Low-energy excitation spectra in the excitonic phase of cobalt oxides

TL;DR

This paper investigates the low-energy excitation spectra of the excitonic phase in cobalt oxides using a multi-orbital Hubbard model, revealing collective excitations like Goldstone and Higgs modes relevant for experiments.

Contribution

It introduces a detailed theoretical analysis of excitonic condensation and collective excitations in cobalt oxides using RPA and mean-field methods.

Findings

Identification of gapless spin-transverse (Goldstone) mode.

Detection of gapful spin-longitudinal (Higgs) mode.

Prediction of excitonic instability in cobalt oxides.

Abstract

We study the excitonic phase and low-energy excitation spectra of perovskite cobalt oxides. Constructing the five-orbital Hubbard model defined on the three-dimensional cubic lattice for the $3d$ bands of Pr$_{0.5}$Ca$_{0.5}$CoO$_3$, we calculate the excitonic susceptibility in the normal state in the random-phase approximation (RPA) to show the presence of the instability toward excitonic condensation. Based on the excitonic ground state with a magnetic multipole obtained in the mean-field approximation, we calculate the dynamical susceptibility of the excitonic phase in the RPA and find that there appear the gapless collective excitation in the spin-transverse mode (Goldstone mode) and the gapful collective excitation in the spin-longitudinal mode (Higgs mode). Experimental relevance is discussed.

Tables

Table 1. Table 1: Nonzero elements of V μ ​ λ ν ​ κ s ​ s ′ superscript subscript 𝑉 𝜇 𝜆 𝜈 𝜅 𝑠 superscript 𝑠 ′ V_{\begin{subarray}{c}\mu\lambda\\ \nu\kappa\end{subarray}}^{ss^{\prime}} , where s = ( σ 1 , σ 2 ) 𝑠 subscript 𝜎 1 subscript 𝜎 2 s=(\sigma_{1},\sigma_{2}) and s ′ = ( σ 1 ′ , σ 2 ′ ) superscript 𝑠 ′ superscript subscript 𝜎 1 ′ superscript subscript 𝜎 2 ′ s^{\prime}=(\sigma_{1}^{\prime},\sigma_{2}^{\prime}) .

	${\sigma }_1={\sigma }_2$ $={\sigma }_1^{\prime }={\sigma }_2^{\prime }$	${\sigma }_1={\sigma }_2$ $\ne {\sigma }_1^{\prime }={\sigma }_2^{\prime }$	${\sigma }_1={\sigma }_2^{\prime }$ $\ne {\sigma }_2={\sigma }_1^{\prime }$
$\mu =\nu =\kappa =\lambda$	–	$-U$	$U$
$\mu =\kappa \ne \nu =\lambda$	–	$-J$	$J$
$\mu =\nu \ne \kappa =\lambda$	$-U+3J$	$-U+2J$	$J$
$\mu =\lambda \ne \nu =\kappa$	$U-3J$	$-J$	$U-2J$

Equations

$$H_0=\sum _{i,\mu ,\sigma }{ϵ}_{\mu }{c}_{i,\mu ,\sigma }^{†}{c}_{j,\mu ,\sigma }+\sum _{i,j}\sum _{\mu ,\nu }\sum _{\sigma }{t}_{ij,\mu \nu }{c}_{i,\mu ,\sigma }^{†}{c}_{j,\nu ,\sigma },$$

$$H_{\mathrm{int}}$$

$$+\frac{U^{\prime }}{2}\sum _{i,\sigma ,{\sigma }^{\prime }}\sum _{\mu \ne \nu }{c}_{i,\mu ,\sigma }^{†}{c}_{i,\mu ,\sigma }{c}_{i,\nu ,{\sigma }^{\prime }}^{†}{c}_{i,\nu ,{\sigma }^{\prime }}$$

$$-\frac{J}{2}\sum _{i,\sigma ,{\sigma }^{\prime }}\sum _{\mu \ne \nu }{c}_{i,\mu ,\sigma }^{†}{c}_{i,\mu ,{\sigma }^{\prime }}{c}_{i,\nu ,{\sigma }^{\prime }}^{†}{c}_{i,\nu ,\sigma }$$

$$+\frac{J^{\prime }}{2}\sum _{i,\sigma }\sum _{\mu \ne \nu }{c}_{i,\mu ,\sigma }^{†}{c}_{i,\nu ,-\sigma }{c}_{i,\mu ,-\sigma }^{†}{c}_{i,\nu ,\sigma },$$

$${\Delta }_{\mu ,\nu }=\sum _{\mathbf{k},\sigma }\sigma ⟨c_{\mathbf{k}+\mathbf{Q},\mu ,\sigma }^{†}{c}_{\mathbf{k},\nu ,\sigma }⟩,$$

$$H^{\mathrm{MF}}=\sum _{{\mathbf{k}}_0,ϵ,\sigma }{E}_{{\mathbf{k}}_0,ϵ,\sigma }{\gamma }_{{\mathbf{k}}_0,ϵ,\sigma }^{†}{\gamma }_{{\mathbf{k}}_0,ϵ,\sigma },$$

$$\begin{gathered} {\chi }_{\begin{array}{c}\lambda \mu \\ \kappa \nu \end{array}}^{s{s}^{\prime }}(\mathbf{q},{\mathbf{q}}^{\prime },\omega )=\frac{i}{N}\sum _{\mathbf{k},{\mathbf{k}}^{\prime }}{\int }_0^{\infty }dt{e}^{i\omega t} \\ \times ⟨[c_{\mathbf{k},\kappa ,{\sigma }_1}^{†}(t)c_{\mathbf{k}+\mathbf{q},\lambda ,{\sigma }_2}(t),c_{{\mathbf{k}}^{\prime }+{\mathbf{q}}^{\prime },\mu ,{\sigma }_1^{\prime }}^{†}{c}_{{\mathbf{k}}^{\prime },\nu ,{\sigma }_2^{\prime }}]⟩, \end{gathered}$$

$${{\chi }_0^{s{s}^{\prime }}}_{\begin{array}{c}\lambda \mu \\ \kappa \nu \end{array}}(\mathbf{q},\mathbf{q}+l\mathbf{Q},\omega )$$

$$=$$

$$\times {\psi }_{\lambda ,m;ϵ}({\mathbf{p}}_0+\mathbf{q},{\sigma }_1){\psi }_{\mu ,m+n+l;ϵ}^{\ast }({\mathbf{p}}_0+\mathbf{q},{\sigma }_2^{\prime })$$

$$\times {\psi }_{\kappa ,m;{ϵ}^{\prime }}^{\ast }({\mathbf{p}}_0,{\sigma }_2){\psi }_{\nu ,m+n;{ϵ}^{\prime }}({\mathbf{p}}_0,{\sigma }_1^{\prime }){\delta }_{{\sigma }_1,{\sigma }_2^{\prime }}{\delta }_{{\sigma }_1^{\prime },{\sigma }_2},$$

$$\begin{gathered} \left(\begin{array}{c}{\chi }^{+-}\\ {\chi }^{\uparrow \uparrow }\\ {\chi }^{\downarrow \uparrow }\end{array}\right)=\left(\begin{array}{c}{\chi }_0^{+-}\\ {\chi }_0^{\uparrow \uparrow }\\ 0\end{array}\right) \\ +\left(\begin{array}{ccc}{\chi }_0^{+-}{V}^{-+}& 0& 0\\ 0& {\chi }_0^{\uparrow \uparrow }{V}^{\uparrow \uparrow }& {\chi }_0^{\uparrow \uparrow }{V}^{\uparrow \downarrow }\\ 0& {\chi }_0^{\downarrow \downarrow }{V}^{\downarrow \uparrow }& {\chi }_0^{\downarrow \downarrow }{V}^{\downarrow \downarrow }\end{array}\right)\left(\begin{array}{c}{\chi }^{+-}\\ {\chi }^{\uparrow \uparrow }\\ {\chi }^{\downarrow \uparrow }\end{array}\right), \end{gathered}$$

$$[{\chi }_{0}V\chi {]}_{\begin{array}{c}\lambda \mu \\ \kappa \nu \end{array}}(\mathbf{q},\omega )=$$

$$\sum _{{\kappa }^{\prime },{\lambda }^{\prime },{\mu }^{\prime },{\nu }^{\prime },m}{{\chi }_0}_{\begin{array}{c}\lambda {\mu }^{\prime }\\ \kappa {\nu }^{\prime }\end{array}}(\mathbf{q},\mathbf{q}+m\mathbf{Q},\omega )V_{\begin{array}{c}{\mu }^{\prime }{\lambda }^{\prime }\\ {\nu }^{\prime }{\kappa }^{\prime }\end{array}}{\chi }_{\begin{array}{c}{\lambda }^{\prime }\mu \\ {\kappa }^{\prime }\nu \end{array}}(\mathbf{q}+m\mathbf{Q},\mathbf{q},\omega )$$

Full text

The Bose–Einstein condensation of fermion pairs is one of the most intriguing phenomena in condensed matter physics. The excitonic phase (EP) is representative of such a pair condensation,[1, 2, 3, 4] where holes in valence bands and electrons in conduction bands spontaneously form pairs owing to attractive Coulomb interaction. After Mott’s prediction of the EP half a century ago,[5] a number of candidate materials for this phase have come to our attention. Among them are the transition-metal chalcogenides 1$T$-TiSe2[6, 7, 8] and Ta2NiSe5,[9, 10, 11] where the electrons and holes on different atoms are considered to form spin-singlet pairs to condense into the EP, which is accompanied by lattice distortion.[12]

Another class of materials includes the perovskite cobalt oxides,[13, 14, 15] where the valence-band holes and conduction-band electrons form spin-triplet pairs in different orbitals on the same atoms. Pr0.5Ca0.5CoO3 (PCCO) is an example in which the “metal-insulator” phase transition is observed at $T_{c}\simeq 80$ K, which is associated with a sharp peak in the temperature dependence of the specific heat and a drop in the magnetic susceptibility below $T_{c}$,[16] together with a valence transition of Pr ions.[17, 18] Some results of experiments indicate that the resistivity is in fact small and nearly temperature independent below $T_{c}$,[19] suggesting that the bands may not be fully gapped. Note that no local magnetic moments are observed, but the exchange splitting of the Pr4+ Kramers doublet occurs,[19] the result of which may therefore be termed as a hidden order, and also that no clear signatures of the spin-state transition are observed in the X-ray absorption spectra.[20, 19]

Kuneš and Augustinsky̌ argued that the anomalies of PCCO can be attributed to the EP transition,[13] whereby they applied the dynamical-mean-field-theory calculation to the two-orbital Hubbard model defined on a two-dimensional square lattice and claimed that the anomalous behaviors of the specific heat, dc conductivity, and spin susceptibility can be explained. They also performed the LDA$+U$ band-structure calculation and showed that the magnetic multipole ordering occurs in PCCO as a result of the excitonic condensation. LaCoO3 under a high magnetic field is another example of the possible realization of the EP,[21] which was substantiated by the theoretical calculations based on the two-orbital Hubbard and related models in two-dimension.[22, 23] In these materials, cobalt ions are basically in the Co3+ valence state with a $3d^{6}$ configuration, where the three $t_{2g}$ orbitals are mostly filled with electrons and the two $e_{g}$ orbitals are nearly empty. The low-spin state is thus favorable for the condensation of excitons.

In this work, motivated by the above development in the field, we will study the EP of PCCO using a realistic Hubbard model, taking into account all five $3d$ orbitals of Co ions arranged in the three-dimensional cubic lattice of the perovskite structure. The noninteracting tight-binding bands are determined from first principles and the electron-electron interactions in the $3d$ orbitals are fully taken into account in each Co ion. We will then study the excitonic fluctuations in the normal state via the calculation of the excitonic susceptibility in the random phase approximation (RPA) and show that the instability toward the EP actually occurs in this model. The ground state of this model is then calculated in the mean-field approximation, whereby we find that the EP with a magnetic multipole order actually occurs. We will also calculate the dynamical susceptibility of both spin-transverse and spin-longitudinal modes in the EP to clarify the presence of the gapless Goldstone and gapful Higgs modes in the excitation spectra. The experimental relevance of our results will be discussed.

The crystal structure of PCCO belongs to the $P_{nma}$ space group, where the CoO6 octahedra are rotated and the cubic perovskite structure is distorted with two independent Co ions in the unit cell,[16] giving rise to complexity in the analysis of the EP in PCCO. We instead make use of the crystal structure of PrCoO3, which is a perfect cubic perovskite with the lattice constant $a=3.82$ Å[24] (hereafter taken as the unit of length). The electronic structure is then calculated from first principles using the WIEN2k code [25]. The obtained band dispersions are illustrated in Fig. 1, where we find that the bands near the Fermi energy come from the Pr $4f$ orbitals (giving narrow dispersions) and Co $3d$ orbitals (giving wide dispersions). For the latter, we find that the valence bands come from the $t_{2g}$ manifold of Co $3d$ orbitals and the conduction bands come from the $e_{g}$ manifold, as indicated in the weight plot. We, moreover, find that the $t_{2g}$ bands and $e_{g}$ bands are orthogonal to each other without hybridization, providing us with an ideal situation for excitonic condensation. We also performed the band-structure calculation of PCCO, arranging the Pr and Ca ions regularly, and confirmed that the bands remain qualitatively unchanged, supporting the validity of the rigid-band approximation. Note that PCCO contains both the Co3+ and Co4+ ions,[19] while PrCoO3 contains only the Co3+ ions and shows no signatures of the phase transition.[26] The change in the valence state of Co ions, which may lead to a better nesting feature of the Fermi surfaces, seems to play an important role in the EP transition. Hereafter, we focus on the $3d$ bands of the Co ions, and assuming that the $4f$ bands of Pr ions act as a bath of electrons and, together with the presence of Ca2+ ions, the valence state of Co ions is kept to be in the $3d^{6}$ configuration for simplicity[13] unless otherwise stated.

Let us now set up the Hamiltonian $H=H_{0}+H_{\rm int}$ for the modeling of the $3d$ electrons of PCCO. The kinetic energy term $H_{0}$ is defined in the tight-binding approximation as

[TABLE]

where $c^{\dagger}_{i,\mu,\sigma}$ is the creation operator of a spin-$\sigma$ $(=\uparrow,\downarrow)$ electron on the orbital $\mu$ at site $i$, $\epsilon_{\mu}$ is the on-site energy of orbital $\mu$, and $t_{ij,\mu\nu}$ is the hopping integral between the orbital $\nu$ at site $j$ and the orbital $\mu$ at site $i$. The orbitals $\mu$ and $\nu$ are labeled as 1 ($d_{xy}$), 2 ($d_{yz}$), 3 $(d_{zx}$), 4 ($d_{x^{2}-y^{2}}$), and 5 ($d_{3z^{2}-r^{2}}$). The 12 molecular orbitals for the $3d$ and $4f$ bands are obtained as the maximally localized Wannier functions,[27, 28] thereby retaining only the $3d$ bands to determine the on-site energies and hopping integrals (up to 6th neighbors). The tight-binding band dispersions thus calculated reproduce the first-principles band structure well, as shown in Fig. 1.

The on-site interaction term is defined as

[TABLE]

where $U,U^{\prime},J$, and $J^{\prime}$ are the intraorbital Coulomb interaction, interorbital Coulomb interaction, Hund’s rule coupling, and pair-hopping interaction, respectively. We assume the atomic-limit relations $U^{\prime}=U-2J$ and $J^{\prime}=J$ for the interaction strengths, and we fix the ratio $J/U$ at $0.1$ in the present calculations.

We apply the mean-field approximation to the interaction terms. We assume the spin-triplet excitonic order in the presence of Hund’s rule coupling [29] and write the order parameters as

[TABLE]

where $c_{\bm{k},\mu,\sigma}$ is the Fourier component of $c_{i,\mu,\sigma}$ with the wave vector $\bm{k}$, and $\bm{Q}$ is an ordering vector. Note that when $\mu$ ($\nu$) is one of the $e_{g}$ orbitals, $\nu$ ($\mu$) is one of the $t_{2g}$ orbitals. All the terms irrelevant to this excitonic ordering are neglected for simplicity. We thus obtain the diagonalized mean-field Hamiltonian,

[TABLE]

where $\gamma_{\bm{k}_{0},\epsilon,\sigma}$ is the canonical transformation of $c_{\bm{k},\mu,\sigma}$ satisfying $c_{\bm{k},\mu,\sigma}=\sum_{\epsilon}\psi_{\mu,m;\epsilon}(\bm{k}_{0},\sigma)\gamma_{\bm{k}_{0},\epsilon,\sigma}$ and $\epsilon$ is the band index. Since the excitonic order enlarges the unit cell, we write the wave vector as $\bm{k}=\bm{k}_{0}+m\bm{Q}$, where $\bm{k}_{0}$ is the wave vector in the reduced Brillouin zone and $m$ is an integer. We carry out the summation with respect to $\bm{k}_{0}$ using the $50\times 50\times 50$ meshes in the reduced Brillouin zone.

We define the dynamical susceptibility as

[TABLE]

where $N$ is the number of ${\bm{k}}$ points used, $c_{\bm{k},\mu,\sigma}(t)$ is the Heisenberg representation of $c_{\bm{k},\mu,\sigma}$, and $s$ denotes a spin pair $(\sigma_{1},\sigma_{2})$, taking the values $\uparrow$, $\downarrow$, $+$, and $-$ for $(\uparrow,\uparrow)$, $(\downarrow,\downarrow)$, $(\uparrow,\downarrow)$, and $(\downarrow,\uparrow)$, respectively. We write Eq. (5) as $\chi(\bm{q},\omega)$ when $\bm{q}=\bm{q}^{\prime}$. The bare susceptibility is given by

[TABLE]

where the summation with respect to $\bm{p}_{0}$ runs over the reduced Brillouin zone. We set $\eta=0.01$ eV.

We calculate the dynamical susceptibilities in the multiorbital RPA, given by

[TABLE]

where the matrix product in the orbital basis is given as

[TABLE]

with the interaction matrix $V$ listed in Table 1.

First, let us discuss the spin-triplet excitonic fluctuations in the normal phase. Figure 2 shows the $\bm{q}$ dependence of the static susceptibility of the excitonic spin-transverse mode $\chi^{+-}_{\begin{subarray}{c}\mu\mu\\ \nu\nu\end{subarray}}(\bm{q},\omega=0)$ calculated in the normal phase, where $\mu$ ($\nu$) is one of the $e_{g}$ ($t_{2g}$) orbitals. We find that, at $\bm{q}=(\pi,\pi,\pi)$, the diverging fluctuations with increasing $U$ toward $1.15$ eV are observed for all the orbital components except $(\mu,\nu)=(5,1)$. This instability toward the EP is caused by the Fermi-surface nesting between the electron pockets of the $e_{g}$ bands located around the $\Gamma$ point of the Brillouin zone and the hole pockets of the $t_{2g}$ bands located around the ${\rm R}(\pi,\pi,\pi)$ point of the Brillouin zone (see Fig. 1). Thus, the EP transition with the ordering vector $\bm{Q}=(\pi,\pi,\pi)$ occurs at the critical value $U_{\mathrm{cr}}=1.15$ eV.

Next, let us solve the mean-field equations to calculate the excitonic order parameter with $\bm{Q}=(\pi,\pi,\pi)$. The obtained orbital components $\Delta_{\mu,\nu}$ are shown in Fig. 3(a), where we find that all the components $\Delta_{\mu,\nu}$ (except $\Delta_{1,5}$) become finite above $U_{\mathrm{cr}}=1.15$ eV. As $U$ increases, $U^{\prime}$ and $J$ also increase, which enhances $\Delta_{\mu,\nu}$. Because the excitons are formed in a single atom, the excitonic spin polarization leads to the magnetic multipole order in real space,[13, 30] as shown in Fig. 3(b). The orbital components of the magnetic multipoles formed between the $\mu$ and $\nu$ orbitals (indicated as $\mu\otimes\nu$) are shown in Fig. 3(c). Reflecting the symmetry of the orbitals, the components of the order parameter satisfy the relations $\Delta_{4,2}=\Delta_{4,3}=\Delta_{4,1}/\sqrt{3}$, $\Delta_{5,2}=-\Delta_{5,3}=\Delta_{4,1}/2$, and $\Delta_{5,1}=0$, where different combinations of the signs are also possible. The last relation indicates that the electrons on the $d_{3z^{2}-r^{2}}$ orbital and holes on the $d_{xy}$ orbital do not form pairs. This is consistent with the result for the calculated excitonic susceptibility in the normal phase [see Fig. 2(b)], where no diverging behavior is observed. We note that the bands in the EP are not fully gapped at $U<1.8$ eV, keeping the system metallic with small Fermi surfaces, which may be consistent with the results of experiment.[19] A full gap opens for larger values of $U>1.8$ eV. We also note that the excitonic order remains finite against the change in the filling of electrons, e.g., between 5.4 and 6.1 per site at $U=1.2$ eV, which is also consistent with experimental results.[19]

Next, let us discuss the excitation spectra in the EP. The calculated excitonic spin-transverse dynamical susceptibilities $\mathrm{Im}\,\chi^{+-}_{\begin{subarray}{c}\mu\mu\\ \nu\nu\end{subarray}}$ are shown in Figs. 4(a)–4(d), where we find the gapless Goldstone mode at $\bm{q}=(\pi,\pi,\pi)$ for all the components except for $(\mu,\nu)=(5,1)$ [see Fig. 4(c)], reflecting the presence/absence of $\Delta_{\mu,\nu}$. The velocities of the collective excitations near $\bm{q}=(\pi,\pi,\pi)$ are the same for all the components. Unlike in the well-known collective mode of the Heisenberg antiferromagnets, the excitonic collective mode does not extend to reach the point $\bm{q}=0$ and $\omega=0$. The gapless collective-mode behavior obtained is consistent with the results of the two-orbital models.[14, 31] Note that the excitations around $\bm{q}=(\pi/2,\pi/2,\pi/2)$ appearing in all the components originate from the particle-hole excitations. The calculated excitonic spin-longitudinal dynamical susceptibilities $\mathrm{Im}\,\chi^{zz}_{\begin{subarray}{c}\mu\mu\\ \nu\nu\end{subarray}}$ defined as $\chi^{zz}_{\begin{subarray}{c}\lambda\mu\\ \kappa\nu\end{subarray}}=\sum_{\sigma,\sigma^{\prime}}\sigma\sigma^{\prime}\chi^{\sigma\sigma^{\prime}}_{\begin{subarray}{c}\lambda\mu\\ \kappa\nu\end{subarray}}$ are also shown in Figs. 4(f)–4(i), where we find the gapful Higgs mode. The broad excitations that have a gap at $\bm{q}=(\pi,\pi,\pi)$ are clearly seen, except for $(\mu,\nu)=(5,1)$ [see Fig. 4(h)], where only the particle-hole excitations are present. The spectra around $\bm{q}=(\pi/2,\pi/2,\pi/2)$ appearing in all the components are again particle-hole excitations. The orbital-diagonal part of the dynamical susceptibilities in the transverse mode, $\sum_{\mu,\nu}\mathrm{Im}\,\chi^{+-}_{\begin{subarray}{c}\mu\nu\\ \mu\nu\end{subarray}}$, and in the longitudinal mode, $\sum_{\mu,\nu}\mathrm{Im}\,\chi^{zz}_{\begin{subarray}{c}\mu\nu\\ \mu\nu\end{subarray}}$, are also shown for comparison in Figs. 4(e) and 4(j), respectively, the results of which reflect the metallic nature of the system.

Finally, let us discuss the possible experimental relevance of our results. Inelastic neutron scattering may be a possible experimental technique for observing the excitations in the spin degrees of freedom, but may enable one to detect only the orbital-diagonal components of the dynamical susceptibility, the intensity of which is in proportion to $\sum_{\mu,\nu}\mathrm{Im}\,\chi_{\begin{subarray}{c}\mu\nu\\ \mu\nu\end{subarray}}^{ss^{\prime}}$. Our corresponding results, which come from the particle-hole transitions, are shown in Figs. 4(e) and 4(j). Because the dynamical susceptibility related to the orbital-off-diagonal excitonic ordering should contain the vertex-nonconserved terms, defined as the terms with $\kappa\neq\lambda$ and $\mu\neq\nu$ in Eq. (5), we need to seek other quantum-beam sources that have the ability to change the orbitals (or orbital angular momentum) in the inelastic scattering processes. To this end, future experimental developments are desired.

In summary, we derived the effective five-orbital Hubbard model defined on the three-dimensional cubic lattice from first principles to describe the electronic states of Pr0.5Ca0.5CoO3 with the cubic perovskite structure. Then, we calculated the static susceptibility of the excitonic spin-transverse mode in the normal phase using the RPA and found that the diverging excitonic fluctuations occur at $\bm{Q}=(\pi,\pi,\pi)$. We calculated the excitonic ground state in the mean-field approximation and found that the magnetic multipole order occurs. We also calculated the dynamical susceptibility in the EP to study the excitation spectra and found that there appear gapless collective excitations in the excitonic spin-transverse mode and gapful collective excitations in the excitonic spin-longitudinal mode.

{acknowledgment}

We thank T. Kaneko and S. Miyakoshi for fruitful discussions. This work was supported in part by Grants-in-Aid for Scientific Research from the Japan Society for the Promotion of Science (Nos. 26400349 and 15H06093). The numerical calculations were carried out on computers at Yukawa Institute for Theoretical Physics, Kyoto University, Japan, and Research Center for Computational Science, Okazaki, Japan.