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The dynamical property of electrons with the tilted Dirac cone was examined using the tilted Weyl equation. The polarization function exhibits cusps and nonmonotonic structures by varying both the frequency and the momentum. A pair of tilted Dirac cones exhibits a new plasmon for the intermediate magnitude of momentum owing to the combined effects of two tilted cones. Dirac electrons with the zero-gap state (ZGS) in organic conductor α-(BEDT-TTF)_{2}I_{3} are examined by calculating the Berry curvature, which displays the peak structure for a pair of Dirac particles between the conduction band and the valence band. The ZGS is theoretically predicted for
3B1;-(BEDT-TTF)_{2}NH_{4}Hg_{4} under uniaxial pressure. Examining the band structure of the stripe charge ordered state of α-(BEDT-TTF)_{2}I_{3} under pressure, we have found a topological transition from a conventional insulator to a new phase of a pair of Dirac electrons with a finite mass. Further, investigating the zero-energy (

_{2}I

_{3}

_{2}NH

_{4}Hg(SCN)

_{4}

The organic conductor α-(BEDT-TTF)_{2}I_{3} (BEDT-TTF=bis(ethylene-dithio)tetrathiafulvalene) [_{0}/v_{c} = 0.8 [

The first discovery of massless Dirac fermion in condensed matter was in graphite [_{2}I_{3} has three unique features: (1) The layered structure with a highly two-dimensional electronic system, which enables us to use powerful experimental methods for bulk material such as NMR [

This paper reviews recent theoretical studies related to the tiled Dirac cone in organic conductors. The outline is as follows. In section 2, dynamical properties such as electron-hole excitation and collective excitation are examined to verify the role of tilting. Although the electronic state has been studied extensively, the dynamical properties associated with the polarization function are not yet clear compared with those in the isotropic case [_{2}I_{3} are examined by calculating the Berry curvature. In addition to the peak structure for a pair of Dirac particles between the conduction band and the valence band, the other neighboring bands show another pair of peaks of Dirac electrons with a tendency toward merging [_{2} NH_{4}Hg(SCN)_{4} under uniaxial pressure [_{2}I_{3} under pressure using an extended Hubbard model [_{2}I_{3} under strong magnetic field [

We examine the following 2x2 effective Hamiltonian [

where the matrix is given by

Here we define ξskR(L) = +(–)v0kx + svc|k|, (s = +,-) and α = v0/vc. From the two contact points corresponding to two valleys of cones, we focus on one, which is given by the state located close to k0. The polarization function per valley is calculated as

where f(ξ) = 1/(1 + exp[(ξ – μ)/_{s}(

The polarization function is calculated on the plane of q and ω for respective regions as shown in

Several regions on the q-ω plane for calculating the polarization function (Equation (3)). From [

The results consist of six regions. These regions 1A, 2A, 3A, 1B, 2B, and 3B are classified into two regions, A and B, corresponding to the process of intraband and interband excitations, respectively. The regions A and B are separated by a solid line expressed as ω_{res} = (1 + cosθ_{q})v_{c}q, which is called the resonance frequency. The resonance frequency is obtained owing to the nesting of the excitations with the linear dispersion, and the polarization function diverges with the chirality factor _{s}(_{+}. The boundary between 2A and 1A (2B and 1B) is given by ω_{A}. The boundary between 2B and 3B is given by ω_{B}. In the case of the isotropic Dirac cone (_{c}q/μ = 2 which separates 1A and 2A (1B and 2B) for the intraband (the interband). In the regions 3A and 1B, the imaginary part vanishes. For the tilted Dirac cone, the boundary between 1A and 2A exhibits a noticeable behavior characterized by the appearance of cusps for the imaginary part as shown in

Normalized imaginary part, Im Π(q,θq,ω) vc2/μ, as a function of ω/μ, for θq = π/2 and α = 0.8. From [

_{c}^{2}/μ, on the plane of v_{c}q/μ and ω/μ for θ_{q}_{res}. In the case of v_{c} q/μ >> 1, where the characteristic energy becomes much larger than the interband energy, Im Π(θ_{q}_{res}) becomes much smaller than that of the interband excitation (ω > ω_{res}) in contrast to the case,v_{c} q/μ = 2. The broad peak in the intraband excitation does not change much for θ_{q}_{q}_{res}.

Normalized imaginary part, Im Π(θq,ω) vc2/μ, on the plane of vcq/μ and ω/μ for θq = π/2. From [

In _{q}

Normalized optical conductivity of Reσ(θ_{q}

The plasma mode is calculated from 1 + v_{q} ReΠ(q,θ_{q}_{pl(1,2)}) = 0 with v_{q} = 2πe^{2}/ε_{0}q. Since the plasma frequency ω_{pl} is located just above the resonance frequency, the solution is expected close to ω_{pl1} =ω^{0}_{pl1} and ω_{pl2}=ω^{0}_{pl2}, respectively. However, the imaginary part is complicated due to the combined effects of Π ^{R}(q,θ_{q}^{L}(q,θ_{q}_{c}/μ and ω/μ with e^{2}/(ε_{0}v_{c}) = 1. The plasma frequency corresponds to ω, which gives a peak of ImΠ_{RPA}(q,θ_{q},ω).

In _{RPA}(q,θ_{q},ω) for θ_{q} = 0 is shown as a function of ω/μwith fixed v_{c}q/μ = 0.5, 0.75, 1, 1.5, 2, 3, 4, 5 and 6, respectively. The dash-dotted line (ω_{pl1}(q,θ_{c}q/μ. The dotted line denotes the location of ω for the novel plasmon, _{pl2}(q,θ_{c}q/μ. Such a plasmon, ω_{pl2}(q,θ_{res }are obtained as the hybridization of the electron of the right cone and that of the left cone. The appreciable peak of Im Π_{RPA}(q,θ_{q},ω) moves from ω_{pl1}(q,θ_{pl2}(q,θ_{pl1}(q,θ_{pl2}(q,θ_{RPA}(q,θ_{q},ω) is shown for θ_{q} = π/4. It is found that the weight for ω_{pl2}(q,θ_{pl2}(q,θ_{q} = π/4, ω_{pl2}(q,θ_{q} = π/2. Furthermore, we find that ω_{pl1}(q,θ^{1/2} for small q, and that ω_{pl2}(q,θ_{res} is proportional to q for intermediate v_{c}q/μ.

Im ΠRPA(q,θq,ω) as a function of ω/μwith fixed vcq/μ = 0.5, 0.75, 1, 1.5, 2, 3, 4, 5 and 6, for θq = 0 (a), and θq =π /4 (b). From [

Here we examine the filtering of the plasma frequency by tuning the angle θ_{q} of the external field with frequency ω and q. _{RPA}(q,θ_{q},ω) when the external frequency is chosen as ω = ω_{pl2}(q,0) (solid line), ω_{pl2}(q,π/8) (dotted line), and ω_{pl2}(q,π/4) (dashed line) for the fixed v_{c}q/μ = 5. For ω = ω_{pl2}(q,0), a pronounced peak with the peak height (about 15) and the width (0.036 π) shows that the plasmon excitation occurs in the narrow region close to θ_{q} = 0. For the choice of ω = ω_{pl2}(q,π/8), the peak appears at θ_{q} = +(–)π/8 where the intensity is reduced less than 1/10 compared with ω = ω_{pl2}(q,0). The peak is further reduced for ω = ω_{pl2}(q,π/4). No peak is expected when the frequency is outside of the regime of the plasma frequency. In terms of the dielectric function ε(q,θ_{q},ω), the location of the peak is determined by Reε(q,θ_{q},ω) = 0 while the height and the width are determined by Im ε(q,θ_{q},ω).Thus the peak (or the double peak) structure gives the filtering of plasmon, which depends on θ_{q}.

θ_{q} dependence of Im Π_{RPA}(q,θ_{q},ω) for v_{c}q/μ = 5 with the fixed ω = ω_{pl2}(q,0) (solid line), ω_{pl2}(q,π/8) (dotted line), and ω_{pl2}(q,π/4) (dashed line). From [

The contact point in a normal state, _{2}I_{3} given by

where α and β(=1, 2, 3,4) are indices for the site of each molecule A(1), A'(2), B(3) and C(4) in the unit cell, and i, j are those for the cell forming a square lattice with N sites. The quantity _{b1}, …, t_{b4} for the direction of the b(-x)-axis and t_{a1}, …, t_{a3} for the direction of the a(y) -axis, respectively.

Crystal structure on a two-dimensional plane with four molecules A(1), A’(2), B(3), and C(4), in the unit cell where the respective bonds represent seven transfer energies tb1, ..., ta3. From [

By introducing site potentials given by I_{1σ} = −I_{2σ}= −∆_{0}, and I_{3σ} = −I_{4σ} = 0, the 4 × 4 Hamiltonian on the site basis and with a Fourier transform is given by

In the above Hamiltonian, k_{x} is replaced by k_{x} + π and then the center corresponds to X point. We obtain four bands E_{j}(_{1} > E_{2} > E_{3} > E_{4}. For ∆_{0} = 0, the inversion symmetry between A and A' is kept, and E_{1} and E_{2} touches at Dirac point, _{0}, where the ZGS is found for the uniaxial pressure P_{a} being larger than 3 kbar. For the calculation of the Berry curvature, we consider the case of a finite value of ∆_{0}, which induces a gap at the Dirac point.

The Berry curvature _{n}(_{n}(

where |n > denotes the eigen vector of _{n}_{1}(_{1} and E_{2}, _{3} and E_{4} is negligibly small within our numerical calculation. We define B_{n}(k) as the component of _{n}(k), which is perpendicular to the k_{x} − k_{y} plane. For the small limit of ∆_{0}, we obtain B_{1} = ∝ (∆_{0})^{−2}.

The energy band and Berry phase under uniaxial P_{a} are calculated using an extrapolation for_{a}) [_{1}(a) and E_{2}(b) for Pa =6 kbar and ∆_{0} = 0.02 eV is shown in

Energy bands of E1(a) and E2(b) for Pa = 6 kbar. From [

The Dirac cone exists at +(-)_{0}, with _{0}= (0.57 π, - 0.3 π), as shown by arrows. A pair of cones is seen close to E_{1}(_{0}) and E_{2}(_{0}) and the cones in the same band are symmetric with respect to the Γ (=(0,0)) point. For E_{1}(_{2}(_{0} for −△_{0} ➝ 0 is realized as the minimum of E_{1}(_{2}(

The corresponding Berry curvature B_{1}(_{0} (+_{0}) is positive (negative). They are antisymmetric with respect to the Γ. Since the curvature exhibits a noticeable peak close to +(–)_{0}, such a peak may be identified as the Dirac particle instead of calculating the contact point.

Berry curvature B_{1}(_{x}–k_{y} plane). From [

We examined the Berry curvature for other bands. The Berry curvature of the second band E_{2}(_{a} in _{0} has a sign opposite to that of E_{1}(_{1}(_{a} smaller than 3 kbar while it becomes rather isotropic for large P_{a}. Such a behavior resembles the emergence of the Dirac particle in the charge ordered state, which has been shown in α-(BEDT-TTF)_{2}I_{3} [_{3}(_{2}(_{4}(

Contour of the Berry curvature, B_{1}(_{2}(_{0} with the positive B_{1}(_{x} > 0 and k_{y} < 0.

In addition to α-(BEDT-TTF)_{2}I_{3}, the ZGS, where a contact point of the Dirac cone coincides with the chemical potential, is predicted for α-(BEDT-TTF)_{2} NH_{4}Hg(SCN)_{4} under uniaxial pressure, P_{c} along the stacking axis [

There are the electron and hole surfaces owing to band-overlap at ambient pressure. The ZGS emerges under the uniaxial pressure along the stacking axis in the conducting plane, for P_{c} > 5 kbar. _{c} = 6 kbar, where the transfer energies are calculated by extrapolation using the data at P_{c} = 0 kbar [_{0} = (−0.25, 0.51) π. The chemical potential coincides with the contact points since there is no band-overlap. The Dirac cones are rather isotropic compared with those of α-(BEDT-TTF)_{2}I_{3}.

The singularity of phase of the wave function, which describes the intrinsic property of the Dirac electron, has been confirmed using the Berry curvature. There is a pair of peaks in the Berry curvature, which corresponds to a pair of Dirac electrons with opposite chirality. The peaks of the Berry curvature are highly anisotropic at ambient pressure, while those become nearly isotropic at P_{c} = 6 kbar.

In addition, the effective Hamiltonian on the Luttinger-Kohn representation has been investigated. It has been found that the Dirac cone, which exists below the chemical potential, is anisotropic and tilted at ambient pressure. The anisotropy of the Dirac electron at ambient pressure is related to the one-dimension-like valley (ridge) structure between two contact points in the conduction (valence) band. The anisotropy and tilting of the Dirac cone are reduced with increasing pressure.

Energy dispersions of the conductionand valence band at P_{c} = 6 kbar of α-(BEDT-TTF)_{2} NH_{4}Hg(SCN)_{4}. From [

We re-examined the band structure of the stripe charge ordered state of α-(BEDT-TTF)_{2}I_{3} under pressure using an extended Hubbard model [_{2}I_{3} is shown in _{αβ,}

where i, j denote site indices of a given unit cell, and a,b = A, A’, B and C are indices of BEDT-TTF sites in the unit cell. In the first term, the transfer energies as a function of a uniaxial pressure (_{a}_{a} = 2 kbar [_{ab}_{a}_{b}

The model describing the electronic system ofα-(BEDT-TTF)_{2}I_{3}. The unit cellconsists of four BEDT-TTF molecules A, A’, B and C with seven transfer energies. The nearestneighbor repulsive interactions are given by Vaand Vb [

The

In traversing this transition line, the dispersion relations stay similar but their relative positions to the Fermi level vary. In this work, by a more detailed analysis of the COM and CO phases, we find a new

The phase diagram on the

This modification of the band structure is described by an effective low energy Luttinger-Kohn Hamiltonian with nine parameters that can be extracted from a numerical Hartree calculation. From a detailed study of this Hamiltonian and the corresponding energy spectrum, we show that the transition is driven by a single quantity det

The existence of a pair of Dirac points is characterized by a special structure of the Berry curvature. In the CO(II) and COM(II) states, the Berry curvature shows two sharp peaks with opposite signs. On the other hand, in CO(I) and COM(I) phases, the Berry curvature becomes very small owing to cancelation of the positive and negative contributions. The existence of the Dirac point is also verified by integrating the Berry curvature over a region limited by a closed energy contour around a single point [

This topological transition could be probed in a magnetic field by the modification of the Landau level structure, therefore by

We have considered the low-energy effective Hamiltonian for ^{2}^{1/2}_{Z}

where _{i}_{στ}^{+ }(c_{i}_{στ}) is the creation (annihilation) operator on the bases of the Wannier functions. The forward-scattering term of the long-range Coulomb interaction, _{0} and breaks the ^{2}α_{L}/l_{L}_{L}, since the large momentum |2_{0}|~_{L} is exchanged. We note that the lattice constant of _{L}

In the absence of an explicit symmetry breaking, such as the Zeeman effect or the above-mentioned backscattering term, no particular spin or valley-pseudo-spin channel is selected, and one may even find an entangled spin-pseudo-spin ferromagnetic state at low temperature. The symmetry-breaking terms may, thus, be viewed as ones that choose a particular channel (spin or valley-pseudo-spin) and direction of a pre-existing ferromagnetic state by explicitly breaking the original

In the spin-polarized state without valley-pseudo-spin polarization (Δ = 0), electrons reside in the spin-up branches of the

Schematic figure of the energy levels in the spin-polarized state (left hand side) and the valley-pseudo-spinferromagnetic state (right hand side). Blue and orange arrows denote the real spins, and μ is the chemical potential. From [

In the presence of the order parameter with a finite amplitude, phase fluctuation exists with the characteristic length of spatial variation much longer than the fictitious lattice spacing. The effect of this phase fluctuation, which has so far been ignored in the mean-field approximation, is treated on the basis of the Wannier functions and the resulting model is similar to the XY model leading to the Kosterlitz-Thouless transition [

In the present review, we have shown the effect of tilting of Dirac cone and the characteristic behavior of the Berry curvature in organic conductors.

The tilting effect is examined by calculating the polarization function of the massless Dirac particle for a finite doping. Using the tilted Weyl equation, the dynamical polarization is calculated to find the anisotropic behavior in optical conductivity and plasma frequency. A new plasma mode appears owing to the combined effect of the two tilted cones, leading to the filtering effect.

We examined the Berry curvature to understand the zero-gap state under the uniaxial pressure. The pronounced peak of the Berry curvature around the Dirac point is obtained by adding a small potential acting on the A and A' sites with opposite signs, which breaks the inversion symmetry. Such a method of calculating the Berry curvature is useful to find the Dirac electron in an organic conductor. In addition to α-(BEDT-TTF)_{2}I_{3}, the zero-gap state is theoretically predicted for the organic conductor α-(BEDT-TTF)_{2} NH_{4}Hg(SCN)_{4} under uniaxial pressure. The band structure of the stripe charge ordered state of α-(BEDT-TTF)_{2}I_{3} is re-examined using an extended Hubbard model. We found a topological transition from a conventional insulator towards a new phase, which is characterized by the emergence of a pair of Dirac electrons with a finite mass.

Finally, by examining the zero-energy _{2}I_{3} under strong magnetic field, we found the tilted-cone-induced XY valley-pseudo-spin ferromagnetic state and Kosterlitz-Thouless transition as an effect of tilting.

The authors are thankful to H. Fukuyama, F. Piechon, G. Montambaux, T. Nishine, and T. Choji for their collaborations in the present work. Y.S. is indebted to the Daiko Foundation for financial aid in the present research. This work was financially supported in part by a Grant-in-Aid for Special Coordination Funds for Promoting Science and Technology (SCF) and for Scientific Research on Innovative Areas 20110002, and by Scientific Research (Nos. 19740205, 22540366, and 23540403) from the Ministry of Education, Culture, Sports, Science and Technology in Japan.