Accurate localization of Kosterlitz-Thouless-type quantum phase transitions for one-dimensional spinless fermions

Abstract

We investigate the charge-density wave (CDW) transition for one-dimensional spinless fermions at half band-filling with nearest-neighbor electron transfer amplitude $t$ and interaction $V$. The model is equivalent to the anisotropic XXZ Heisenberg model for which the Bethe Ansatz provides an exact solution. For $V>V_{\mathrm{c}}=2t$, the CDW order parameter and the single-particle gap are finite but exponentially small, as is characteristic for a Kosterlitz-Thouless transition. It is notoriously difficult to locate such infinite-order phase transitions in the phase diagram using approximate analytical and numerical approaches. Second-order Hartree-Fock theory is qualitatively applicable for all interaction strengths, and predicts the CDW transition to occur at $V_{\mathrm{c},2}^{(2)}\approx 1.5t$. Second-order Hartree Fock theory is almost variational because the density of quasi-particle excitations is small. We apply the density-matrix renormalization group (DMRG) for periodic boundary conditions for system sizes up to 514 sites which permits a reliable extrapolation of all physical quantities to the thermodynamic limit, apart from the critical region. We investigate the ground-state energy, the gap, the order parameter, the momentum distribution, the quasi-particle density, and the density-density correlation function to locate $V_{\mathrm{c}}$ from the DMRG data. Tracing the breakdown of the Luttinger liquid and the peak in the quasi-particle density at the band edge permits us to reproduce $V_{\mathrm{c}}$ with an accuracy of one percent.

arXiv commentMain text: 26 pages, 17 figures; supplemental material: 16 pages, 3 figures

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