Optical Conductivity on Charge Order Transition in Organic Dirac Electron System α-( BEDT-TTF ) 2 I 3

The optical conductivity in the charge order phase is calculated in the extended Hubbard 1 model describing an organic Dirac electron system α-(BEDT-TTF)2I3 using the mean field theory 2 and the Nakano-Kubo formula. A peak structure due to interband excitation being characteristic in 3 two-dimensional Dirac electron system is found above the charge order gap. It is shown that the peak 4 structure originates from the Van Hove singularities of the conduction and valence bands, where 5 those singularities are located at a saddle point between two Dirac cones in momentum space. The 6 frequency of the peak structure exhibits drastic change in the vicinity of the charge order transition. 7

It has been observed that the optical gap determined by the optical conductivity decreases monotonously as P increases and reaches almost zero at about P c = 12kbar [20,21], while the resistivity gap reaches almost zero at about 7kbar [22].In order to explain such metallic behavior in the presence of the CO gap, metallic channels owing to edges and domain walls in the CO have been studied using the extended Hubbard model [23][24][25][26].It has been shown that the massive DE phase with the gapless edge states emerges in the intermediate region between the massless DE phases and the trivial CO.
Although the CO gap induced by the inversion symmetry breaking exists in both the massive DE phase and the trivial CO, a pair of Dirac cones with a finite gap at incommensurate momentum, ±k D , merges at a time reversal invariant momentum (TRIM) at the transition between these two phases [11,14,27,28].Such a drastic change in the band structure is expected to give rise to a characteristic in the optical conductivity.
In the present paper, the optical conductivity is calculated in the band structure determined by the mean-field theory in the extended Hubbard model for the two dimensional electron system in the organic conductor α-(BEDT-TTF) 2 I 3 , where the CO transition is controlled by the nearest-neighbor Coulomb repulsion, V a .It is shown that a peak structure emerges above the CO gap in the optical conductivity.The frequency of the peak structure, ω peak , owing to interband excitation between two Van Hove singularities in the conduction and valence bands, rapidly moves as a nonmonotonic function of V a , while the frequency of the CO gap, ω CO , increases monotonically as V a increases.
Those Van Hove singularities originate from the saddle points between two Dirac cones.The optical conductivity exhibits a characteristic strong peak when two Dirac cones merge in the presence of a large CO gap.This paper is described as follows.An extended Hubbard model for two dimensional electron system in α-(BEDT-TTF) 2 I 3 , the mean field theory, and the optical conductivity are described in section 2. Numerical results are shown in section 3. Section 4 and 5 are devoted to discussion and summary.

Formulation
The extended Hubbard model [1,2] has been used in theoretical studies for α-(BEDT-TTF) 2 I 3 , in order to take the on-site Coulomb repulsion, U, and the nearest-neighbor Coulomb repulsions, V αβ , into account.
where a iασ and t (iα,jβ) represent the annihilation operators and the transfer energies with unit cells i, j, spins σ, and sublattices α, β = A, A , B and C of α-(BEDT-TTF) 2 I 3 .
Hereafter, the energies are given in eV.The tight binding model for α-(BEDT-TTF) 2 I 3 [29][30][31][32][33] is shown in Fig. 1(a).The sublattice A and A' are equivalent due to the inversion symmetry in the massless DE phase.The transfer energies given by the first-principle calculation [32]: t b1 = 0.1241, t b2 = 0.1296, t b3 = 0.0513, t b4 = 0.0152, t a1 = −0.0267,t a2 = −0.0511,t a3 = 0.0323, t a1 = 0.0119, t a3 = 0.0046, and t a4 = 0.0060.The nearest neighbor interaction V a in the stacking direction is used for controlling the CO transition, because this is the most sensitive as a function of P [34][35][36], while we treat U = 0.4 and V b = 0.05 as constants.The temperature T = 0.001 is fixed in the present paper.The lattice constants, k B and h are taken as unity.The system size in numerical calculations is The mean-field Hamiltonian H MF [10] is where φ ασ is the Hartree potential, n ασ = a † iασ a iασ is the electron number, and δ is a vector between unit cells.The energy eigenvalues ξ γσ (k) and the wave functions Φ αγσ (k) are given by with the band index γ = 1, 2, 3, 4. The conduction and valence bands correnspond to ξ 1σ (k) and ξ 2σ (k), respectively, since the Fermi energy is located between these two bands.The electron number n ασ is given by where the Fermi distribution function is  The Green function G αβσ (ω, k) and the density of state ρ(ω) are given by where N L is the number of lattice points.
The optical conductivity is calculated by the Nakano-Kubo formula based on linear response theory.It is represented by where the velocity matrix v γγ σ is calculated by The optical conductivities at several V a are shown in Fig. 2(a).These values are divided by the universal conductivity σ 0 = πe 2 /2h [37].In the massless DE phase for V a = 0.18, the optical conductivity almost reaches a universal constant for T < ω < Λ, where Λ ∼ = 0.01 is a energy scale of the linear dispersion as shown in previous studies for the massless DE [37][38][39][40].When V > V c1 a = 0.198, a frequency of the CO gap, ω CO , increases as V a increases as shown in Fig. 2(b), where ω CO is defined as a flexion point of the shoulder structure in the optical conductivity.It is found that a peak structure appears above ω CO and its frequency, ω peak , rapidly decreases as V a increases for V a < V c2 a = 0.212, which is defined by the merging of the Dirac cones in the conduction band [25,26].ω peak rebounds after falling to ω CO , and the optical conductivity shows a strong peak.Those two frequencies exhibits asymptotic behavior for V a > V c2 a .Figure 3 shows the density of states ρ(ω), where the Fermi energy is defined as zero.When a , there is a valley due to the Dirac cones.A CO gap, ω CO , opens at the Fermi energy for In order to analyze the behavior of ω peak and ω CO , the band structure is intensively examined in

Discussion
The drastic change of the optical conductivity near CO transition is characterized not only the CO gap, but also the peak structure as shown in Fig. 2 Peer-reviewed version available at Crystals 2018, 8, 137; doi:10.3390/cryst8030137 in Fig. 3 and band structure shown in Fig. 4, it is elucidated that the origin of the peak structure is the VHSs at the M-points between two tilted Dirac cones.The nonmonotonic V a -dependence of ω peak is due to both the merging of the Dirac cones with the CO gap and the tilting of the Dirac cones.Those characteristic may be measured in the low frequency region of the optical conductivity [20], which can provide many important information on existence, merging, and tilting of the two-dimensional massive Dirac electrons in the CO phase.

Conclusions
The optical conductivity in the vicinity of the CO transition has been investigated using the

Figure 1 .
Figure 1.Schematic figure of a unit cell with transfer energies, on-site Coulomb repulsion U and the nearest-neighbor Coulomb repulsions, V a and V b (a).The conduction band (purple) and valence band (green) in the massive DE phase for V a = 0.18 (b).

)Figure 1
Figure 1(b) shows the conduction and valence bands in the massive DE phase.There is a pair of massless Dirac cones at incommensurate momenta, ±k D .The Fermi energy is located at the degenerate points of the two bands.There are several saddle points at the TRIMs near the Fermi energy, e. g., the M-point in the conduction band (Mc-VHS), the M-point in the valence band (Mv-VHS), and the Y-point in the valence band (Yv-VHS).

Fig. 4 .
Fig. 4.Both Mc-VHS and Mv-VHS exist at the M-point (the saddle point) between two Dirac cones.The Dirac cones in the conduction band (the purple band) merge at the M-point at V c2 a in the presence of large CO gap as shown in Figs.4(b) and 4(e), leading to the absorption of the Mc-VHS into the upper edge of the CO gap.On the other hand, the massive Dirac cones in the valence band (the green band) do not merge against relatively larger V a as shown in Figs.4(c) and 4(f), since the Dirac conesare tilted[11,12].Thus the Mv-VHS and the lower edge of the CO gap shows asymptotic behavior.

Figure 4 .
Figure 4.The contour plots of the conduction band for V a = 0.205 (a), V a = V c2 a = 0.212 (b), and V a = 0.220 (c).The band structures as a function of k x for V a = 0.205 (d), V a = V c2 a = 0.212 (e), and V a = 0.220 (f), where the CO gaps open between the conduction bands (the purple bands) and the valence bands (the green bands).Tilting of the Dirac cones causes electron-hole asymmetric behavior as the CO gap increases.

Nakano-
Kubo formula and the mean-field theory in the extended Hubbard model describing the Dirac electrons in α-(BEDT-TTF) 2 I 3 .It has been found that a peak structure above the CO gap emerges due to the two dimensional Dirac cones.It has been also shown that the drastic change of the peak structure in the vicinity of the CO transition indicates the merging of the massive Dirac electrons.