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We derive the Luttinger liquid (LL) to charge density wave (CDW) transition condition for the spinless one-dimensional Holstein model by using different approaches in the anti-adiabatic and
the adiabatic regimes. In the anti-adiabatic regime, we derive an effective polaronic interaction Hamiltonian which is exact to second order in perturbation with the small parameter being the polaron size parameter [2]. We map our effective polaronic Hamiltonian onto a next-to-nearest-neighbor interaction anisotropic Heisenberg spin model and study the mass gap
and the power-law exponent of the spin-spin correlation function to obtain the phase transition in the anti-adiabatic region.
In the adiabatic regime, the LL to CDW transition condition is obtained exactly to second order in a novel blocked perturbative approach with the small parameter being the ratio of the electron-phonon coupling constant and the adiabaticity parameter [1]. In this adiabatic region, we correct the mean-field criterion for Peierls instability by replacing the static non-interacting susceptibility at twice the Fermi momentum with the dynamic one evaluated at the phonon frequency. At non-half-filling, we present the phase diagram showing the surprising result that the CDW occurs in a more restricted region of the two parameter (polaron size and adiabaticity) space than at half-filling [1]. [1]"Phase transition and phase diagram at a general filling in the spinless one-dimensional Holstein Model", Sanjoy Datta and Sudhakar Yarlagadda, Phys. Rev. B 75, p. 035124 (2007). [2] "Many-Polaron Effects in the Holstein Model", Sanjoy Datta, Arnab Das, and Sudhakar Yarlagadda, Phys. Rev. B 71, p. 235118 (2005). |
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