Weyl Mott Insulator

# Weyl Mott Insulator

Takahiro Morimoto RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama, 351-0198, Japan    Naoto Nagaosa RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama, 351-0198, Japan Department of Applied Physics, The University of Tokyo, Tokyo, 113-8656, Japan
July 2, 2019
###### Abstract

Relativistic Weyl fermion (WF) often appears in the band structure of three dimensional magnetic materials and acts as a source or sink of the Berry curvature, i.e., the (anti-)monopole. It has been believed that the WFs are stable due to their topological indices except when two Weyl fermions of opposite chiralities annihilate pairwise. Here, we theoretically show for a model including the electron-electron interaction that the Mott gap opens for each WF without violating the topological stability, leading to a topological Mott insulator dubbed Weyl Mott insulator (WMI). This WMI is characterized by several novel features such as (i) energy gaps in the angle-resolved photo-emission spectroscopy (ARPES) and the optical conductivity, (ii) the nonvanishing Hall conductance, and (iii) the Fermi arc on the surface with the penetration depth diverging as approaching to the momentum at which the Weyl point is projected. Experimental detection of the WMI by distinguishing from conventional Mott insulators is discussed with possible relevance to pyrochlore iridates.

###### pacs:
72.10.-d,73.20.-r,73.43.Cd

Introduction — Weyl fermions (WFs) in solids attract recent intensive interests from the viewpoint of their novel quantum transport properties and chiral anomaly. The WF is described by the 2-component spinors originating from 4-component Dirac spinor when the mass is zero in the Dirac equation. The realization of WFs in condensed matters has been recently established FangScience (); Murakami (); Wan11 (). In magnetic materials, the time-reversal symmetry is broken and the energy dispersion of Bloch wavefunction has no Kramer’s degeneracy. In this case, the band crossings between the two bands are described by a 2 2 Hamiltonian as

 H(k)=ε0(k)+∑i=1,2,3σihi(k), (1)

with 2 2 Pauli matrices . Three conditions of the band crossing for can be satisfied in general by appropriately choosing the three components of the crystal momentum . Weyl points sometimes exist exactly at the Fermi energy when dictated by some symmetry and topology of the Bloch wavefunctions, for example, in Dirac semimetals Young (); Liu (); Xu-Na3Bi (); Neupane (); Borisenko (). More recently, experimental discovery of Weyl semimetals in an inversion broken material TaAs has been reported TaAs1 (); TaAs2 (); Ding ().

Weyl fermion plays an important role in the context of the Berry phase, which is defined by ( : the periodic part of the Bloch wave function with the band index and the momentum ) and acts as the vector potential in the momentum space. The Berry curvature is the emergent magnetic field, and can be enhanced near the band crossing points. When one expand Eq. (1) around the band crossing point (Weyl point, which we assume to be ), there appears the WF described with , by an appropriate choice of the coordinate ’s, where is the Fermi velocity. The sign specifies the chirality of the WF, and the Berry curvature of the lower eigenstates of the Hamiltonian in Eq. (1) is obtained as

 b−k=η2k|k|3, (2)

which diverges as and the total flux penetrating the surface enclosing the Weyl point is given by . This indicates that the WF acts as the magnetic monopole (anti-monopole) for (); the magnetic charge plays a role of topological index. Strong Berry curvature leads to the enhanced anomalous Hall effect FangScience () as well as the chiral magnetic effect which results in the negative magneto-resistance Burkov ().

Figure 1 shows the schematic figure of the three dimensional first Brillouin zone in which two WFs exist along the direction. One can define the Chern number

 Ch(kz)=∫dkxdky2πbz(k), (3)

for the plane of fixed . When we consider as a function of , there appears the jump by at , i.e., -coordinate of the Weyl points. Therefore, due to the periodicity of by , we need the pair of and  Nielsen1 (); Nielsen2 (). The existence of a single (an odd number of) WF is also excluded. Therefore, the annihilation of a single WF is prohibited, i.e., the only way to destroy the WFs is to annihilate a pair of WFs with opposite chiralities either by making the two WFs approach to each other in the momentum space or by introducing a scattering between two WFs with some density-wave-type order. The former scenario is actually proposed for the transition between the Weyl metal and insulator in pyrochlore compounds WWK12 (). The latter one is also discussed intensively Maciejko (); Sekine (). Meanwhile, effects of the electron correlation have been discussed for WFs by several methods including random phase approximation Abrikosov71 () and, more recently, cluster perturbation theory Witczak-Krempa14 (). However, the possibility of the Mott gap opening at each WF has never been explored thus far to the best of the present authors’ knowledge.

In this paper, we study the effect of the electron correlation on WFs by using a simple model which is exactly solvable. It is shown that the Mott gap due to open at each WF without the pair annihilation, while the topological properties are kept unchanged. Namely, the magnetic charge of the WF is unchanged with the role of poles in Green’s function being replaced by its zeros. The Hall conductance remains nonvanishing and the Fermi arc on the surface remains, while the Green’s function and the optical conductivity show the gap. Therefore, this Mott insulating state is identified as a topological Mott insulator, and we name it “Weyl Mott Insulator (WMI)”. The experimental detection of this new state is also discussed.

Model and Green’s function — The model we study is given by the Hamiltonian

 H=∑k[ψ†kh(k)⋅σψ+12U(nk−1)2], (4)

where is the two-component spinor, and . The most peculiar nature of this model arises from the electron-electron interaction which is local in , i.e., the Hamiltonian is decomposed into independent -sectors. In the real space, this corresponds to the non-local interaction in the limit of forward scattering. A similar idea has been explored to study the Mott transition Hatsugai () and the spin-charge separation Baskaran (). This locality of the interaction in enables the exact solution of this problem. One can introduce the unitary transformation satisfying with and a new spinor , and then obtain

 U(k)†HU(k)=∑k[hk(nk+−nk−)+12U(nk−1)2], (5)

with and . There are four eigenstates and eigenenergies: (i) with , (ii) with , (iii) with , and (iv) with .

Using these solutions, one can easily obtain the thermal Green’s function in the zero temperature limit as

 ^G−1(k,iω)=iω^1−heff(k)⋅σ (6)

where with . (For details, see Supplementary Information SI.) As can be seen from Eq. (6), the energy dispersions of the poles are , where the Mott gap of exists even at the Weyl point with as shown in Fig. 2(a), which can be measured in the angle-resolved photoemission spectroscopy (ARPES). It should be noticed that Eq. (6) is derived from the exact Green’s function and is not a result of some mean-field approximation. Thus, the WFs disappear due to the electron correlation without the pair annihilation.

Topological properties — The topological index for the interacting electronic systems can be defined in terms of Green’s function Gurarie (). For the (2+1)D case, it is given by

 Ch(kz) =εαβγ6∫∞−∞dω∫d2k(2π)2 ×tr[(^G−1∂kα^G)(^G−1∂kβ^G)(^G−1∂kγ^G)], (7)

where run over , and is the totally antisymmetric tensor. Plugging Eq. (6) into Eq. (7), one obtains

 (8)

where the -integral is over and for fixed . Since , does not change in spite of the gap opening at Weyl points. The Green’s function in Eq. (6) depends on the direction in which approaches to the Weyl point and still plays a role of a source (sink) of the Berry curvature. At exactly , has a zero at when one averages over the direction of . Namely, the role of a pole is replaced by a zero in the topological properties of Green’s function Gurarie (). Because of the bulk-edge correspondence, nonzero topological index indicates that the existence of the surface states on the side surface, i.e., the Fermi arc. (For details, see Supplementary Information SII.)

Therefore, the present insulating state is topological and we call it “Weyl Mott insulator (WMI)” distinct from the usual antiferromagnetic Mott insulator (AFI).

Optical conductivity — Now we study the conductivity, which is given by the two-particle correlation function. We consider a single WF with the Fermi velocity described by . It is crucial to distinguish between the nonzero momentum transfer and exactly . In the former case, the double occupancy of the electrons will be created, while not in the latter case, which brings about the singularity or discontinuity at . This reflects the long-range nature of our Coulomb interaction in Eq. (4). For , the particle-hole continuum starts from for the transition of an electron from to .

For the optical conductivity, the momentum of the incident light is finite although small, and hence we take the limit . In this limit, the optical conductivity at zero temperature for a single WF is obtained as (See Supplementary Information SIII for details)

 σ(\boldmathq→\boldmath0,ω)=e212hvFω(ω−U)2θ(ω−U), (9)

where and . The optical conductivity shows a Mott gap of and its asymptotic behavior for large is given by which coincides with the well known result for a free WF Hosur (). In Fig. 3, the optical conductivity is plotted for various temperatures (See Supplementary Information SIII for the derivation). As the temperature increases, peaks at and appear, as denoted by bars whose heights represent mutual ratios of the weights of the peaks. The peak at is a Drude peak for finite temperatures, while the peak at arises from an intraband contribution in which a WF at scattered to within the same band feels a Coulomb repulsion . The appearance of the peak at in for WMIs contrasts to the absence of such a peak for AFIs, because the peak at originates from the correlation effect. In addition, the appearance of the in-gap absorption indicates the fragile nature of the Mott gap compared with the single-particle band gap.

For the case of exactly , only the vertical transitions within the same -sector contribute to the conductivity as indicated by the red line in Fig. 2(b). Since no double occupancy is created in this case, no energy cost of occurs. Therefore, the conductivity is given by Eq. (9) with .

Discussion — Now the relevance of the present results to realistic systems is discussed. There are clear differences between the WMI and the AFI due to their topological nature: (i) The Hall conductance is finite in WMI while it is zero in AFI. (ii) Correspondingly, the Fermi arc on the surface remains in WMI while not in AFI. Also importantly, the phase transition between the WMI and AFI is possible once the former exists as explained below. Figure 4(a) shows the phase diagram of the present model. The horizontal axis is the separation between the two WFs which is controlled by, e.g., the strength of the antiferromagnetic long range order parameter . When , the phase transition occurs from the Weyl semimetal to the AFI by the pair annihilation of WFs at . Once the interaction is switched on, we always opens the gap and the system becomes the WMI as long as is finite. Along the phase transition line at , , the pair annihilation of the two zeros of the Green’s function occurs, which is distinct from that at where the two poles collide and pair-annihilate. Here one must consider the peculiarity of the present model. The effect of the long range Coulomb interactions is marginally irrelevant as in the case of quantum electrodynamics (QED) Isobe (). As for the short-range Coulomb interaction , it is irrelevant. This means that there must be a finite range of Coulomb interaction within which the WFs remains gapless and stable. On the other hand, the strong limit in the lattice model corresponds to the localized electron and hence a trivial AFI. Therefore, the conjectured phase diagram of a more realistic model is given in Fig. 4(b), where the successive transitions from the Weyl semimetal to the WMI, and from the WMI to the AFI occurs as the strength of the interaction increases. (The separation of two WFs is reduced also as the interaction increases and hence the trajectory should goes as and simultaneously increase.)

Now we discuss the Green’s function and the two-particle correlation function for realistic electron-electron interactions given by

 HC=∑k,k′,qV(q)c†k+q,σc†k′−q,σ′ck′,σ′ck,σ. (10)

The self-energy of the Green’s function in the second order in is given in Supplementary Information SIV. It is concluded that the gap of the spectral function is stable and remains nonzero. The two-particle correlation functions such as , on the other hand, is gapless at for the Hamiltonian in Eq. (4). For a finite size system of sites, the number of poles forming this gapless excitation in the two particle correlation function [the red line in Fig. 2(b)] is of (which is the number of where the excitation can be created). In general, the number of poles for a collective excitation is of , while that of a continuous excitation arising as a pair of single-particle excitations is of . This infers that the gapless excitations at corresponds to a collective excitation. For realistic interactions, the discontinuity at should be removed. In this case, we conjecture that the vertical transition [red line in the inset of Fig. 4(a)] turns into a collective mode with a linear dispersion as shown in the blue line in the inset in the WMI phase of Fig. 4(b).

These considerations offer a different scenario to interpret the phase diagram of pyrochlore iridates IrO Wan11 (); WWK12 (). As the radius of the rare-earth ion is reduced, the correlation strength increases. A recent optical measurement in NdIrO has revealed the opening of the Mott gap of the order of 0.05eV Ueda1 (). A transport experiment also discovered the metallic domain wall states even in the Mott insulating state, i.e., the bulk is insulating while the domain wall is metallic Ueda2 (). As the correlation is further reduced, these surface metallic states also disappear. One scenario is proposed by Yamaji et al. Yamaji () based on the mean field theory. As an alternative scenario, one can consider the two types of Mott insulators, i.e., the WMI and the AFI, and the disappearance of the metallic domain wall states signals the phase transition between the two phases. Namely, since two domains of the antiferromagnetic order correspond to opposite signs of and hence the two-dimensional chiral surface modes are expected to appear at the domain boundary in the WMI phase. However, we note that this requires a symmetry lowering to violate the cancellation of the Chern vectors pointing toward four momentum directions equivalent to (1,1,1) which makes zero in the cubic symmetric case Yang (). The smoking-gun experiment for the WMI should be the ARPES to detect the Fermi arc even in the Mott insulating phase as mentioned above.

Acknowledgment — We thank Y. Tokura, B.-J. Yang, L. Balents, L. Fu, and W. Witczak-Krempa for fruitful discussions. This work was supported by Grant-in-Aid for Scientific Research (No. 24224009, and No. 26103006) from the Ministry of Education, Culture, Sports, Science and Technology (MEXT) of Japan and from Japan Society for the Promotion of Science.

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Supplementary Information for “Weyl Mott insulator”

## Appendix SI Green’s function

We derive the Green’s function for the Hamiltonian given by

 H=ψ†\boldmathk\boldmathh(% \boldmathk)⋅\boldmathσψ\boldmathk+12U(n\boldmathk−1)(n\boldmathk−1), (S1)

where

 ψ\boldmathk =⎛⎝c\boldmathk↑c\boldmathk↓⎞⎠, (S2)

are Pauli matrices acting on spin degrees of freedom, is the density operator , and is the magnitude of the repulsive interaction. The repulsive interaction is infinite-ranged in the real space, which can be represented in a local way in the momentum space. Thanks to the locality in the momentum space, the Green’s function can be exactly computed for this Hamiltonian as follows.

First we perform a unitary transformation

 ⎛⎝c\boldmathk↑c\boldmathk↓⎞⎠ =U(\boldmathk)⎛⎝b\boldmathk+b\boldmathk−⎞⎠ (S3)

that diagonalizes the single-particle part of the Hamiltonian as

 U†(\boldmathk)[\boldmathh(% \boldmathk)⋅\boldmathσ]U(\boldmathk) =h(\boldmathk)σz. (S4)

Then the Green’s function is transformed as

 Gαβ(\boldmathk,τ) ≡−⟨Tτc\boldmathkα(τ)c†\boldmathkβ⟩=−Uαa(\boldmathk)Uβa′(\boldmathk)⟨Tτb\boldmathka(τ)b†\boldmathka′⟩, (S5)

where denotes the time ordering and .

In this basis, the Hamiltonian is diagonalized as

 H|0⟩ =U2|0⟩, (S6) Hb†\boldmathk−|0⟩ =−h(\boldmathk)b†\boldmathk−|0⟩, (S7) Hb†\boldmathk+|0⟩ =h(\boldmathk)b†\boldmathk+|0⟩, (S8) Hb†\boldmathk−b†\boldmathk+|0⟩ =U2b†\boldmathk−b†\boldmathk+|0⟩, (S9)

where is the vacuum state and . Thus the expectation value is written as

 ⟨b\boldmathka(τ)b†\boldmathka′⟩ =1Z[⟨0|b\boldmathka(τ)b†% \boldmathka′|0⟩e−βU2 +⟨0|b\boldmathk−b\boldmathka(τ)b†ka′b†\boldmathk−|0⟩eβh(\boldmathk) +⟨0|b\boldmathk+b\boldmathka(τ)b†\boldmathka′b†\boldmathk+|0⟩e−βh(\boldmathk) +⟨0|b\boldmathk+b\boldmathk−bka(τ)b†\boldmathka′b†\boldmathk−b†\boldmathk+|0⟩e−βU2], (S10) Z =eβh(\boldmathk)+e−βh(\boldmathk)+2e−βU2. (S11)

In the right hand side of the equation for , the forth term vanishes and other terms are nonzero when . Thus we obtain

 ⟨b\boldmathk+(τ)b†\boldmathk+⟩ =1Z(⟨0|b\boldmathk+(τ)b†% \boldmathk+|0⟩e−βU2+⟨0|b% \boldmathk−b\boldmathk+(τ)b†\boldmathk+b†\boldmathk−|0⟩eβh(\boldmathk)) =1Z(eτ(−h+U2)−βU2+eτ(−h−U2)+βh), (S12) ⟨b\boldmathk−(τ)b†\boldmathk−⟩ =1Z(⟨0|b\boldmathk−(τ)b†% \boldmathk−|0⟩e−βU2+⟨0|b% \boldmathk+b\boldmathk−(τ)b†\boldmathk−b†\boldmathk+|0⟩e−βh(\boldmathk)) =1Z(eτ(h+U2)−βU2+eτ(h−U2)−βh). (S13)

Therefore, the imaginary-time Green’s function is given by

 G++(\boldmathk,iωn) =−∫β0dτeiωnτ⟨b% \boldmathk+(τ)b†\boldmathk+⟩ =1Z⎛⎝e−βh+e−βU2iωn−h+U2+eβh+e−βU2iωn−h−U2⎞⎠, (S14) G−−(\boldmathk,iωn) =−∫β0dτeiωnτ⟨b% \boldmathk−(τ)b†\boldmathk−⟩ =1Z⎛⎝e−βh+e−βU2iωn+h−U2+eβh+e−βU2iωn+h+U2⎞⎠. (S15)

### si.1 Green’s function for T=0

In the zero temperature (), the above Green’ function reduces to

 G++(\boldmathk,iωn) =1iωn−h−U2, (S16) G−−(\boldmathk,iωn) =1iωn+h+U2. (S17)

In the original basis, the Green’s function is given by

 G(\boldmathk,iωn) =1iωn−(h+U2)\boldmathn⋅\boldmathσ, n =\boldmathhh. (S18)

## Appendix SII Fermi arc

In this section, we study the Fermi arc in the WMIs. Nonvanishing topological indices for the WMIs indicate that the Fermi arc remains in the WMIs, which we verify in the following. Since our model [Eq. (S1)] is diagonalized at each -point and hence we can consider an effective two-dimensional model for each -sector when the system is periodic in the -direction. Because of the bulk-boundary correspondence, we expect that “edge channels” for each form a Fermi arc. More explicitly, one can obtain the surface bound state from the effective Hamiltonian

 Heff =[h(k)+U2]n(k)⋅σ, h(k) =vF(kx,ky,kz), (S19)

by replacing the momenta with the derivatives . Away from the plane , the surface state is almost unchanged from the noninteracting case. The nontrivial issue is how the surface state behaves as approaches . Specifically, the problem is whether the penetration depth of the surface states diverges or not with . Intuitively, the finite gap indicates that the length scale remains finite, i.e., . However, it turns out not when one studies the effective Hamiltonian in Eq. (S19) and the asymptotic behavior of the surface bound state as (here we assume ) by tentatively taking the limit of . In this limit, , which indicates that the penetration depth diverges with . In any case, the length scale is determined by even when we take into account of the higher orders in . Therefore, the surface bound states penetrate into the bulk as approaches to .

## Appendix SIII Optical conductivity

We study the optical conductivity for a single WF described . In the following, we set the Fermi velocity , which can be always restored by the dimension analysis.

### siii.1 Matrix elements

Here we calculate matrix elements that we will need in evaluation of conductivities, i.e., . We first parameterize the direction of the momentum as

 \boldmathn=(sinθcosϕ,sinθcosϕ,cosθ). (S20)

Then the wave functions that diagonalize the Hamiltonian are written as

 |+⟩ =⎛⎝cosθ2eiϕsinθ2⎞⎠, |−⟩ =⎛⎝−sinθ2eiϕcosθ2⎞⎠. (S21)

The matrix elements are given by

 ⟨+|σx|+⟩ =sinθcosϕ, (S22) ⟨−|σx|−⟩ =−sinθcosϕ, (S23) ⟨+|σx|−⟩ =cosθcosϕ+isinϕ, (S24) ⟨+|σy|+⟩ =sinθsinϕ, (S25) ⟨+|σy|−⟩ =−icosϕ+cosθsinϕ. (S26)

In the evaluation of the optical conductivity, we need

 ∫sinθdθdϕ⟨±|σx|±⟩⟨±|σx|±⟩ =4π3, (S27) ∫sinθdθdϕ⟨+|σx|−⟩⟨−|σx|+⟩ =8π3. (S28)

In the evaluation of the Hall conductivity as a function of , we need

 ∫dϕ⟨+|σx|+⟩⟨+|σy|+⟩ =0, (S29) ∫dϕ⟨+|σx|−⟩⟨−|σy|+⟩ =2πicosθ=2πikzk. (S30)

### siii.2 Zero temperature

We first focus on the conductivity for the zero temperature. The Green’s function is given by

 G(iωm) =1(iωm)2−(k+U2)2[iωm+(k+U2)\boldmathn⋅\boldmathσ], (S31)

with . The optical conductivity is given by

 σ(ω) =Re[Q(ω+iϵ)−iω], (S32) Q(iΩ) =lim\boldmathq→\boldmath0∫d3\boldmathk(2π)3∑iωmtr[G(% \boldmathk,iωm)σxG(\boldmathk+\boldmathq,iωm+iΩ)σx]. (S33)

The integrand of reads

 ∑iωmtr[G(\boldmathk,iωm)σxG(\boldmathk,iωm+iΩ)σx] =∑iωm1(iωm+iΩ)2−(k+U2)21(iωm)2−(k+U2)2 tr[(iωm+iΩ+(k+U2)\boldmathn⋅\boldmathσ)σx(iωm+(k+U2)\boldmathn⋅% \boldmathσ)σx] =∑iωm2(iωm+iΩ)2−(k+U2)21(iωm)2−(k+U2)2[(iωm+iΩ)iωm+(1+U2k)2(k2x−k2y−k2z)] =∑iωm2(iωm+iΩ)2−(k+U2)21(iωm)2−(k+U2)2[(iωm+iΩ)iωm−13(k+U2)2]. (S34)

By using the formula

 ∑iωm[(iωm+iΩ)iωm−abc][(iωm+iΩ)2−a2][(iωm)2−b2] =12(a+b)(1−c)[iΩ−(a+b)][iΩ+(a+b)] (S35)

for , we perform the summation over for the above equation and obtain

 ∑iωmtr[G(\boldmathk,iωm)σxG(\boldmathk,iωm+iΩ)σx] =83(k+U2)[iΩ−(2k+U)][iΩ+(2k+U)]. (S36)

After the analytic continuation , only the pole at contributes to the imaginary part of the -integral. Thus, we obtain

 Im[Q(ω)] =43π2∫∞0k2dk(k+U2)ω+(2k+U)Im[1ω+iϵ−(2k+U)] =−124π(ω−U)2θ(ω−U), (S37)

where we used the formula . Hence, the optical conductivity for the zero temperature is given by

 σ(ω)=−Im[Q(ω)]ω=124πω(ω−U)2θ(ω−U). (S38)

By restoring the unit of and the Fermi velocity , we end up with

 σ(ω)=e212hvFω(ω−U)2θ(ω−U). (S39)

#### siii.2.1 Poles of σ(\boldmathq,ω)

Let us study the locus of the poles of the two-particle correlation function that contribute to the conductivity for nonzero . From Eq. (S35) and setting and , the poles of can be read off as . By using the formula and restoring the Fermi velocity , the lower bound of the poles is given by

 ω=U+vF|\boldmathq|. (S40)

### siii.3 Finite temperature

In this section, we calculate the optical conductivity in the finite temperature. In doing so, we consider contributions from interband and intraband transitions separately as

 σ(ω)=σinter(ω)+σ% intra(ω). (S41)

#### siii.3.1 Interband transition

The interband contribution to is given by

 Qinter(iΩ)=1(2π)3∫k2dkAinter(iΩ), (S42)

where

 Ainter(iΩ) ≡∫sinθdθdϕ∑iωm[G++(%\boldmath$k$,iωm)⟨+|σx|−⟩G−−(\boldmath% k,iωm+iΩ)⟨−|σx|+⟩ +G++(\boldmathk,iωm+iΩ)⟨+|σx|−⟩G−−(\boldmathk,iωm)⟨−|σx|+⟩] =8π3Z−2∑s=±1[⎛⎝eβk+e−βU22k+U+isΩ+e−βk+e−βU22k+isΩ⎞⎠e−βU2+⎛⎝eβk+e−βU22k+isΩ+e−βk+e−βU22k−U+isΩ⎞⎠e−βk +⎛⎝eβk+e−βU2−(2k+U)+isΩ+e−βk+e−βU2−2k+isΩ⎞⎠eβk+⎛⎝eβk+e−βU2−2k+isΩ+e−βk+e−βU2−(2k−U)+isΩ⎞⎠e−βU2]. (S43)

This is reduced to

 Qinter(ω) =13π2Z−2∑s=±1∫k2dk⎛⎝eβk+e−βU22k+U+isΩ+e−βk+e−βU22k+isΩ⎞⎠(−eβk+e−βU2) +⎛⎝eβk+e−βU22k+isΩ+e−βk+e−βU22k−U+isΩ⎞⎠(e−βk−e−βU2). (S44)

After the analytic continuation , poles that contribute to the imaginary part of the -integral are

 k=ω−U2,ω2,ω+U2,U−ω2. (S45)

Thus the interband contribution to the optical conductivity at the finite temperature is given by

 σinter(ω) =Im[Qinter(ω)−iω] =−16πω[(eβω−U2+e−βU2)(−eβω−U2+e−βU2)(ω−U2)2Z(ω−U2)−2θ(ω−U) +2(e−βω+U2−eβω−U2)(ω2)2Z(ω2)−2 +(e−βω+U2+e−βU2)(e−βω+U2−e−βU2)(ω+U2)2Z(ω+U2)−2<