Degeneracy of Ground State in Two-Dimensional Electron-Lattice System

Hamano, Tetsuya

doi:10.1143/jpsj.70.1849

Cited by 17 publications

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References 4 publications

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“…It has been shown 4,5) that the state is the ground state of the half filled tight-binding model on the 2D square lattice described by the HamiltonianHere the lattice distortion at the site r is denoted by v(r) = (v x (r), v y (r)) and is defined by the difference of the lattice displacements u(r) aswhere e x(y) denotes the unit vector in the x(y) direction. In the present note, the spin degrees of freedom are neglected for simplicity.…”

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Berry's Phase in the Multimode Peierls States

Kawarabayashi

Watanabe

2008

J. Phys. Soc. Jpn.

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In the present short note, we show that Berry's phase associated with the adiabatic change of local variables in the Hamiltonian 1-3) can be used to characterize the multimode Peierls state, which has been proposed as a new type of the ground state of the two-dimensional(2D) systems with the electron-lattice interaction.4, 5) It is called "multimode" since its lattice distortion pattern consists of more than one Fourier components. It has been shown 4,5) that the state is the ground state of the half filled tight-binding model on the 2D square lattice described by the HamiltonianHere the lattice distortion at the site r is denoted by v(r) = (v x (r), v y (r)) and is defined by the difference of the lattice displacements u(r) aswhere e x(y) denotes the unit vector in the x(y) direction. In the present note, the spin degrees of freedom are neglected for simplicity. The third term in the Hamiltonian describes the elastic energy of the lattice within the harmonic approximation. In this model, the relevant parameter for the electron-lattice coupling is λ = α 2 /(t 0 K). This 2D Peierls model is a simple generalization of the Su-Schrieffer-Heeger model 6, 7) originally introduced in one dimension for the analysis of solitons in polyacetylene. The model has been widely adopted for taking into account the effect of the electron-phonon coupling in 2D systems.8, 9) The conventional Peierls instability expected from the shape of the Fermi surface is the instability with the wave vector (π, π). It has been shown, 4,5) however, that the state having the lattice distortion pattern with the wave vector (π, π) is not the true ground state of the model at half filling, and that the true ground state exhibits a complex distortion pattern with more than one Fourier components including (π, π), which is called the multimode Peierls state. With the additional components, which turn out to be parallel to (π, π), the energy gap opens in the whole range of the Fermi surface. Note that for the lattice distortion described only by the (π, π) mode, there exist gapless points at the Fermi surface. More interestingly, there are many multimode patterns giving the same ground state energy.5) This huge degeneracy is removed by introducing an anisotropy, 10) for instance, by replacing the hopping amplitude t x = (t 0 − αv x (r)) and t y = (t 0 − αv y (r)) with (1 + κ/2)t x and (1 − κ/2)t y , respectively. The parameter κ(0 < κ < 2) determines the strength of anisotropy. It has been pointed out that the real space characterization of the distortion patterns is useful to understand the relations between those degenerated distortion patterns. 11)In spite of these interesting properties of the Peierls instability in two dimensions, the property of the electron wave function has not been discussed in the present model. The reason is that the electron density is uniform and is exactly equal to n = 1/2 due to the electronhole symmetry, irrespective of the lattice distortion patterns. Namely, the electron density does not show any structure related to ...

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“…It has been shown 4,5) that the state is the ground state of the half filled tight-binding model on the 2D square lattice described by the Hamiltonian…”

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confidence: 99%

Berry's Phase in the Multimode Peierls States

Kawarabayashi

Watanabe

2008

J. Phys. Soc. Jpn.

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“…The ground state of 2D SSH model has multimode lattice distortions with the wave vector Q π πµ and those parallel to it. In addition, there are infinite number of degenerate states with non-equivalent distortion patterns corresponding to many different combinations of the Fourier components of the distortion for the wave vectors parallel to Q [4].The effects of the e-e interaction on this multimode Peierls state can be discussed using the 2D PeierlsHubbard (2D PH) model. The Hamiltonian of the 2D PH model is described as follows and e x´yµ the lattice displacements at the site r and the unit lattice vector for respective direction, respectively, and the parameters t 0 , α, K and U are the electron transfer integral between nearest-neighbor sites, the e-l coupling constant, the force constant describing the ionic coupling strength and the strength of the on-site e-e interaction between up and down spins, respectively.…”

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“…1 in terms of the absolute values of the Fourier components x q and y q . We assume the system size to be L ¢L 8 ¢8 and 12 ¢12, the e-l coupling constant to be λ 0 5 and the distortion being composed of the Fourier components for Q and Q 2 which is one of the simplest distortion patterns among many degenerate multimode Peierls states [4]. In the multimode Peierls state, it is known that x q y q .…”

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Correlation Effect on the Two-Dimensional Peierls Phase

Chiba¹

2006

AIP Conference Proceedings

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Abstract. The effect of the electron-electron (e-e) interaction on the Peierls lattice distortions due to the electron-lattice (e-l) interaction is studied in the two-dimensional Peierls-Hubbard model, treating the fluctuation of e-e interaction around the Hartree-Fock solution within the 2nd order perturbation theory. In our previous work, using the Hartree-Fock approximation, we found multimode Peierls lattice distortions with wave vectors, the nesting vector Q and those parallel to it, are not affected by an e-e interaction if it is weak compared with the e-l coupling. The phase transition between the BOW (bond order wave) with multimode lattice distortions and the SDW (spin density wave) with the wave vector Q behaves as the 1st order transition. The property of the multimode BOW is found to change drastically when we consider the fluctuation effect within the 2nd order perturbation theory. The Fourier components of the multimode BOW increase gradually as the e-e interaction parameter is increased. Especially the Fourier component with the wave vector equal to the smallest reciprocal lattice vector in the presence of the multimode BOW is most strongly affected by the fluctuation effect.Keywords: multimode Peierls phase, 2D Peierls-Hubbard model, electron correlation, BOW-SDW transition, perturbation theory PACS: 71.30.+h, 71.45.Lr, 75.30.Fv, 63.20.Kr Competition between the Peierls lattice distortion and the electron-electron (e-e) interaction is widely discussed in the 1D Peierls-Hubbard model with a half-filled electronic band. In this system the lattice distortion which has a wave number Q π plays a very important role in the all range of e-e interaction parameter [1].We are interested in the same problem in the 2D system, since the ground state of the 2D electron-lattice (e-l) system, especially in the 2D SSH (Su-SchriefferHeeger) model with a half-filled electronic band, is an unusual Peierls state [2] in contrast to the Peierls state expected from that of the 1D system [3]. The ground state of 2D SSH model has multimode lattice distortions with the wave vector Q π πµ and those parallel to it. In addition, there are infinite number of degenerate states with non-equivalent distortion patterns corresponding to many different combinations of the Fourier components of the distortion for the wave vectors parallel to Q [4].The effects of the e-e interaction on this multimode Peierls state can be discussed using the 2D PeierlsHubbard (2D PH) model. The Hamiltonian of the 2D PH model is described as follows and e x´yµ the lattice displacements at the site r and the unit lattice vector for respective direction, respectively, and the parameters t 0 , α, K and U are the electron transfer integral between nearest-neighbor sites, the e-l coupling constant, the force constant describing the ionic coupling strength and the strength of the on-site e-e interaction between up and down spins, respectively.In our previous paper [6] we used the Hartree-Fock approximation (HFA) as the easiest approach to discuss the e-e i...

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Phonon softening in triangular lattice

Miyoshi

Terai

2010

Physica B: Condensed Matter

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Degeneracy of Ground State in Two-Dimensional Electron-Lattice System

Cited by 17 publications

References 4 publications

Berry's Phase in the Multimode Peierls States

Berry's Phase in the Multimode Peierls States

Correlation Effect on the Two-Dimensional Peierls Phase

Phonon softening in triangular lattice

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