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Dynamical properties of interacting charge system on frustrated lattices
Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎
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Dynamical properties of interacting charge system on frustrated lattices Makoto Naka a, Hiroshi Hashimoto b, Sumio Ishihara b,c,n a
RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan Department of Physics, Tohoku University, Sendai 980-8578, Japan c Core Research for Evolutional Science and Technology (CREST), Sendai, 980-8578, Japan b
art ic l e i nf o
Keywords: Charge order Optical spectra Photo-excitation
a b s t r a c t Dynamical properties in interacting fermion systems on geometrically frustrated lattices are introduced. As a minimal model for the electronic charge order phenomena, we adopt the so-called spin-less fermion V –t model where the hoppings and interactions of spin-less fermions are taken into account in frustrated lattices. By analyzing this model, we examine (1) dielectric and optical properties in a double triangular lattice, (2) a collective excitation in a dimer-molecular system on a triangular lattice, and (3) a photoexcited state in a triangular lattice. We demonstrate that frustration effects influence directly dynamical properties in these charge systems, and explain successfully recent some experimental results. & 2014 Published by Elsevier B.V.
1. Introduction It is widely accepted that electronic charge order (CO) phenomena are ubiquitously seen in a wide range of solids, such as transition-metal oxides, molecular organic solids, and rare-earth compounds. Below the CO transition temperature, that is abbreviated as TCO , electronic charge distribution lowers the symmetry of the system above TCO , and a new periodicity of the charge alignment appears. The charge ordering and melting influence directly dielectric, transport, and optical properties in the materials. The origin of the CO phenomena is usually attributed to the electron–lattice interaction and the long-range Coulomb interaction between electrons. In particular, the CO transition in the low-dimensional electron–lattice coupled systems, and that in the degenerate electron gas are termed the Peierls and Wigner transitions, respectively. The CO phenomena on frustrated geometrical lattices are termed the frustrated charge systems. This concept was first proposed by Anderson for the Verwey transition in magnetite Fe3O4 where an equal amount of Fe3 þ and Fe4 þ is supposed to be aligned in the frustrated pyrochlore lattice structure [1,2]. Recently, the frustrated charge systems have attracted much attention of a number of researchers, since non-trivial CO structures as well as exotic quantum phenomena are expected [3,4]. In contrast to the static aspects of the frustrated charge systems, the dynamical properties have not been well established. This might be
attributable to the facts that both the experimental and theoretical methods which can access directly to a wide range of energy and momentum in the charge dynamics have been limited. In this paper, we introduce the theoretical examinations of the dynamical properties in some frustrated charge systems. As minimal models which describe the CO phenomena, we adopt the interacting fermion systems termed the spin-less fermion V –t model [5–8]. This is explicitly given by
/ =−
⎛
⎞
⎝
⎠
∑ tij⎜⎜ci†cj + H . c.⎟⎟ + ∑ Vijninj , (ij)
(ij)
(1)
ci†
where and ci, respectively, are the creation and annihilation operators for a spin-less fermion at site i, and ni = ci†ci is a number operator. The first and second terms represent the inter-site hoppings and the Coulomb interactions, respectively. We apply this model to the three kinds of the frustrated charge systems, and examine (1) dielectric and optical propertied in a double triangular lattice, (2) a collective excitation in a dimer-molecular system on a triangular lattice, and (3) a photo-excited state in a triangular lattice. It is found that charge frustration effects influence directly not only the static properties but also the excitation spectra as well as the real-time dynamics, and explain recent some experimental results.
2. Charge dynamics in a double triangular lattice for iron oxides n
Corresponding author at: Department of Physics, Tohoku University, Sendai 980-8578, Japan. E-mail address: [email protected] (S. Ishihara).
In the first section, we introduce the charge dynamics in the spin-less V –t model in a paired triangular lattice where the
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Please cite this article as: M. Naka, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.11.042i
M. Naka et al. / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
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a
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c
Fig. 2. (a) Optical conductivity spectra. Bold and dotted lines are for the CO1/3 and CO1/2 phases, respectively. (b) Dynamical charge correlation functions in CO1/3 phase. Bold and dotted lines are for N (+)(q, ω) and N (−)(q, ω) , respectively. Fig. 1. (a) Schematic paired triangular lattice and interactions, and (b) CO1/3 structure. Filled and open circles are for the charge rich and poor sites, respectively. (c) A schematic CO phase diagram at T ¼ 0 in the plane of t –V (after Ref. [11]).
fermion number per site is 1/2. We consider the double triangular lattices shown in Fig. 1(a) . We introduce the Coulomb interactions between the inter-layer NN sites (VcNN ), the intra-layer NN sites (VabNN ) and the inter-layer next NN sites (VcNNN ). The transfer integrals are considered between the inter-layer NN sites (tcNN ) and the intra-layer NN sites (tabNN ). A unit of the energy parameters is taken to be t. This model was introduced as a minimal model for an electronic ferroelectricity LuFe2O4 where nominal valence of Fe is þ2.5 and an equal number of Fe2 þ and Fe3 þ exists [9–12]. The experimentally observed dielectric anomalies are believed to be due to a CO of Fe 3d electrons with 3-fold periodicity without inversion symmetry; electron number densities on the upper and lower triangular planes are different with each other [13–15]. In the present paired triangular lattice, this CO is represented schematically in the charge alignment along a direction in a triangular lattice as ⋯○○⋯ in the upper plane and ⋯○○○○⋯ in the lower plane, where and ○ represent charge rich and poor sites, respectively. This CO is termed CO1/3 from now on (see Fig. 1(b)). A phase diagram at zero temperature (T ) on the plane of the electron transfer t ≡ tabNN = tcNN and one of the Coulomb interaction VcNNN is schematically shown in Fig. 1(c) [11,12]. At t¼0, this CO1/3 only appears at VcNNN = 0.6 , which is recognized as a frustration point, and the electron transfer stabilizes this polar CO phase. That is, CO1/3 is the “quantum CO” and other COs, i.e. CO1/2 with the 2-fold periodicity and CO1/4 with the 4-fold periodicity, are the classical CO due to the Coulomb interactions. Now we introduce the charge dynamics in CO1/3. We use the cluster mean-field method; the Hamiltonian in a finite-size cluster, where the mean fields are applied at the edge sites, is solved, and
the mean fields are obtained self-consistently with the solution inside of the cluster. At the edge sites, the hopping terms in the Hamiltonian are neglected, and the Coulomb interaction terms are decoupled. The exact diagonalization method based on the Lanczos algorithm is adopted in the clusters, the size of which is taken up to 12 2 sites. In Fig. 2(a), the optical conductivity spectra at T¼ 0 are presented. We define the optical conductivity by
σμ(ω) = −
⎛ 〈0|j μ |m〉〈m|j μ |0〉 e2 〈0|j μ |m〉〈m|j μ |0〉 ⎞⎟ Im ∑ ⎜⎜ , + Nω E E i ω − + + η ω + Em − E0 + iη ⎟⎠ m 0 m ⎝
(2)
where |m〉 and Em are the m-th eigen states and the eigen energy, respectively, j μ = i ∑〈ij〉 tijRijμci†cj + H . c . is a current operator with
the Cartesian coordinate μ, and a position vector Rijμ connecting sites i and j, and an infinitesimal constant η. As shown in the figure, multiple-peak structures are confirmed in CO1/3. The spectra are decomposed into the three components, located around ω ¼ 0.5, 2 and 3. This is highly in contrast to the results in CO1/2 where the spectra are located in a narrow region around 1.5 < ω < 2. These characteristic broad spectra are attributed to the specific CO structure in the CO1/3 phase. There are two-inequivalent sites: charge rich (poor) sites surrounded by six intra-plane NN poor (rich) sites, and the sites surrounded by three intra-plane NN poor and rich sites (see Fig. 1(b)). As a result, three kinds of the charge excitations occur. The multiple-peak structures in the optical conductivity spectra are attributable to these excitations. On the other hand, no inequivalent sites exist in CO1/2. The charge fluctuations in finite momenta are examined by calculating the dynamical charge-correlation function defined as
Please cite this article as: M. Naka, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.11.042i
M. Naka et al. / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎
N (l)(q, ω) = − Im
∑ m
for
nq(−)
l = ( + , − ). −1
〈0|n−(lq) |m〉〈m|nq(l)|0〉 ω − Em + E0 + iη We
,
(3) nq(+) = N−1 ∑i nie−iq·ri
introduce
and
−iq·ri
where si ¼1 and 1 in the cases where i lo= N ∑i sinie cates in the upper and lower planes, respectively. The dynamical correlation functions at T¼ 0 in CO1/3 are shown in Fig. 2(b). Large peak intensities around ω ¼0 are attributed to the charge ordering, corresponding to the super-lattice peaks. A dispersion-like feature is seen in the lowest edges of the spectra.
3. Collective charge excitation in a triangular lattice for a dimer-type Mott insulator Next we show the results of the V –t model on a triangular lattice where the dimer-molecule degree of freedom is taken into account. This is motivated from the κ-type BEDT-TTF molecular solids. A schematic lattice structure is shown in Fig. 3(a), where the BEDT-TTF molecules are denoted by ovals. Two molecules form the so-called dimer molecular units, and are located on the sites in a triangular lattice. The V –t model Hamiltonian in this system is explicitly given by
⎛ ⎞ / Vt = tA ∑ ⎜⎜cia† cib + H . c .⎟⎟ + VA ∑ nia′ nib′ ⎠ i ⎝ i +
⎞ ⎛ tijμμ ′⎜⎜ci†μcjμ + H . c .⎟⎟ + ′ ⎠ ⎝ 〈ij〉μμ ′
∑
∑ 〈ij〉μμ ′
Vijμμ ′ni′μn′ jμ , ′
(4)
where ciμ is an annihilation operator for a spin-less fermion at the i-th dimer with molecule μ, and ni′μ( ≡ ci†μciμ) is a number operator. We introduce the intra-dimer transfer (tA), the intra-dimer Coulomb interaction (VA), the inter-dimer transfer (tijμμ ′) , and the interdimer Coulomb interactions (Vijμμ ′) (see Fig. 3(a)). We take V=Vp, and set tA as a unit of the energy parameters. As shown in Fig. 3(a), the two BEDT-TTF molecules form a dimer unit where the bonding and antibonding orbitals are constructed from the BEDT-TTF molecular orbitals. Since one hole
a
3
exists per dimer in the chemical formula, the antibonding molecular orbital is occupied by one hole. When the charge distributions are equivalent in two molecules in a dimer, the system is identified as a Mott insulator, termed a dimer-Mott insulator in the strong coupling region [16]. On the other hand, there is a possibility that the charge distributions in the two molecules inside of a dimer are not equivalent. When the electric dipole moments in the dimer units are aligned, a macroscopic electric polarization appears. This is termed the polar CO phase [17–20]. It is proposed that recently discovered dielectric anomalies in κ-(BEDTTTF)2Cu2(CN)3 are attributed to this electronic electric dipole moments [21]. The static electronic structures in this model are analyzed by the mean field approximation and the classical Monte Carlo simulations [17]. As shown in Fig. 3(b), with increasing V, there is a critical V ( ≡ Vc), at which the phase transition from the DM phase to the polar CO phase occurs. Now we consider the charge excitation in the dimer Mott phase and the polar CO phase. The intra-dimer charge excitations become a collective charge excitations through the inter-dimer hoppings and Coulomb interactions. The optical conductivity spectra, σx, y(ω), in the V –t model are calculated by the Lanczos method, and the contour maps of the intensities are plotted on the planes of ω and V (see Fig. 4). Spectra around 1.1 ≲ ω ≲ 1.4 in σx(ω) and 0.4 ≲ ω ≲ 1.2 in σy(ω) are attributed to the intra-dimer charge excitations. Clear polarization dependence and softening around the phase boundary are observed. These low-energy excitations in σx(ω) and σy(ω) are identified as the optical and acoustic mode excitations, respectively, because of the two dimers in a unit cell. The finite-excitation energy at the phase boundary is due to the finite size effect, and the energy of the acoustic mode becomes zero at the phase boundary. This soft collective mode is connected to the electric-polarization flip in the CO phase and the intra-dimer bonding-antibonding excitation, the so-called dimer-excitation, in the dimer-Mott phase.
4. Photo-induced dynamics in charge ordered phases in a triangular lattice In this section, we introduce the real-time charge dynamics in the V –t model on a triangular lattice induced by the ultrafast optical pulse [22–25]. We adopt the V –t model on a
a
b b
Fig. 3. (a) A schematic lattice structure of the κ-type BEDT-TTF system, and interactions introduced in the Hamiltonian. Ovals represent the BEDT-TTF molecules. (b) Schematic phase diagram in the plane of T –V and charge distributions in the dimer-Mott and polar CO phases (after [17]). Shaded regions represent the charge clouds.
Fig. 4. Contour maps for the optical conductivity spectra in the plane of ω and V. The spectra with the x and y polarizations are plotted in (a) and (b), respectively.
Please cite this article as: M. Naka, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.11.042i
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Fig. 5. (a) Schematic triangular lattice and interactions. (b) A schematic calculated CO phase diagram in the V − V ′ plane (after Ref. [18]). The horizontal, 3-fold, and vertical CO structures are also shown. Filled and open circles are for charge rich and poor sites, respectively.
two-dimensional triangular lattice shown in Fig. 5(a), where anisotropies in the fermion hopping and the Coulomb interactions are represented as (t , t′), and (V , V ′), respectively. The number of fermion per site is fixed to be 1/2. This model has been studied in terms of the CO phenomena in organic molecular materials [5–7]. The ground-state phase diagram is obtained on the plane of the nearest-neighbor (NN) Coulomb-interactions on the anisotropic triangular-lattice bonds, i.e. V and V ′ as shown in Fig. 5(b). Two insulating CO states, called the horizontal-type and vertical-type COs, are realized in the classical limit of no fermion hopping, for V < V ′ and V > V ′, respectively, and the two phases compete with each other at V = V ′. When a finite fermion hopping is included, a metallic 3-fold CO phase appears between the two COs. The optical pump-pulse is introduced as the Peierls phase into the transfer integral as tij → tijeiA(τ)·Rij where A(τ) is the vector potential at time τ, and R ij is a relative position vector connecting sites i and j. The time dependence of the vector potential is assumed to be a Gaussian form given by −1
A(τ) = A p e ( 2π τp) e−τ
2
/2τp2
cos(ωpτ),
(5)
with amplitude Ap, frequency ωp, damping factor τp, and unit vector e . The electronic structures are calculated by using the exact diagonalization method based on the Lanczos algorithm. Time evolution of the wave function Ψ (τ) is obtained by using the quasieigenstates in the Lanczos algorithm as [26–28] M
|Ψ (τ + δτ)〉 =
∑ e−iϵ jδτ |ψj〉〈ψj|Ψ (τ)〉, j=1
(6)
where ϵj and |ψj〉, respectively, are the eigenvalues and eigenstates. We focus on the real time charge dynamics in the horizontaltype and vertical-type COs. The pump-photon frequencies are tuned around the optical excitation peaks in the two COs. The transient CO structures are monitored by calculating the charge correlation functions defined by
C (k) = N −2 ∑ 〈(ni − 1/2)(nj − 1/2)〉e ik·Rij , i, j
(7)
Fig. 6. Charge correlation functions after photo-irradiation (a) in the horizontal CO, and (b) in the vertical CO as functions of the pump-photon amplitudes. The 6-fold and diagonal CO are other types of COs realized in the present model Hamiltonian.
where the momentum k is defined in the Brillouin zone. The charge correlation functions after the pump-pulse irradiation (τ > > τp) in the horizontal and vertical CO phases are plotted as functions of the pump-photon amplitudes in Fig. 6. In the horizontal-CO phase, after irradiation of pump-pulse, the correlation function for the horizontal CO rapidly decreases and is interchanged by the correlation for the 3-fold CO. That is, the photo-induced phase change occurs in the case of the strong pump-photon amplitude. On the other hand, in the vertical-CO phase, the correlation for the vertical-CO phase is the largest one even in the case of the strong pump-photon amplitude. This difference is attributed to the frustration effects in the photo-excited states: in the horizontal-CO phase, there are a number of low energy charge configurations around the frustration point. This is highly in contrast to the vertical-CO phase, where the possible low-energy excitation processes are limited in comparison with the horizontal-CO phase. This fact implies that the vertical-CO phase is robust under the photo-excitations, and is merely weakened by photo-irradiation.
5. Summary In summary, we present the dynamical properties in the three frustrated CO systems: (1) dielectric and optical propertied in a double triangular lattice, (2) a collective excitation in a dimermolecular system on a triangular lattice, and (3) a photo-excited state in a triangular lattice. Essences of the charge frustration are the large number of the degenerate states and exotic quantum states due to releasing frustration and competition. We show that these directly influence the excitation spectra as well as the real time dynamics in interacting fermion systems. Beyond the present analyses of the spin-less fermion models, couplings to the spin, orbital and lattice degrees of freedom may provide rich variety of the interacting electron systems with multi-degrees of freedom on geometrical lattices.
Please cite this article as: M. Naka, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.11.042i
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Acknowledgments We thank H. Seo, H. Matsueda, T. Watanabe, J. Nasu, K. Iwano, K. Ishii, S. Iwai, N. Ikeda, T. Sasaki, S. Koshihara, and H. Okamoto for their helpful discussions. This work was supported by KAKENHI from the Ministry of Education, Science and Culture of Japan (No. 26287070). Some of the numerical calculations were performed using the supercomputing facilities at ISSP, the University of Tokyo.
References [1] P.W. Anderson, Phys. Rev. 102 (1956) 1008. [2] E.J.W. Verwey, P.W. Haaymann, Physica 8 (1941) 979. [3] Special topics: novel states of matter induced by frustration, J. Phys. Soc. Jpn. 79 (2010) 001101–001112. [4] H.T. Diep (Ed.), Frustrated Spin Systems, World Scientific, New Jersey, 2004. [5] H. Seo, J. Merino, H. Yoshioka, M. Ogata, J. Phys. Soc. Jpn. 75 (2006) 051009. [6] J. Merino, H. Seo, M. Ogata, Phys. Rev. B 71 (2005) 125111. [7] C. Hotta, N. Furukawa, A. Nakagawa, K. Kubo, J. Phys. Soc. Jpn. 75 (2006) 123704. [8] S. Nishimoto, M. Shingai, Y. Ohta, Phys. Rev. B 78 (2008) 035113. [9] A. Nagano, M. Naka, J. Nasu, S. Ishihara, Phys. Rev. Lett. 99 217202 (2007).
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M. Naka, A. Nagano, S. Ishihara, Phys. Rev. B 77 (2008) 224441. T. Watanabe, S. Ishihara, J. Phys. Soc. Jpn. 78 (2009) 113702. T. Watanabe, S. Ishihara, J. Phys. Soc. Jpn. 79 (2010) 114714. Y. Yamada, S. Nohdo, N. Ikeda, J. Phys. Soc. Jpn. 66 (1997) 3733. N. Ikeda, H. Ohsumi, K. Ohwada, K. Ishii, T. Inami, K. Kakurai, Y. Murakami, K. Yoshii, S. Mori, Y. Horibe, H. Kito, Nature 436 (2005) 1136. J. de Groot, T. Mueller, R.A. Rosenberg, D.J. Keavney, Z. Islam, J.-W. Kim, M. Angst, Phys. Rev. Lett. 108 (2012) 187601. H. Kino, H. Fukuyama, J. Phys. Soc. Jpn. 64 (1995), 1877, 2726, 4523. M. Naka, S. Ishihara, J. Phys. Soc. Jpn. 79 (2010) 063707. C. Hotta, Phys. Rev. B 82 (2010) R31044. H. Gomi, T. Imai, A. Takahashi, M. Aihara, Phys. Rev. B 82 (2010) 035101. S. Dayal, R.T. Clay, H. Li, S. Mazumdar, Phys. Rev. B 83 (2011) 245106. M. Abdel-Jawad, I. Terasaki, T. Sasaki, N. Yoneyama, N. Kobayashi, Y. Uesu, C. Hotta, Phys. Rev. B 82 (2010) 125119. K. Nasu, Photo-Induced Phase Transitions, World Scientific, New Jersey, 2004. Special topics: photo-induced phase transitions and their dynamics, J. Phys. Soc. Jpn. 75 (2006) 001101–001108. K. Yonemitsu, N. Maeshima, Phys. Rev. B 76 (2007) 075105. K. Iwano, Phys. Rev. Lett. 102 (2009) 106405. T.J. Park, J.C. Light, J. Chem. Phys. 85 (1986) 5870. P. Prelovsek, J. Bonca, arxiv:1111.5931. H. Hashimoto, H. Matsueda, H. Seo, S. Ishihara, J. Phys. Soc. Jpn. 83 (2014) 123703.
Please cite this article as: M. Naka, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.11.042i
Contents lists available at ScienceDirect
Physica B journal homepage: www.elsevier.com/locate/physb
Dynamical properties of interacting charge system on frustrated lattices Makoto Naka a, Hiroshi Hashimoto b, Sumio Ishihara b,c,n a
RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan Department of Physics, Tohoku University, Sendai 980-8578, Japan c Core Research for Evolutional Science and Technology (CREST), Sendai, 980-8578, Japan b
art ic l e i nf o
Keywords: Charge order Optical spectra Photo-excitation
a b s t r a c t Dynamical properties in interacting fermion systems on geometrically frustrated lattices are introduced. As a minimal model for the electronic charge order phenomena, we adopt the so-called spin-less fermion V –t model where the hoppings and interactions of spin-less fermions are taken into account in frustrated lattices. By analyzing this model, we examine (1) dielectric and optical properties in a double triangular lattice, (2) a collective excitation in a dimer-molecular system on a triangular lattice, and (3) a photoexcited state in a triangular lattice. We demonstrate that frustration effects influence directly dynamical properties in these charge systems, and explain successfully recent some experimental results. & 2014 Published by Elsevier B.V.
1. Introduction It is widely accepted that electronic charge order (CO) phenomena are ubiquitously seen in a wide range of solids, such as transition-metal oxides, molecular organic solids, and rare-earth compounds. Below the CO transition temperature, that is abbreviated as TCO , electronic charge distribution lowers the symmetry of the system above TCO , and a new periodicity of the charge alignment appears. The charge ordering and melting influence directly dielectric, transport, and optical properties in the materials. The origin of the CO phenomena is usually attributed to the electron–lattice interaction and the long-range Coulomb interaction between electrons. In particular, the CO transition in the low-dimensional electron–lattice coupled systems, and that in the degenerate electron gas are termed the Peierls and Wigner transitions, respectively. The CO phenomena on frustrated geometrical lattices are termed the frustrated charge systems. This concept was first proposed by Anderson for the Verwey transition in magnetite Fe3O4 where an equal amount of Fe3 þ and Fe4 þ is supposed to be aligned in the frustrated pyrochlore lattice structure [1,2]. Recently, the frustrated charge systems have attracted much attention of a number of researchers, since non-trivial CO structures as well as exotic quantum phenomena are expected [3,4]. In contrast to the static aspects of the frustrated charge systems, the dynamical properties have not been well established. This might be
attributable to the facts that both the experimental and theoretical methods which can access directly to a wide range of energy and momentum in the charge dynamics have been limited. In this paper, we introduce the theoretical examinations of the dynamical properties in some frustrated charge systems. As minimal models which describe the CO phenomena, we adopt the interacting fermion systems termed the spin-less fermion V –t model [5–8]. This is explicitly given by
/ =−
⎛
⎞
⎝
⎠
∑ tij⎜⎜ci†cj + H . c.⎟⎟ + ∑ Vijninj , (ij)
(ij)
(1)
ci†
where and ci, respectively, are the creation and annihilation operators for a spin-less fermion at site i, and ni = ci†ci is a number operator. The first and second terms represent the inter-site hoppings and the Coulomb interactions, respectively. We apply this model to the three kinds of the frustrated charge systems, and examine (1) dielectric and optical propertied in a double triangular lattice, (2) a collective excitation in a dimer-molecular system on a triangular lattice, and (3) a photo-excited state in a triangular lattice. It is found that charge frustration effects influence directly not only the static properties but also the excitation spectra as well as the real-time dynamics, and explain recent some experimental results.
2. Charge dynamics in a double triangular lattice for iron oxides n
Corresponding author at: Department of Physics, Tohoku University, Sendai 980-8578, Japan. E-mail address: [email protected] (S. Ishihara).
In the first section, we introduce the charge dynamics in the spin-less V –t model in a paired triangular lattice where the
http://dx.doi.org/10.1016/j.physb.2014.11.042 0921-4526/& 2014 Published by Elsevier B.V.
Please cite this article as: M. Naka, et al., Physica B (2014), http://dx.doi.org/10.1016/j.physb.2014.11.042i
M. Naka et al. / Physica B ∎ (∎∎∎∎) ∎∎∎–∎∎∎
2
a
a
b b
c
Fig. 2. (a) Optical conductivity spectra. Bold and dotted lines are for the CO1/3 and CO1/2 phases, respectively. (b) Dynamical charge correlation functions in CO1/3 phase. Bold and dotted lines are for N (+)(q, ω) and N (−)(q, ω) , respectively. Fig. 1. (a) Schematic paired triangular lattice and interactions, and (b) CO1/3 structure. Filled and open circles are for the charge rich and poor sites, respectively. (c) A schematic CO phase diagram at T ¼ 0 in the plane of t –V (after Ref. [11]).
fermion number per site is 1/2. We consider the double triangular lattices shown in Fig. 1(a) . We introduce the Coulomb interactions between the inter-layer NN sites (VcNN ), the intra-layer NN sites (VabNN ) and the inter-layer next NN sites (VcNNN ). The transfer integrals are considered between the inter-layer NN sites (tcNN ) and the intra-layer NN sites (tabNN ). A unit of the energy parameters is taken to be t. This model was introduced as a minimal model for an electronic ferroelectricity LuFe2O4 where nominal valence of Fe is þ2.5 and an equal number of Fe2 þ and Fe3 þ exists [9–12]. The experimentally observed dielectric anomalies are believed to be due to a CO of Fe 3d electrons with 3-fold periodicity without inversion symmetry; electron number densities on the upper and lower triangular planes are different with each other [13–15]. In the present paired triangular lattice, this CO is represented schematically in the charge alignment along a direction in a triangular lattice as ⋯○○⋯ in the upper plane and ⋯○○○○⋯ in the lower plane, where and ○ represent charge rich and poor sites, respectively. This CO is termed CO1/3 from now on (see Fig. 1(b)). A phase diagram at zero temperature (T ) on the plane of the electron transfer t ≡ tabNN = tcNN and one of the Coulomb interaction VcNNN is schematically shown in Fig. 1(c) [11,12]. At t¼0, this CO1/3 only appears at VcNNN = 0.6 , which is recognized as a frustration point, and the electron transfer stabilizes this polar CO phase. That is, CO1/3 is the “quantum CO” and other COs, i.e. CO1/2 with the 2-fold periodicity and CO1/4 with the 4-fold periodicity, are the classical CO due to the Coulomb interactions. Now we introduce the charge dynamics in CO1/3. We use the cluster mean-field method; the Hamiltonian in a finite-size cluster, where the mean fields are applied at the edge sites, is solved, and
the mean fields are obtained self-consistently with the solution inside of the cluster. At the edge sites, the hopping terms in the Hamiltonian are neglected, and the Coulomb interaction terms are decoupled. The exact diagonalization method based on the Lanczos algorithm is adopted in the clusters, the size of which is taken up to 12 2 sites. In Fig. 2(a), the optical conductivity spectra at T¼ 0 are presented. We define the optical conductivity by
σμ(ω) = −
⎛ 〈0|j μ |m〉〈m|j μ |0〉 e2 〈0|j μ |m〉〈m|j μ |0〉 ⎞⎟ Im ∑ ⎜⎜ , + Nω E E i ω − + + η ω + Em − E0 + iη ⎟⎠ m 0 m ⎝
(2)
where |m〉 and Em are the m-th eigen states and the eigen energy, respectively, j μ = i ∑〈ij〉 tijRijμci†cj + H . c . is a current operator with
the Cartesian coordinate μ, and a position vector Rijμ connecting sites i and j, and an infinitesimal constant η. As shown in the figure, multiple-peak structures are confirmed in CO1/3. The spectra are decomposed into the three components, located around ω ¼ 0.5, 2 and 3. This is highly in contrast to the results in CO1/2 where the spectra are located in a narrow region around 1.5 < ω < 2. These characteristic broad spectra are attributed to the specific CO structure in the CO1/3 phase. There are two-inequivalent sites: charge rich (poor) sites surrounded by six intra-plane NN poor (rich) sites, and the sites surrounded by three intra-plane NN poor and rich sites (see Fig. 1(b)). As a result, three kinds of the charge excitations occur. The multiple-peak structures in the optical conductivity spectra are attributable to these excitations. On the other hand, no inequivalent sites exist in CO1/2. The charge fluctuations in finite momenta are examined by calculating the dynamical charge-correlation function defined as
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N (l)(q, ω) = − Im
∑ m
for
nq(−)
l = ( + , − ). −1
〈0|n−(lq) |m〉〈m|nq(l)|0〉 ω − Em + E0 + iη We
,
(3) nq(+) = N−1 ∑i nie−iq·ri
introduce
and
−iq·ri
where si ¼1 and 1 in the cases where i lo= N ∑i sinie cates in the upper and lower planes, respectively. The dynamical correlation functions at T¼ 0 in CO1/3 are shown in Fig. 2(b). Large peak intensities around ω ¼0 are attributed to the charge ordering, corresponding to the super-lattice peaks. A dispersion-like feature is seen in the lowest edges of the spectra.
3. Collective charge excitation in a triangular lattice for a dimer-type Mott insulator Next we show the results of the V –t model on a triangular lattice where the dimer-molecule degree of freedom is taken into account. This is motivated from the κ-type BEDT-TTF molecular solids. A schematic lattice structure is shown in Fig. 3(a), where the BEDT-TTF molecules are denoted by ovals. Two molecules form the so-called dimer molecular units, and are located on the sites in a triangular lattice. The V –t model Hamiltonian in this system is explicitly given by
⎛ ⎞ / Vt = tA ∑ ⎜⎜cia† cib + H . c .⎟⎟ + VA ∑ nia′ nib′ ⎠ i ⎝ i +
⎞ ⎛ tijμμ ′⎜⎜ci†μcjμ + H . c .⎟⎟ + ′ ⎠ ⎝ 〈ij〉μμ ′
∑
∑ 〈ij〉μμ ′
Vijμμ ′ni′μn′ jμ , ′
(4)
where ciμ is an annihilation operator for a spin-less fermion at the i-th dimer with molecule μ, and ni′μ( ≡ ci†μciμ) is a number operator. We introduce the intra-dimer transfer (tA), the intra-dimer Coulomb interaction (VA), the inter-dimer transfer (tijμμ ′) , and the interdimer Coulomb interactions (Vijμμ ′) (see Fig. 3(a)). We take V=Vp, and set tA as a unit of the energy parameters. As shown in Fig. 3(a), the two BEDT-TTF molecules form a dimer unit where the bonding and antibonding orbitals are constructed from the BEDT-TTF molecular orbitals. Since one hole
a
3
exists per dimer in the chemical formula, the antibonding molecular orbital is occupied by one hole. When the charge distributions are equivalent in two molecules in a dimer, the system is identified as a Mott insulator, termed a dimer-Mott insulator in the strong coupling region [16]. On the other hand, there is a possibility that the charge distributions in the two molecules inside of a dimer are not equivalent. When the electric dipole moments in the dimer units are aligned, a macroscopic electric polarization appears. This is termed the polar CO phase [17–20]. It is proposed that recently discovered dielectric anomalies in κ-(BEDTTTF)2Cu2(CN)3 are attributed to this electronic electric dipole moments [21]. The static electronic structures in this model are analyzed by the mean field approximation and the classical Monte Carlo simulations [17]. As shown in Fig. 3(b), with increasing V, there is a critical V ( ≡ Vc), at which the phase transition from the DM phase to the polar CO phase occurs. Now we consider the charge excitation in the dimer Mott phase and the polar CO phase. The intra-dimer charge excitations become a collective charge excitations through the inter-dimer hoppings and Coulomb interactions. The optical conductivity spectra, σx, y(ω), in the V –t model are calculated by the Lanczos method, and the contour maps of the intensities are plotted on the planes of ω and V (see Fig. 4). Spectra around 1.1 ≲ ω ≲ 1.4 in σx(ω) and 0.4 ≲ ω ≲ 1.2 in σy(ω) are attributed to the intra-dimer charge excitations. Clear polarization dependence and softening around the phase boundary are observed. These low-energy excitations in σx(ω) and σy(ω) are identified as the optical and acoustic mode excitations, respectively, because of the two dimers in a unit cell. The finite-excitation energy at the phase boundary is due to the finite size effect, and the energy of the acoustic mode becomes zero at the phase boundary. This soft collective mode is connected to the electric-polarization flip in the CO phase and the intra-dimer bonding-antibonding excitation, the so-called dimer-excitation, in the dimer-Mott phase.
4. Photo-induced dynamics in charge ordered phases in a triangular lattice In this section, we introduce the real-time charge dynamics in the V –t model on a triangular lattice induced by the ultrafast optical pulse [22–25]. We adopt the V –t model on a
a
b b
Fig. 3. (a) A schematic lattice structure of the κ-type BEDT-TTF system, and interactions introduced in the Hamiltonian. Ovals represent the BEDT-TTF molecules. (b) Schematic phase diagram in the plane of T –V and charge distributions in the dimer-Mott and polar CO phases (after [17]). Shaded regions represent the charge clouds.
Fig. 4. Contour maps for the optical conductivity spectra in the plane of ω and V. The spectra with the x and y polarizations are plotted in (a) and (b), respectively.
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4
a
a
b
b
Fig. 5. (a) Schematic triangular lattice and interactions. (b) A schematic calculated CO phase diagram in the V − V ′ plane (after Ref. [18]). The horizontal, 3-fold, and vertical CO structures are also shown. Filled and open circles are for charge rich and poor sites, respectively.
two-dimensional triangular lattice shown in Fig. 5(a), where anisotropies in the fermion hopping and the Coulomb interactions are represented as (t , t′), and (V , V ′), respectively. The number of fermion per site is fixed to be 1/2. This model has been studied in terms of the CO phenomena in organic molecular materials [5–7]. The ground-state phase diagram is obtained on the plane of the nearest-neighbor (NN) Coulomb-interactions on the anisotropic triangular-lattice bonds, i.e. V and V ′ as shown in Fig. 5(b). Two insulating CO states, called the horizontal-type and vertical-type COs, are realized in the classical limit of no fermion hopping, for V < V ′ and V > V ′, respectively, and the two phases compete with each other at V = V ′. When a finite fermion hopping is included, a metallic 3-fold CO phase appears between the two COs. The optical pump-pulse is introduced as the Peierls phase into the transfer integral as tij → tijeiA(τ)·Rij where A(τ) is the vector potential at time τ, and R ij is a relative position vector connecting sites i and j. The time dependence of the vector potential is assumed to be a Gaussian form given by −1
A(τ) = A p e ( 2π τp) e−τ
2
/2τp2
cos(ωpτ),
(5)
with amplitude Ap, frequency ωp, damping factor τp, and unit vector e . The electronic structures are calculated by using the exact diagonalization method based on the Lanczos algorithm. Time evolution of the wave function Ψ (τ) is obtained by using the quasieigenstates in the Lanczos algorithm as [26–28] M
|Ψ (τ + δτ)〉 =
∑ e−iϵ jδτ |ψj〉〈ψj|Ψ (τ)〉, j=1
(6)
where ϵj and |ψj〉, respectively, are the eigenvalues and eigenstates. We focus on the real time charge dynamics in the horizontaltype and vertical-type COs. The pump-photon frequencies are tuned around the optical excitation peaks in the two COs. The transient CO structures are monitored by calculating the charge correlation functions defined by
C (k) = N −2 ∑ 〈(ni − 1/2)(nj − 1/2)〉e ik·Rij , i, j
(7)
Fig. 6. Charge correlation functions after photo-irradiation (a) in the horizontal CO, and (b) in the vertical CO as functions of the pump-photon amplitudes. The 6-fold and diagonal CO are other types of COs realized in the present model Hamiltonian.
where the momentum k is defined in the Brillouin zone. The charge correlation functions after the pump-pulse irradiation (τ > > τp) in the horizontal and vertical CO phases are plotted as functions of the pump-photon amplitudes in Fig. 6. In the horizontal-CO phase, after irradiation of pump-pulse, the correlation function for the horizontal CO rapidly decreases and is interchanged by the correlation for the 3-fold CO. That is, the photo-induced phase change occurs in the case of the strong pump-photon amplitude. On the other hand, in the vertical-CO phase, the correlation for the vertical-CO phase is the largest one even in the case of the strong pump-photon amplitude. This difference is attributed to the frustration effects in the photo-excited states: in the horizontal-CO phase, there are a number of low energy charge configurations around the frustration point. This is highly in contrast to the vertical-CO phase, where the possible low-energy excitation processes are limited in comparison with the horizontal-CO phase. This fact implies that the vertical-CO phase is robust under the photo-excitations, and is merely weakened by photo-irradiation.
5. Summary In summary, we present the dynamical properties in the three frustrated CO systems: (1) dielectric and optical propertied in a double triangular lattice, (2) a collective excitation in a dimermolecular system on a triangular lattice, and (3) a photo-excited state in a triangular lattice. Essences of the charge frustration are the large number of the degenerate states and exotic quantum states due to releasing frustration and competition. We show that these directly influence the excitation spectra as well as the real time dynamics in interacting fermion systems. Beyond the present analyses of the spin-less fermion models, couplings to the spin, orbital and lattice degrees of freedom may provide rich variety of the interacting electron systems with multi-degrees of freedom on geometrical lattices.
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Acknowledgments We thank H. Seo, H. Matsueda, T. Watanabe, J. Nasu, K. Iwano, K. Ishii, S. Iwai, N. Ikeda, T. Sasaki, S. Koshihara, and H. Okamoto for their helpful discussions. This work was supported by KAKENHI from the Ministry of Education, Science and Culture of Japan (No. 26287070). Some of the numerical calculations were performed using the supercomputing facilities at ISSP, the University of Tokyo.
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