Phase diagram of one-dimensional Hubbard-Holstein model at quarter-filling

TL;DR

This paper derives an effective Hamiltonian for the one-dimensional Hubbard-Holstein model at quarter-filling, mapping out its phase diagram and identifying phase transitions driven by electron-electron and electron-phonon interactions.

Contribution

It introduces a non-perturbative approach to derive the effective Hamiltonian and employs a modified Lanczos method to study phase transitions in the model.

Findings

Transition from AF cluster to singlet phase with increasing U/t.

Discontinuous transition to charge-density-wave phase at high U/t and g.

Identification of various correlated phases depending on interaction strengths.

Abstract

We derive an effective Hamiltonian for the one-dimensional Hubbard-Holstein model, valid in a regime of both strong electron-electron (e-e) and electron-phonon (e-ph) interactions and in the non-adiabatic limit ($t/\omega_0 \leq 1$), by using a non-perturbative approach. We obtain the phase diagram at quarter-filling by employing a modified Lanczos method and studying various density-density correlations. The spin-spin AF (antiferromagnetic) interactions and nearest-neighbor repulsion, resulting from the e-e and the e-ph interactions respectively, are the dominant terms (compared to hopping) and compete to determine the various correlated phases. As e-e interaction $(U/t)$ is increased, the system transits from an AF cluster to a correlated singlet phase through a discontinuous transition at all strong e-ph couplings $2 \leq g \leq 3$ considered. At higher values of $U/t$ and moderately strong e-ph interactions ($2 \leq g \leq 2.6$), the singlets break up to form an AF order and then to a paramagnetic order all in a single sublattice; whereas at larger values of $g$ ($> 2.6$), the system jumps directly to the spin disordered charge-density-wave (CDW) phase.