented.

II. MODEL AND FORMULATION

A. Effective model based on first-principlescalculation

First-principles calculation was performed to derive aneffective model using the X-ray crystal structural dataof α-(BETS)2I3 at 30K under ambient pressure [29]. Asthe exchange-correlation function, GGA was used in theQuantum espresso (QE) package [55, 56]. We used SG15optimized norm-conserving Vanderbilt (ONCV) fully rel-ativistic pseudopotentials to consider the effect of SOCin the first-principles calculation [57]. We set the cut-off energies of the wave functions and charge densities as80 and 320 Ry, respectively, and the mesh of wave num-ber k as 5 × 5 × 3. After first-principles calculation, toobtain the transfer integrals with SOC, maximally local-ized Wannier functions (MLWFs) were created using theWannier90 code [58]. Eight bands near the Fermi energywere selected to construct the MLWFs, and the initial co-ordinates of MLWFs were set at the center of the BETSmolecule in the unit cell.

Figure 1(a) shows the real-space distribution of theMLWFs in each site of the unit cell. The unit cell ofα-(BETS)2I3 has four BETS molecules labeled A, A’, B,and C, where the A and A’ sites are crystallographicallyequivalent. The B and C sites, and the center of A andA’ sites include inversion symmetric points. Figure 1(b)

(a)

-0.6

-0.4

-0.2

0

0.2

0.4

(0,0,0) (-π,π,0) (0,π,0) (0,0,0) (-π,0,0) (-π,π,0)

Ene

rgy

[eV

]

QEWannier90

(b)

FIG. 1. (Color online) (a) Crystal structure and real-spacedistribution of the MLWFs for α-(BETS)2I3 under ambientpressure at 30 K in a unit cell drawn by VESTA [59]. (b)Energy band structure with SOC effect obtained by first-principles calculation.

ta1', Va1'ta3', Va3'ta4', Va4'

tb1', Vb1'tb3', Vb3'tb4', Vb4'

(b)

BA

C

A’

BA

C

A’B

AC

A’

BA

C

A’

(a)

C

C

BA’

BA

C

CA

A’

A’

AB

A’

ta1, Va1ta2, Va2ta3, Va3

tb1, Vb1tb2, Vb2tb3, Vb3tb4, Vb4

a

b(x)

(y)

FIG. 2. (Color online) Schematic of the lattice structureof α-(BETS)2I3 and definition of the transfer integrals andCoulomb interactions for the (a) N.N. sites, and (b) N.N.N.sites. The shaded pink parallelogram denotes a the unit cell.

shows the energy band structure with SOC calculatedusing QE and the Wannier interpolation performed usingWannier90. The energy origin is set at the Fermi energy.

Next, we calculated the long-range Coulomb inter-action considering the screening effect using the con-strained random phase approximation (cRPA) methodin the RESPACK code [60]. We set the energy cutoff ofthe dielectric function as 5.0 Ry.

Figure 2 (a) and (b) show the two-dimensional (2D)

3

TABLE I. List of transfer integrals without (with) SOC t(δ)α,β,σ

(t(δ)α,β,σ

SOC), and effective Coulomb interactions W

(δ)α,β (units

of meV). Here, sgn(σ) is +1(σ =↑) or −1(σ =↓).

Re [t(δ)α,β,σ] Re [t

(δ)α,σ;β,σ]SOC Im [t

(δ)α,σ;β,σ]SOC

ta1 -10.12 -9.345 sgn(σ)×1.365ta2 -16.31 -16.80 sgn(σ)×0.206ta3 51.08 50.22 sgn(σ)×0.614tb1 138.1 136.5 sgn(σ)×12.06tb2 158.7 154.1 sgn(σ)×19.46tb3 65.84 63.77 sgn(σ)×8.866tb4 18.65 17.92 sgn(σ)×4.205t′a1 14.09 13.88 sgn(σ)×0.064t′a3 4.527 4.425 0.0t′a4 21.89 21.68 0.0t′b1 -1.289 -1.276 sgn(σ)×(-0.186)t′b3 -1.280 -1.161 0.0t′b4 2.494 2.636 0.0

Re [W(0)α,α] Re [W

(δ)α,β ]

UA 1383 Va1 580.5UA′ 1383 Va2 596.2UB 1396 Va3 566.7UC 1359 Vb1 579.9

Vb2 572.9Vb3 537.8Vb4 556.9V ′a1 329.8V ′a3 328.1V ′a4 329.1V ′b1 326.9V ′b3 323.9V ′b4 332.4

lattice structures of α-(BETS)2I3. The N.N. and N.N.N.components of the transfer integrals and the Coulombinteractions considered in our calculation are also illus-trated. It is to be noted that in this calculation, spinpolarization was not considered. However, as the energyscale of the spin polarization is small compared to theCoulomb interactions, its effect on the Coulomb interac-tions may not be considerable.

The values of the transfer integrals and Coulomb inter-actions are listed in Table I. The first column of the Ta-ble I presents the notations of the inter-molecular trans-fer integrals and interactions shown in Figs. 2 (a) and(b). The values of the real part of the transfer inte-

grals without (with) SOC, Re [t(δ)α,β,σ] (Re [t

(δ)α,β,σ]SOC),

are listed in the second (third) column. The forth col-umn shows the imaginary part of the transfer integrals

with SOC Im [t(δ)α,β,σ]SOC. Inter-plane transfer integrals

are not considered in this study because they are con-siderably smaller than those in the intra-plane (approx-imately 1/1000 order of the intra-plane components).The bottom of the second column lists the values of thereal part of the effective direct integral Re [W

(δ)α,β ], cal-

(a)

(b) -1

0

1

2

3

4

5

Cou

lom

b in

tera

ctio

n [e

V] VA(R) (BETS)

WA(R) (BETS)

0

5

10

0 10 20

Cou

lom

b in

tera

ctio

n [e

V]

|R| [angst]

V(R) (Al)W(R) (Al)

FIG. 3. (Color online) Range |R|-dependence of the bare di-rect integral Vα(R) and static effective direct integral Wα(R)at the A site (α = A) for (a) α-(BETS)2I3 at 30 K, and (b)result of aluminium (Al) as an example.

culated using RESPACK. Here, δ = (δb, δa) is the rel-ative lattice vector in the a-b plane, and α and β arethe indices of the molecules in the unit cell (A, A’, B,

C). The onsite components Uα = Re[W(0)α,α] and the av-

erage values of the N.N. and N.N.N. components areVa = 1

3

∑3n=1 Van = 581.1 meV, Vb = 1

4

∑4n=1 Vbn =

561.9 meV, V ′a = 13

∑4n=1 V

′an = 329.0 meV, and Vb =

13

∑4n=1 V

′bn = 327.7 meV. As Vb/Va = 0.967 ' 1 and

V ′b /V′a = 0.996 ' 1, there is charge geometrical frustra-

tion in α-(BETS)2I3. This charge geometrical frustrationeffect is one of characteristic features of organic conduc-tors [61].

Figure 3 (a) shows the range(|R|)-dependence of thebare direct integral Vα(R) and static effective direct in-tegral Wα(R) for α = A, evaluated using RESPACK.It is to be noted that even when the screening effect isconsidered, the long-range components of Wα(R) havefinite values. Previous studies [62, 63] have indicatedthat the Coulomb interaction shifts by a constant valuewhen the dimensional down-folding method is used toreduce the three-dimensional Hamiltonian to two dimen-sions. However, α-(BETS)2I3 has a Dirac cone and doesnot have a Fermi surface. Therefore, the screening ef-fect is expected to be weaker and it is more likely thatthe long-range components will survive. Moreover, basedon calculations using the Weyl model, it has been sug-gested that long-range Coulomb interaction survives inDirac electron systems, even when the screening effect isconsidered [42–44]. For comparison, we plotted Vα(R)and Wα(R) for aluminium in Fig. 3 (b). In this case, the

4

effective Coulomb interaction is considerably decreasedby the screening effect, and even the N.N. componentbecomes zero.

Based on the above first-principles calculation results,we constructed a 2D extended Hubbard model [15]:

H =∑R,δ

∑α,β

∑σ1,σ2

t(δ)SOCα,σ1;β,σ2

c†R,α,σ1cR+δ,β,σ2

+ λ∑R,α

UαnR,α,↑nR,α,↓

+λ

2

∑R,δ

∑α,β

∑σ1,σ2

V(δ)α,βnR,α,σ1

nR+δ,β,σ2, (1)

where R is the coordinate of the unit cell, α and β arethe site indices, and σ1 and σ2 are the spin indices (↑, ↓).t(δ)SOCα,σ1;β,σ2

is the transfer integral between (α, σ1) and

(β, σ2) separated by δ. Site potentials t(0)A = t

(0)A′ = 4.470

eV, t(0)B = 4.465 eV, and t

(0)C = 4.477 eV are excluded in

our model because their contribution to the energy band

is negligible. c†R,α,σ1(cR,α,σ1

) is the creation (annihila-

tion) operator for the α site with spin σ1 in the unit cell

located at R. Uα and V(δ)α,β are the onsite and inter-site

Coulomb interactions, respectively, given by the static ef-

fective direct integrals W(δ)α,β calculated using RESPACK.

λ (0 < λ < 1) is multiplied with W(δ)α,β in our model to

control the strength of W(δ)α,β . We also define the num-

ber operator as nR,α,σ = c†R,α,σcR,α,σ. In the following,the lattice constants, Boltzmann constant kB , and thePlanck constant ~ are considered to be unity. Electron-volt (eV) is used as the unit of energy throughout thispaper.

B. Electronic state using Hartree-Fockapproximation

We treat Eq. (1) within the Hartree-Fock (HF) approx-imation in the wave number space. Fourier inverse trans-

form, cR,α,σ1= N

−1/2cell

∑k ck,α,σ1

eik·R, was performedon Eq. (1). Here, Ncell is the total number of unit cellsand k = (kx, ky) indicates the wave-number vector. The

Hartree-Fock Hamiltonian is as follows:

HHF = HT +HU +HV

HT =∑k

∑α,β

∑σ1,σ2

∑δ

t(δ)SOCα,σ1;β,σ2

eik·δc†k,α,σ1ck,β,σ2

HU = λ∑k

∑α

Uα

[∑σ1

〈nα,σ1〉c†k,α,σ1

ck,α,σ1

− 1

Ncell

∑q

(〈c†k−q,α,↑ck′,α,↓〉c†k′+q,α,↓ck,α,↑

+〈c†k′+q,α,↓ck,α,↑〉c†k−q,α,↑ck′,α,↓

)]HV =

λ

2

∑k

∑α,β

∑σ1,σ2

∑δ

V(δ)α,β

[2〈nβ,σ2

〉c†k,α,σ1ck,α,σ1

− 1

Ncell

∑q

e−iq·δ(〈c†k−q,α,σ1

ck′,β,σ2〉c†k′+q,β,σ2

ck,α,σ1

+〈c†k′+q,β,σ2ck,α,σ1

〉c†k−q,α,σ1ck′,β,σ2

)],

(2)

where σ1 = −σ1, and 〈nα,σ1〉 = 〈c†0,α,σ1

c0,α,σ1〉 is the

charge density of site α and spin σ1. We consider onlythe order that occurs within the unit cell and assumetranslation symmetry to exclude long-range orders thathave not been observed experimentally. These treatmentgive q = 0 and k′ = k. With these simplifications, theFourier transform of the order parameter in HF approx-imation is given by

〈c†k,α,σ1ck,β,σ2

〉 =∑δ

〈c†0,α,σ1cδ,β,σ2

〉e−ik·δ,

〈c†0,α,σ1cδ,β,σ2

〉 =1

Ncell

∑k,ν,σ

dα,σ1;ν,σ(k)d∗β,σ2;ν,σ(k)

1 + exp (Eν,σ(k)/T )eik·δ.

(3)

Thus, HHF for each k can be finally expressed as follows:

Hα,σ1;β,σ2(k) =∑δ

t(δ)SOCα,σ1;β,σ2

eik·δc†k,α,σ1ck,β,σ2

+ λUαδα,βδσ1,σ2

{〈nα,σ1

〉c†k,α,σ1ck,α,σ1

− δb(0)α,σ1;α,σ1

c†k,α,σ1ck,α,σ1

}+λ

2

∑δ

V(δ)α,β

{2〈nβ,σ2

〉c†k,α,σ1ck,α,σ1

−[(∑

δ′

δb(δ′)α,σ1;β,σ2

e−ik·δ′

)c†k,β,σ2

ck,α,σ1 + h.c.

]}, (4)

where the off-diagonal site component of the order pa-

rameter, δb(δ)α,σ1;β,σ2

≡ 〈c†0,α,σ1cδ,β,σ2

〉 (α 6= β), modulatesthe transfer integrals.

We diagonalized Hα,σ1;β,σ2(k) numerically using eigen-

vector dα,σ1;ν,σ(k) and obtained the energy eigenvalues.

5

For convenience, we define eigenvalues Eν,σ(k) as

Eν,σ(k) = Eν,σ(k)− µ

=

⟨∑α,β

∑σ1,σ2

d∗α,σ1;ν,σ(k)Hα,σ1;β,σ2(k)dβ,σ2;ν,σ(k)

⟩− µ,

(5)

where µ is the chemical potential calculated to satisfy3/4-filling. Eν,σ(k) are the eigenvalues with band in-dex ν obtained by numerical diagonalization [E1,σ(k) >E2,σ(k) > E3,σ(k) > E4,σ(k)].

The Berry curvature Bzν,σ(k) is calculated as

Bzν,σ(k) =∑ν′ 6=ν

vxν,ν′,σ(k)vyν′,ν,σ(k)

i [Eν,σ(k)− Eν′,σ(k)]2 + c.c., (6)

where the velocity matrix vην,ν′,σ(k) along the a-axis (η =

a(y)) and b-axis (η = b(x)) is given by

vην,ν′,σ(k) =∑α,β

∑σ1,σ2

d∗α,σ1;ν,σ(k)

×∂Hα,σ1;β,σ2(k)

∂kηdβ,σ2;ν′,σ(k). (7)

The Onsager phase factor γ and Berry phase φB wereobtained based on the semiclassical theory [64–68]:

γ ≡ 1

2− φB

2π,

φB =

∫SF

Bzν,σ(k)dS.(8)

∫SF

indicates surface integration on the Fermi surface. In

γ calculation, we used an electron-doped band of approx-imately 0.005, which is of the same order as the carrierdoping value in the experiment [37].

C. Conductivity

The DC conductivity was calculated using the Nakano-Kubo formula [13, 51–54]:

ση(ω) =1

iω

[QRη (ω)−QRη (0)

], (9)

QRη (ω) =e2

Ncell

∑k,ν,ν′,σ

|vην,ν′,σ(k)|2χ0ν,ν′,σ(k, ω), (10)

χ0ν,ν′,σ(k, ω) = − f(Eν,σ(k))− f(Eν′,σ(k))

Eν,σ(k)− Eν′,σ(k) + ~ω + i0+, (11)

where η = b(x) and a(y) are the axes and ω is the fre-quency. 0+ is set as 0+ = 5× 10−4, in this study. In thelimit of ω → 0, we can obtain the longitudinal conduc-

tivity along the a-axis (η = a(y)) and b-axis (η = b(x)):

ση =

∫dω

(− dfdω

)Φη(ω), (12)

Φη(ω) =2e2

Ncell

∑k,ν,σ

∣∣vην,σ(k)∣∣2 τν,σ(ω,k)δ(~ω − Eν,σ(k)).

(13)

We treat the effect of elastic scattering between electronsand the impurities originating from the lack and disorderof I3− molecules, using T -matrix approximation. Theimpurity potential term Himp is defined by

Himp =V0

Ncell

∑k,q,α,σ

Nimp∑i

e−iq·ric†k+q,α,σck,α,σ, (14)

and we treat it using the perturbation theory for Green’sfunction. Here, V0 is the strength of the potential and ri(i = 1, · · · , Nimp) is the coordinate of the i-th impurity.The damping constant γν,σ(ω,k) is obtained by calcula-tion as follows:

γν,σ(ω,k) =~

2τν,σ(ω,k)= −ImΣRν,σ(ω,k)

= cimp

|dα,ν,σ(k)|2{πV 2

0 ρEσ (ω)

}1 + {πV0ρEσ (ω)}2

, (15)

where cimp =Nimp

Ncell= 0.02 � 1 is impurity density and

ρEσ (ω) represents the density of state for spin σ:

ρEσ (ω) =∑

k,α,σ1,ν

|dα,σ1;ν,σ(k)|2δ(~ω − Eν,σ(k))

In this study, the DC conductivity is normalized to theuniversal conductivity σ0 = 4e2/πh.

D. Calculation of the spin fluctuation using RPA

We investigated the effect of spin fluctuation on theNMR properties, e.g., the Knight shift and 1/T1T in thehigh-T Dirac electron phase in RPA using eq. (1) [7, 23,25].

The bare Green’s function on the site representation isdefined as

Gα,σ1;β,σ2(k, εn) ≡

∑ν,σ

dα,σ1;ν,σ(k)d∗β,σ2;ν,σ(k)

iεn − Eν,σ(k), (16)

where εn = (2n + 1)πT is the Matsubara frequency.In the linear response theory, the bare susceptibility

6

χ0α,β(q, ωm) can be calculated as follows:

χ0α,σ1;β,σ2

(q, ωm)

= − T

Ncell

∑k,n

Gα,σ1,β,σ1(k + q, εn + ωm)Gβ,σ2,α,σ2(k, εn)

=1

Ncell

∑k

∑σ,σ′

4∑ν,ν′=1

F ν,σ;ν′,σ′

α,σ1;β,σ2(k, q)χ0

ν,σ;ν′,σ′(q, ωm),

(17)

χ0ν,σ;ν′,σ′(q, ωm) = − f(Eν,σ(k + q))− f(Eν′,σ′(k))

Eν,σ(k + q)− Eν′,σ′(k) + iωm,

(18)

where Ncell is the system size and ωm = 2mπT . F (k, q)indicates the form factor represented by

F ν,σ;ν′,σ′

α,σ1;β,σ2(k, q) = dα,σ1,ν,σ(k + q)d∗β,σ1;ν,σ(k + q)

×dβ,σ2;ν′,σ′(k)d∗α,σ2;ν′,σ′(k). (19)

In RPA, the spin susceptibility χS and the transversespin susceptibility χ± in the absence of an external fieldand in presence of spin symmetry are calculated as fol-lows.

χS(q, ω) = χ±(q, ω)

=(I − χ0(q, ω)U

)−1

χ0(q, ω), (20)

where I is the unit matrix and Uαβ = Uα(λU )δα,β . More-over, to estimate the contribution of the intra- and inter-band components to the spin fluctuation, we divide thebare susceptibility into two components [25]:

χ0,Intraα,σ1;β,σ2

(q, ω) =1

Ncell

∑k

∑σ,σ′

∑ν=ν′

F ν,σ;ν′,σ′

α,σ1;β,σ2(k, q)

×χ0ν,σ;ν′,σ′(q, ω), (21)

χ0,Interα,σ1;β,σ2

(q, ω) =1

Ncell

∑k

∑σ,σ′

∑ν 6=ν′

F ν,σ;ν′,σ′

α,σ1;β,σ2(k, q)

×χ0ν,σ;ν′,σ′(q, ω), (22)

where χ0,Intra and χ0,Inter are the intra- and inter-bandcomponents of the bare susceptibility χ0. Thus, theintra-band component of the spin susceptibility is cal-culated as follows:

χS,Intra =(I − χ0,IntraU

)−1

χ0,Intra. (23)

The inter-band component χS,Inter is also obtained basedon definition χS,Inter = χS−χS,Intra. Applying analyticalcontinuation iωm → ~ω + i0+, the site-resolved Knightshift Kα and 1/T1T in RPA are obtained as follows:

Kα =∑β

Re[χSα,β(q = 0, ω = 0)

], (24)

and

1/T1T =∑q

∑α

Im[χ±α,α(q, ω0)

]ω0

, (25)

0

0.1

0.2

0.3

Δ[e

V]

1

1.2

1.4

1.6

1.8

2

〈n α〉

0 0.2 0.4 0.6 0.8 1λ

w/ SOCw/o SOC

α = Aα = A'α = Bα = C

(a)

(b)

(c)

λC

-0.05

-0.025

0

0.025

0.05

Im [

δb

] Im [ δbb1, ↓' ] Im [ δbb4, ↓ ]

Im [ δbb4, ↑ ]Im [ δbb1, ↑' ]

FIG. 4. (Color online) λ-dependence of the (a) energy gap∆, (b) imaginary part of the order parameter Im[δb′b1,σ] andIm[δbb4,σ], and (c) charge density 〈nα〉.

where the frequency ω0 is infinitely close to zero and isset as ω0 = 0.001 in this study.

III. NUMERICAL RESULTS

A. Electronic state at low temperature

In this subsection, the numerical results for the elec-tronic state with HF approximation are shown. We setthe initial states randomly and investigated the electronicstate with the lowest energy, other than the charge, spin,and bond orders, with HF approximation. Throughoutthis subsection, T = 1× 10−4.

Figure 4(a) shows the λ-dependence of the energy gapat the Dirac point ∆ with and without SOC. In the ab-sence of SOC, ∆ opens at λ > λC , and phase transi-tion from the zero gap Dirac electronic state (ZGS) tothe topological Mott insulator (TMI) state occurs. Onthe other hand, in the presence of SOC, the electronicsystem becomes a topological insulator (TI) at λ = 0.With the increase in λ, ∆ increases continuously. In Fig.4(b), the λ-dependence of the imaginary part of the order

7

parameter Im[δb′b1,σ] and Im[δbb4,σ] with SOC are plot-

ted. Due to the contribution of SOC, δb′b1,↑ (δbb4,↑) and

δb′b1,↓ (δbb4,↓) have opposite signs. With the increase in

λ, Im[δb] gradually increases above λ > 0. In Fig. 4(c),the λ-dependence of the charge density 〈nα〉 is shown. Atλ = 0, charge disproportion due to the chemical potentialis observed, and as λ increases, the charge densities con-tinuously become uniform. It is to be noted that whenSOC is considered, the TMI state caused by the contri-bution of Coulomb interaction and SOC [42–50] appearsin the region above λ > 0. The difference between the TIand TMI states is summarized as follows: The TI statehas a slight gap of approximately 2 meV due to the con-tribution of the SOC alone [32, 41]. The gap in the TIstate is enhanced by the contribution of Coulomb inter-action when λ > 0, and we refer to this state as the TMIstate with SOC. In the TMI state with SOC, phase mod-ulation of the transfer integrals occurs, as shown in Fig.4 (b). On the other hand, the TMI state without SOCat λ > λC is induced by the contribution of Coulombinteraction alone [42–50]. In this state, the sign of orderparameter δb does not depend on the degrees-of-freedomof the spin. The TMI state with and without SOC isanalogous to a ferromagnet with and without an exter-nal magnetic field.

Figures 5(a) and (b) show the energy eigenvaluesEν,σ=↑(k) near the Fermi energy (ν = 1, 2) in the TI(λ = 0) and TMI (λ = 0.5) states, respectively, withSOC. The energy gap in the TI state is approximately 2meV as shown in the inset of Fig. 5(a). The Berry curva-ture Bzν,σ(k) in the TMI state with SOC (λ = 0.5) is plot-ted in Figure 5 (c) for each spin. Bzν,σ(k) has two peaksoriginating from two massive Dirac cones in the Brillouinzone. The spin Chern number becomes finite because thetwo peaks of Bzν,σ(k) have the same sign, which invert ac-cording to the spin index σ. These wavenumber and spindependencies on Bzν,σ(k) are the almost same as those inthe TI state [41].

B. Stability of the TMI state in the presence ofCoulomb interaction

Next, to investigate the relationship between the sta-bility of the TMI state in α-(BETS)2I3 and the values ofthe N.N. and N.N.N. interactions, we represent the N.N.and N.N.N. Coulomb interactions by parameter V andV ′, respectively, and draw the phase diagram. Through-out in this subsection, the onsite Coulomb interaction Uand T are fixed at (U, T ) = (0.5, 1× 10−4), unless other-wise stated.

We first draw the V -V ′ phase diagram. The calculationresult is shown in Fig. 6. V ′C indicates the value ofV ′ at which ZGS to TMI phase transition occurs whenSOC is absent. V ′D is defined as the value of V ′ atwhich the Onsager phase factor γ without SOC becomes0.25, as described below. In Fig. 6, it can be observedthat the CO state appears in V > 0.12 and V ′ < V ′CO

ν = 1

ν = 2

-π 0 π -π 0 π-1

-0.5

0

0.5

1

kx ky

Bz ν

=1,σ

(k)

[eV

]

σ = ↑

σ = ↓

(b)

(a)

-π 0 π-π0π

kx

ky

Eν,σ

= ↑ (k)

[eV

]

-0.2

0

0.2

(c)

-π 0 π-π0π

kx

ky

Eν,σ

= ↑ (k)

[eV

]

-0.4-0.2

00.20.4

Eν,σ

= ↑ (k)

[eV

]

-π 0 π-0.02

-0.01

0

0.01

0.02

kx

~ 2 meV

FIG. 5. (Color online) Energy band structure Eν,σ(k) in the(a) TI (λ = 0) and (b) TMI (λ = 0.5) states near the Fermienergy (ν = 1, 2) for σ =↑. Eν,↑(k) and Eν,↓(k) are degener-ated. Inset in (a) is the magnified view of Eν,σ(k) in the TIstate in the −0.02 < Eν,σ(k) < 0.02 energy range. (c) Berrycurvature Bzν,σ(k) in the TMI phase for ν = 1.

(upper-left region). However, with the increase in V ′,the TMI state is stabilized above V ′ > V ′CO and V ′ >V ′C or V ′D. This indicates that V ′ plays a significantrole in stabilizing the TMI state. As indicated in thelist of Coulomb interactions in Fig. 2, V ′/V is expectedto be large in α-(BETS)2I3. This tendency favors therealization of the TMI state.

Next, we calculate the phase of order parameter ϕ =

tan−1 Im[δb]Re[δb] in the unit cell and investigate the possibility

of a local magnetic field caused by the TMI state. Fig-ures 7 (a) and (b) show the schematic of the unit cell ofα-(BETS)2I3 and the loop patterns to calculate the sum-mation of ϕ. These loops include only N.N. and N.N.N.bonds. The V ′-dependence of the summation of phases ineach loop of the unit cell ϕn (n = 1, · · · , 10) at V = 0.18

8

0

0.1

0.2

0.3

0 0.1 0.2 0.3

V [

eV]

TMI

V'C

V'CO

CO

V'D

TI / ZGS

V' [eV]

FIG. 6. (Color online) V -V ′ phase diagram for (U, T ) =(0.5, 10−4). V ′C indicates the value of V ′ that opens theenergy gap when SOC is absent. For V ′ < V ′CO, the in-version symmetry is broken (〈nA〉 6= 〈nA′〉) and horizontalstripe charge order appears. V ′D is defined as V ′ at whichthe absolute value of the Onsager phase factor |γ| withoutSOC becomes 0.25.

are plotted in Figs. 7 (c) and (d). The sign of ϕn dependson the spin degrees-of-freedom when SOC is considered.The sum of ϕn in the unit cell becomes zero because can-cellation occurs: ϕ1 = −ϕ4, ϕ2 = −ϕ5, ϕ3 = −ϕ6 (loopsincluding only N.N. bonds) and ϕ7 = −ϕ10, ϕ8 = −ϕ9

(loops including N.N. and N.N.N. bonds). Therefore, theTMI state due to V ′ and SOC in this study differs froma current state suggested in cuprates super conductors[69, 70] in which the summation of the phases in the unitcell is finite and there is a net current.

The V ′-dependence of the absolute value of the On-sager phase factor |γ| at V = 0.1 is plotted in Fig. 8.Here, V ′D is determined as the value at which |γ| with-out SOC becomes 0.25. The value of |γ| becomes zeroas V ′ decreases, and increases to 0.5 when V ′ is suffi-ciently large. This behavior is consistent with previouslyreported experimental results [37] in which the phase fac-tor changes from 0.5 to zero as the pressure P decreases,where the P -dependence can be considered as the changein V ′ associated with the change in P . |γ| does not be-come zero even when the system is in the TMI phasebecause of the following. The Berry phase φB is cal-culated by the surface integral of the Berry curvature onFermi surface SF as shown in Eq. (8).In a massive Diracelectron system such a TMI, the peak of the Berry cur-vature Bzν,σ(k) decreases with the increase in the energygap ∆, and Bzν,σ(k) widens and spreads in the Brillouinzone as shown in Fig. 5 (b). Therefore, when the spreadof Bzν,σ(k) becomes sufficiently larger than the integralrange SF with the increase in ∆ due to V ′ (see Figs. 8(b) and (c)), φB decreases and becomes zero (|γ| becomes

B

A

C

C

A’

A

A’

φ1

φ4

φ5φ3

φ2

φ6

(a)

(c)

-π/2

-π/4

0

π/4

π /2

0 0.1 0.2 0.3

φn

-π/2

-π/4

0

π/4

π/2

φn

φ3

φ6

φ2

φ5 φ1φ4

(d)

φ8

φ7

φ9

φ10

(b)

B

A A

A’ A’

AA

C

φ7φ8

φ9φ10

V' [eV]

FIG. 7. (Color online) Schematic of the unit cell and def-inition of the phases around the (a) three N.N. bonds and(b) three N.N. bonds and one N.N.N. bond. (c) and (d) V ′-dependence of the phases on the loops shown in (a) and (b):ϕn (n = 1, ..., 10) at (U, V, T ) = (0.5, 0.18, 10−4).

0.5).

Finally, we investigated the T -dependence of the elec-tronic state with HF approximation and calculated theDC conductivity using T -matrix approximation. Figure9 (a) shows the T -dependence of the energy gap ∆ at(V, V ′) = (0.1, 0.16). With SOC and without Coulombinteraction, ∆ is constant, (∆ ' 0.002) for 0 < T < 0.03.This small gap is considered as the energy gap in theTI state of α-(BETS)2I3 [32]. On the other hand, whenconsidering U , V , and V ′ in the Hamiltonian withoutSOC, the TMI state in which order parameter Im[δb] hasa finite value appears, and ∆ increases sharply below thecritical temperature TC due to ZGS to TMI phase transi-tion. When both SOC and Coulomb interaction are con-sidered, ∆ ∼ 0.01 at T = 0.03, which is approximatelyfive times that of the case without Coulomb interaction.On decreasing T , ∆ gradually increases toward Tc dueto the effect of TI to TMI crossover and has a constantvalue ∆ ∼ 0.038, which is twice that of the case withoutSOC. This result indicates that the contribution of SOCand V ′ renders the TI as well as TMI states more stable.

9

0

0.5

0 0.1 0.2 0.3

|γ|

V' [eV]

w/ SOC

w/o SOC

V' D

(a)

(b)

(c)

FIG. 8. (Color online) (a) V ′-dependence of the abso-lute value of the Onsager phase factor |γ| at (U, V, T ) =(0.5, 0.1, 10−4). V ′D is defined as V ′ at which the abso-lute value of |γ| without SOC becomes 0.25. (b) and (c)Illustration of the Berry curvature Bzν,σ(k) and energy bandEν,σ(k) near the Dirac point for two ∆ values: ∆� |EF| and∆ > |EF|.

In Figs. 9 (b) and (c), the T -dependence of the DCconductivity ση/σ0 in units of the universal conductivityis plotted along the b-axis (η = 1) and a-axis (η = 2)directions. ση(T )/σ0 without SOC shows quantum con-duction at T > TC and sharply decreases at T < TC, asshown in Fig. 9 (b). On the other hand, ση(T )/σ0 withSOC (Fig. 9 (c)) decreases continuously near T = TC

0

0.01

0.02

0.03

0.04

0 0.01 0.02 0.03

Δ[e

V]

w/ SOCw/o SOC

only SOC

TC

T [eV]

(a)

(b)

(c)

10-6

10-5

10-4

10-2

10-1

100

102

10-4 10-3 10-2

T [eV]

10-5

10-4

10-2

10-1

100

102

σ η(T

)/σ

0σ η

(T)/σ

0

w/o SOC

w/ SOC

σ1/σ0 σ2/σ0

TC

σ1/σ0 σ2/σ0

FIG. 9. (Color online) (a) T -dependence of the energy gap∆ at (U, V, V ′) = (0.5, 0.1, 0.16). T -dependence of the DCconductivity σ/σ0 in units of the universal conductivity σ0 =4e2/πh at (U, V, V ′) = (0.5, 0.1, 0.16) when (b) SOC is notconsidered and (c) SOC is considered.

( ≃ 50 K)

Dirac electron with small SOC gapInsulating state

High-TLow-TTMI TI

Cross over

FIG. 10. (Color online) Illustration of the T -dependence ofthe electronic state obtained by calculation with HF approx-imation, with onsite U , N.N. sites V , and N.N.N. sites V ′.

with the decrease in T reflecting the gentle T -dependenceof the energy gap due to crossover.

To summarize this subsection, we presented the T -dependence of the electronic state obtained with HF ap-proximation (Fig. 10). As T is increased, TMI to TIcrossover occurs, and a TI state with small gap due toSOC appears at high-T . The TMI state, which is mainlycaused by the N.N. interaction V ′ and SOC, is a strongcandidate of the insulating state on α-(BETS)2I3 at low-T .

10

0

1

2

3

40

1

2

0 0.0025 0.005 0.0075 0.01T [eV]

0

1

2

KC

KA

KB

-0.5

0

0.5

1

0 0.0025 0.005 0.0075 0.01

KB

T [eV]

intra + inter

intra

inter

(a)

(b)

(c)

FIG. 11. (Color online) T -dependence of the site-resolvedKnight shift Kα ≡

∑β Re

[χSα,β(0, 0)

]at U = 0.13 (thick

line) and U = 0 (thin line) for (a) α = A, (b) α = C, and (c)α = B. Inset in (c) is the T -dependence of KB divided into twocomponents: intra-band (dotted line) and inter-band (one-dotchain line).

C. Spin fluctuations in the high-T Dirac electronphase

In this subsection, we calculate the spin susceptibil-ity using RPA and discuss the relationship with NMR inthe high-T TI state. Previous studies on α-(ET)2I3 [23–25] have shown that the ferrimagnetic (FM) spin fluc-tuation observed in the site-resolved Knight shift [24] isinduced by U , where only the site-resolved Knight shiftat B site KB , defined in Eq. (24), becomes negative withthe increase in U , and the other components always re-main positive. It has been reported that this behavioris caused by the inter-band electron-hole excitation en-hanced by U [23–25]. In this study, we investigate thepossibility of FM spin fluctuation in α-(BETS)2I3 [71].

Figures 11(a)-(c) show the T -dependence of Kα at theα = A, B, and C sites obtained using RPA at U = 0.13(thick line) and U = 0 (thin line) as an example. AsRPA overestimates the magnitude of the Coulomb inter-

-500

-250

0

250

500

0 0.0025 0.005 0.0075 0.01

Re[χ

SA

,β(q

= 0

,ω=

0)]

χSA, A

χSA, A'

T [eV]

-π0 π -π

0π0

20

40

qx qy

Re[χS

A, A

(q,ω

= 0

)]

-40-2002040

-π0 π -π

0π-40

-20

qxqy

Re[χS

A, A

'(q,ω

= 0

)]

-40-20020400

(a) (b)

(c)

FIG. 12. (Color online) Momentum q-dependence of thereal-part of the site-resolved spin susceptibility of the Asite Re

[χSA,β(q, 0)

]at (U, T ) = (0.46, 0.007) for (a) β = A

and (b) β = A′. (c) T -dependence of Re[χSA,A(0, 0)

]and

Re[χSA,A′(0, 0)

].

action, we consider the U value, which is smaller thanthose used in previous subsections (U < 0.5) for com-parison with the experimental results. As shown in Figs.11 (a) and (b), KA and KC are enhanced when U isconsidered, and become zero with the decrease in T ow-

ing to the cancellation of each component of Re[χSA,β

]and Re

[χSC,β

](β =A, A′, B, and C). In contrast, KB

decreases and becomes negative below T ∼ 0.0075 as

shown in Fig.11(c). The T -dependencies of Re[χS,Intra

B

],

Re[χS,Inter

B

], and Re

[χSB]

are also plotted in the inset of

Fig. 11 (c). It is indicated that Re[χS,Inter

B

]becomes neg-

ative for 0 < T < 0.01, and causes Re[χSB]

to becomenegative. This behavior is qualitatively similar to thatobserved in α-(ET)2I3[24]. This is because α-(BETS)2I3

at high-T has a characteristic wavenumber dependence ofthe square of the absolute value of the eigenvector sucha zero line for B and C sites, which is similar to that ofα-(BEDT-TTF)2I3 under high pressure.

Next, we calculated spin susceptibility consideringstronger interaction and investigated the kind of spinsusceptibility that was enhanced. Figures 12(a) and(b) show the momentum q-dependence of the spin sus-

ceptibility Re[χSA,β

]at ω = 0 for β = A,A′ with

(U, T ) = (0.46, 0.007). In the strong interaction case,Re[χSA,A

]and Re

[χSA,A′

]exhibit a peak at q = 0 re-

flecting the Fermi point in the Dirac electron system.

11

10-1

100

101

102

103

0.001 0.01

1/T

1T [

a.u.

]

T [eV]

(1/T1T )0

(1/T1T )FM

(1/T1T )AF

FIG. 13. (Color online) T -dependence of 1/T1T at U =0 ((1/T1T )0), U = 0.13 ((1/T1T )FM), and U = 0.46((1/T1T )AF).

Re[χSA,A

]and Re

[χSA,A′

]at q = 0 have almost the same

absolute values with opposite signs. Furthermore, weplotted the T -dependence of Re

[χSA,A(q = 0, ω = 0)

]and

Re[χSA,A′(q = 0, ω = 0)

](Fig. 12 (c)). With the decrease

in T , they are enhanced, and diverge positively and neg-atively toward T ∼ 0.006. This result indicates thatthe spin fluctuation inducing the antiferromagnetic (AF)spin order between A and A′ sites in the unit cell is en-hanced as T decreases, when strong Coulomb interactionis considered. This AF spin ordered state correspondsto the spin ordered massive Dirac electron (SMD) statementioned in our previous study [41].

Finally, the T -dependence of 1/T1T =∑q

∑α Im

[χ±α,α(q, ω0)

]/ω0 at ω0 = 0.001 is dis-

played in Fig. 13 for three different U values: (1/T1T )0

for U = 0, (1/T1T )FM for U = 0.13 where FM spin fluc-tuation is dominant, and (1/T1T )AF for U = 0.46 whereAF spin fluctuation is dominant. It can be observed thatthe FM spin fluctuation occurring at weak U (= 0.13)causes only a slight increase in the value of 1/T1T andhas no significant contribution to 1/T1T (see (1/T1T )0

and (1/T1T )FM). However, AF spin fluctuation causesa clear change in 1/T1T . When a strong U (= 0.46) isconsidered and AF spin fluctuation occurs, 1/T1T tendsto increase and diverges as T decreases.

IV. SUMMARY AND DISCUSSION

In this study, first-principles calculation was first per-formed to construct an extended Hubbard model withtransfer integrals considering SOC, and onsite, N.N. site,and N.N.N. site Coulomb interactions (U, V, V ′). As α-(BETS)2I3 has the Dirac cone near the Fermi energy and

does not have Fermi surface, the long-range componentsof Coulomb interaction are expected to survive even whenthe screening effect is considered [42–44]. In fact, we con-firmed the value of V ′ is the almost of the same order asV in α-(BETS)2I3.

Next, the electronic state in the low-T insulating statewas investigated using HF approximation with Coulombinteraction obtained by RESPACK, and the TMI statewas found as one of the candidate states without charge,spin, and bond orders. Furthermore, for investigatingthe stability of the TMI state against the change in Vand V ′, we draw the V -V ′ phase diagram and found thatthe TMI state was stabilized by the contribution of V ′

and that the energy gap due to the TMI state was en-hanced by the contribution of V ′ and SOC. Moreover,the Onsager phase factor was calculated based on thesemiclassical theory [64–68] and compared with a recentexperiment [37]. Considering the pressure dependenceas the change in V ′ associated with the change in pres-sure, we showed that the Onsager phase factor changedfrom 0.5 to zero at low-T due to the spread of the Berrycurvature in the wavenumber space in a massive Diracelectron system and slight shift of the Fermi energy dueto carrier doping. This behavior is consistent with therecent Shubnikov–de Haas oscillation experiment[37]. Inaddition, using T -matrix approximation and the Nakano-Kubo formula, we showed that when the insulating stateat low-T is assumed to be the TMI state, the experi-mental result [36, 37, 72] where the DC conductivity de-creases sharply at T < 50K can be understood withoutany contradiction.

Finally, the effect of spin fluctuations on the NMRproperties were investigated using the transfer integralswith SOC obtained by first-principles calculations andRPA with U . For a case weak interaction, when T de-creased, only ferrimagnetic spin fluctuation appeared ow-ing to the characteristic wave function of α-(BETS)2I3

[71]. This behavior is similar to the ferrimagnetic spinfluctuation observed in α-(BEDT-TTF)2I3 [24]. Whilefor a case strong interaction, the antiferromagnetic spinfluctuation is enhanced. In this case, the components ofsites A-A and A-A′ of the spin susceptibility at q = 0diverged positively and negatively with the decrease inT . This divergence corresponds to the emergence of theSMD phase as suggested in a previous study [41], butis not consistent with the results of NMR experiments[39, 40].

In our calculations, the contribution of V ′ to the emer-gence of the TMI state was significant. Recent studieshave reported that the TMI state induced by Coulombinteraction appears even in real materials such as digi-tal transition metal oxide hetero structures [73–76]. α-(BETS)2I3 is also considered to be a system in which theTMI state due to V ′ and SOC appears by the calcula-tion with HF approximation. In the TMI state, the totalphase shift in the unit cell becomes zero; hence, this stateis different from a current state suggested in cuprates su-per conductors [69, 70] in which the summation of the

12

phases in the unit cell is finite and there is a net current.

A recent NMR experiment reported that time rever-sal symmetry breaking was not observed, and 1/T1T wasproportional to the power of T and varied continuouslynear 50 K [40]. In our calculation using RPA, ferrimag-netic spin fluctuation has no significant effect on the T -dependence of 1/T1T and this is consistent with exper-imental result. The ferrimagnetic spin polarization hasbeen observed by a site-resolved NMR experiment in α-(BEDT-TTF)2I3 [24] and α-(BETS)2I3 [71].

In future research, the effect of the TMI state on T1 canbe calculated using the Weyl model. Theoretical studieswith the Dirac Hamiltonian as well as NMR experimentshave shown that T1 in the TI state is affected by orbitalcurrent and that the power of T changes [77, 78]. Itwould be interesting to investigate the presence or ab-sence of similar effects in the TMI state of α-(BETS)2I3.In recent studies on the TI state, an exotic state lo-calized only at the intersection of the edges of the TIstate, has been reported, and such materials are calledhigher-order topological Mott insulators [73, 79–86]. Fu-ture research should investigate whether the TMI statesuggested in this study is a higher-order topological Mottinsulator, using models such as a cylindrical system that

considers the real space structure. Moreover, it is nec-essary to calculate physical quantities such as the See-beck and Nerunst coefficients [72]. Finally, there aremany unclear points on the low-T insulating state of α-(BETS)2I3; hence, it is possible that mean-field approx-imation is not a sufficient calculation method. Calcula-tions considering electron correlation effects using vertexcorrection [87], variational Monte Carlo method [88], andfunctional renormalization group theory [69, 70] will alsobe addressed in future.

ACKNOWLEDGMENTS

The authors would like to thank S. Onari, Y. Ya-makawa, and H. Kontani for the fruitful discussions. Wewould also like to thank H. Sawa, T. Tsumuraya, and S.Kitou for their valuable comments. We would also like toexpress our gratitude to N. Tajima and Y. Kawasugi forinformative discussions from the experimental aspects.The computation in this work was performed using thefacilities of the Supercomputer Center, Institute for SolidState Physics, University of Tokyo. This work was sup-ported by MEXT/JSPJ KAKENHI under grant numbers21H01041, 19J20677, 19H01846, and 15K05166.

[1] P. R. Wallace, Phys. Rev. 71, 622 (1947).[2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang,

M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, andA. A. Firsov, Nature 438, 197 (2005).

[3] P. A. Wolff, J. Phys. Chem. Solids 25, 1057 (1964).[4] H. Fukuyama and R. Kubo, J. Phys. Soc. Jpn. 28, 570

(1970).[5] K. Kajita, T. Ojiro, H. Fujii, Y. Nishio, H. Kobayashi,

A. Kobayashi, and R. Kato, J. Phys. Soc. Jpn. 61, 23(1992).

[6] N. Tajima, M. Tamura, Y. Nishio, K. Kajita, and Y. Iye,J. Phys. Soc. Jpn. 69, 543 (2000).

[7] A. Kobayashi, S. Katayama, K. Noguchi, and Y. Suzu-mura, J. Phys. Soc. Jpn. 73, 3135 (2004).

[8] S. Katayama, A. Kobayashi, and Y. Suzumura, J. Phys.Soc. Jpn. 75, 054705 (2006).

[9] A. Kobayashi, S. Katayama, Y. Suzumura, and H.Fukuyama, J. Phys. Soc. Jpn. 76, 034711 (2007).

[10] M. O. Goerbig, J.-N. Fuchs, G. Montambaux, and F.Piechon, Phys. Rev. B 78, 045415 (2008).

[11] K. Kajita, Y. Nishio, N. Tajima, Y. Suzumura, and A.Kobayashi, J. Phys. Soc. Jpn. 83, 072002 (2014).

[12] N. Tajima, S. Sugawara, M. Tamura, Y. Nishio, and K.Kajita, J. Phys. Soc. Jpn. 75, 051010 (2006).

[13] N. H. Shon and T. Ando, J. Phys. Soc. Jpn. 67, 2421(1998).

[14] H. Fukuyama, and R. Kubo, J. Phys. Soc. Jpn. 28, 570(1970).

[15] H. Seo, J. Phys. Soc. Jpn. 69, 805 (2000).[16] T. Takahashi, Synth. Met. 26, 133-134 (2003).[17] T. Kakiuchi, Y. Wakabayashi, H. Sawa, T. Takahashi,

and T. Nakamura, J. Phys. Soc. Jpn. 76, 113702 (2007).

[18] Y. Tanaka and M. Ogata, J. Phys. Soc. Jpn. 85, 104706(2016).

[19] K. Ishikawa, M. Hirata, D. Liu, K. Miyagawa, M.Tamura, and K. Kanoda, Phys. Rev. B 94, 085154(2016).

[20] R. Beyer, A. Dengl, T. Peterseim, S. Wackerow, T. Ivek,A. V. Pronin, D. Schweitzer, and M. Dressel, Phys. Rev.B 93, 195116 (2016).

[21] D. Liu, K. Ishikawa, R. Takehara, K. Miyagawa, M.Tamura, and K. Kanoda, Phys. Rev. Lett. 116, 226401(2016).

[22] D. Ohki, Y. Omori, and A. Kobayashi, Phys. Rev. B 100,075206 (2019).

[23] A. Kobayashi and Y. Suzumura, J. Phys. Soc. Jpn. 82,054715 (2013).

[24] M. Hirata, K. Ishikawa, K. Miyagawa, M. Tamura, C.Berthier, D. Basko, A. Kobayashi, G. Matsuno, and K.Kanoda, Nat. Commun. 7, 12666 (2016).

[25] G. Matsuno, and A. Kobayashi, J. Phys. Soc. Jpn. 86,014705 (2017).

[26] G. Matsuno and A. Kobayashi, J. Phys. Soc. Jpn. 87,054706 (2018).

[27] M. Hirata, K. Ishikawa, G. Matsuno, A. Kobayashi, K.Miyagawa, M. Tamura, C. Berthier, and K. Kanoda, Sci-ence 358, 1403 (2017).

[28] D. Ohki, M. Hirata, T. Tani, K. Kanoda, and A.Kobayashi, Phys. Rev. Research 2, 033479 (2020).

[29] S. Kitou, T. Tsumuraya, H. Sawahata, F. Ishii, K. Hiraki,T. Nakamura, N. Katayama, and H. Sawa, Phys. Rev. B103, 035135 (2021).

[30] T. Osada, J. Phys. Soc. Jpn. 87, 075002 (2018).[31] S. M. Winter, K. Riedl, and R. Valenti, Phys. Rev. B 95,

13

060404(R) (2017).[32] T. Tsumuraya, Y. Suzumura, Eur. Phys. J. B 94, 17

(2021).[33] R. Kondo, S. Kagoshima, N. Tajima, and R. Kato, J.

Phys. Soc. Jpn. 78, 114714 (2009).[34] P. Alemany, J. P. Pouget, and E. Canadell, Phys. Rev. B

85, 195118 (2012).[35] T. Morinari and Y. Suzumura, J. Phys. Soc. Jpn. 83,

094701 (2014).[36] M. Inokuchi, H. Tajima, A. Kobayashi, T. Ohta, H.

Kuroda, R. Kato, T. Naito, and H. Kobayashi, Bull.Chem. Soc. Jpn. 68, 547 (1995).

[37] Y. Kawasugi, H. Masuda, M. Uebe, H. M. Yamamoto,R. Kato, Y. Nishio, and N. Tajima, Phys. Rev. B 103,205140 (2021).

[38] Y. Kawasugi, H. Masuda, J. Pu, T. Takenobu, H. M.Yamamoto, R. Kato, and N. Tajima, Crystals 11, 791(2021).

[39] K. Hiraki, S. Harada, K. Arai, Y. Takano, T. Takahashi,N. Tajima, R. Kato, and T. Naito, J. Phys. Soc. Jpn. 80,014715 (2011).

[40] S. Fujiyama, H. Maebashi, N. Tajima, T. Tsumuraya, H-B. Cui, M. Ogata, and R. Kato, arXiv:2104.13547 [cond-mat.mes-hall] (2021).

[41] D. Ohki, K. Yoshimi, and A. Kobayashi, Phys. Rev. B102, 235116 (2020).

[42] D. V. Khveshchenko, J. Phys. Condens. Matter 21,075303 (2009).

[43] V. N. Kotov, B. Uchoa, V. M. Pereira, F. Guinea, andA. H. Castro Neto, Rev. Mod. Phys. 84, 1067 (2012).

[44] M. Hirata, A. Kobayashi, C. Berthier, and K. Kanoda,Rep. Prog. Phys. 84, 036502 (2021).

[45] F. D. M. Haldane, Phys. Rev. Lett. 61, 2015 (1988).[46] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801

(2005).[47] S. Raghu, X. L. Qi, C. Honerkamp, and S. C. Zhang,

Phys. Rev. Lett. 100, 156401 (2008).[48] Y. Omori, G. Matsuno, and A. Kobayashi, J. Phys. Conf.

Proc. 1, 012119 (2014).[49] C. Weeks and M. Franz, Phys. Rev. B 81, 085105 (2010).[50] A. Dauphin, M. Muller, and M. A. Martin-Delgado,

Phys. Rev. A 86, 053618 (2012).[51] P. Streda and L. Smrcka, Phys. Stattus Solidi B 70, 537

(1975).[52] I. Proskurin, M. Ogata, and Y. Suzumura, Phys. Rev. B

91, 195413 (2015).[53] A. Ruegg, S. Pilgram, and M. Sigrist, Phys. Rev. B 77,

245118 (2008).[54] Y. Omori, G. Matsuno, and A. Kobayashi, J. Phys. Soc.

Jpn. 86, 074708 (2017).[55] J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev.

Lett. 77, 3865 (1996).[56] P. Giannozzi, S. Baroni, N. Bonini et al., J. Phys. Con-

dens. Matter 21, 395502 (2009).[57] M. Schlipf and F. Gygi, Comput. Phys. Commun. 196,

36 (2015).[58] A. A. Mostofi, J. R. Yates, G. Pizzi, Y. S. Lee, I. Souza,

D. Vanderbilt, and N. Marzari, Comput. Phys. Commun.178, 9, 685-699 (2008).

[59] K. Momma and F. Izumi, J. Appl. Cryst. 44, 1272-1276(2011).

[60] K. Nakamura, Y. Yoshimoto, Y. Nomura, T. Tadano,M. Kawamura, T. Kosugi, K. Yoshimi, T. Misawa, and

Y. Motoyama, Comput. Phys. Commun. 261, 107781(2021).

[61] K. Yoshimi, M. Naka, and H. Seo, J. Phys. Soc. Jpn. 89,034003 (2020).

[62] K. Nakamura, Y. Yoshimoto, and M. Imada, Phys. Rev.B 86, 205117 (2012).

[63] T. Misawa, K. Yoshimi , and T. Misawa, Phys. Rev. Re-search 2, 032072(R) (2020).

[64] G. P. Mikitik and Yu. V. Sharlai, Phys. Rev. Lett. 82,2147 (1999).

[65] G. P. Mikitik and Yu. V. Sharlai, Phys. Rev. B 85, 033301(2012).

[66] A. A. Taskin and Y. Ando, Phys. Rev. B 84, 035301(2011).

[67] A. R. Wright and R. H. McKenzie, Phys. Rev. B 87,085411 (2013).

[68] M. O. Goerbig, Rev. Mod. Phys. 83, 1193 (2011).[69] R. Tazai, Y. Yamakawa, and H. Kontani, Phys. Rev. B

103, L161112 (2021).[70] H. Kontani, Y. Yamakawa, R. Tazai, and S. Onari, Phys.

Rev. Research 3, 013127 (2021).[71] T. Sekine, (2020), (private communication).[72] N. Tajima, (2019), (private communication).[73] S. Rachel, Rep. Prog. Phys. 81, 116501 (2018).[74] D. Xiao, W. Zhu, Y. Ran, N. Nagaosa, and S. Okamoto,

Nat. Commun. 2, 596 (2011).[75] A. Ruegg and G. A. Fiete, Phys. Rev. B 84, 201103(R)

(2011).[76] A. Ruegg, C. Mitra, A. A. Demkov, and G. A. Fiete,

Phys. Rev. B 85, 245131 (2012).[77] T. Hirosawa, H. Maebashi, and M. Ogata, J. Phys. Soc.

Jpn. 86, 063705 (2017).[78] W. A. MacFarlane, C. B. L. Tschense, T. Buck, K. H.

Chow, D. L. Cortie, A. N. Hariwal, R. F. Kiefl, D.Koumoulis, C. D. P. Levy, I. McKenzie, F. H. McGee,G. D. Morris, M. R. Pearson, Q. Song, D. Wang, Y. S.Hor, and R. J. Cava, Phys. Rev. B 90, 214422 (2014).

[79] Y. Hatsugai and I. Maruyama, Eur. Phys. Lett. 95, 20003(2011).

[80] K. Hashimoto, X. Wu, and T. Kimura, Phys. Rev. B 95,165443 (2017).

[81] W. A. Benalcazar, B. A. Bernevig, and T. L. Hughes,Science 357, 61 (2017).

[82] F. Schindler, A. M. Cook, M. G. Vergniory, Z.Wang, S. S. P. Parkin, B. A. Bernevig, and T.Neupert, Science Advances 4, 6, eaat0346 (2018),DOI:10.1126/sciadv.aat0346.

[83] K. Kudo, T. Yoshida, and Y. Hatsugai, Phys. Rev. Lett.123, 196402 (2019).

[84] H. Araki, T. Mizoguchi, and Y. Hatsugai, Phys. Rev. B99, 085406 (2019).

[85] F. Schindler, Z. Wang, M. G. Vergniory, A. M. Cook,A. Murani, S. Sengupta, A. Y. Kasumov, R. Deblock, S.Jeon, I. Drozdov, H. Bouchiat, S. Gu’eron, A. Yazdani,B. A. Bernevig, and T. Neupert, Nature Physics 14, 918(2018).

[86] M. Hohenadler and F. F. Assaad, J. Phys. Condens. Mat-ter 25, 143201 (2013).

[87] K. Yoshimi, H. Maebashi, and T. Kato, J. Phys. Soc.Jpn. 78, 104002 (2009).

[88] T. Misawa, S. Morita, K. Yoshimi et al., Comp. Phys.Commun. 236, 447-462 (2019).