Berry's Phase in the Multimode Peierls States

Abstract

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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