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International Journal of Modern Physics B

PHASE SEPARATION IN THE TWO-DIMENSIONAL HUBBARD MODEL

M. YU, Hua Lin

Published Jan 30, 2005• 2 citations
Publisher

Abstract

In this paper, we study phase separation in the two-dimensional single-band Hubbard model with the unrestricted Hartree-Fock(UHF) method and the restricted Hartree-Fock (RHF) method. We perform the calculation for square lattices and rectangle lattices. It is observed that the stripe phase exists and it depends on three aspects: geometry of the lattice, Coulomb interaction U and band filling n. To gain more physical insights, we consider the Hubbard model with spin dependent hoppings: t↑ and t↓, and study the effect of varying [Formula: see text] on the phase separation.

DOI
10.1142/s021797920502844x

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In this paper, we study phase separation in the two-dimensional single-band Hubbard model with the unrestricted Hartree-Fock(UHF) method and the restricted Hartree-Fock (RHF) method. We perform the calculation for square lattices and rectangle lattices. It is observed that the stripe phase exists and it depends on three aspects: geometry of the lattice, Coulomb interaction U and band filling n. To gain more physical insights, we consider the Hubbard model with spin dependent hoppings: t↑ and t↓, and study the effect of varying [Formula: see text] on the phase separation.
The wave-function renormalization constant Z has been calculated for the one-band Hubbard model on a square lattice. Near half-filling the Hamiltonian has been solved on finite clusters (up to 16\ifmmode\times\else\texttimes\fi{}16) by means of the unrestricted Hartree-Fock (UHF) approximation. Some checks of the UHF results have been made on 4\ifmmode\times\else\texttimes\fi{}4 clusters by means of the Lanczos method for U=\ensuremath{\infty}. At high dilutions (two electrons) exact calculations have been carried out. The ground-state wave functions and energies for the different cases here considered are discussed. The system shows nonconventional behavior (Z=0) near half-filling. For a given value of U, Z vanishes at low dopings, and becomes finite as doping is increased. For very large U the value of Z remains finite for all dopings, excluding half-filling. These results are cast into a plausible phase diagram.
We propose an analytical method for the construction of Hartree-Fock phase diagrams for the (fermion) Hubbard model and various generalizations thereof. Such phase diagrams are traditionally constructed numerically, but we argue that, by using asymptotic techniques, it is possible to obtain analytic formulas approximating the curves separating the different phases to very high accuracy. To illustrate the new method, we apply it to the two-dimensional Hubbard model on the square lattice at zero temperature. This yields formulas for the Hartree-Fock phase boundaries that agree with, but also improve on, earlier numerical results. In particular, our results provide the first rigorous proof of the existence of mixed phases in this model.
We present an unbiased mean-field analysis of magnetic and charge orders in the two-dimensional Hubbard model on a square lattice, at both zero and finite temperatures. Unrestricted Hartree-Fock calculations on large finite lattices are complemented by solutions restricted to N\'eel and circular spiral order in the thermodynamic limit. The magnetic states are classified by a systematic scheme based on the dominant Fourier components of the spin texture. On finite lattices a whole zoo of ordering patterns appears. We show that many of these states are finite-size artifacts related to the limited choice of ordering wave vectors on a finite lattice. In the thermodynamic limit only three classes of states with a relatively simple structure survive: N\'eel, circular spiral, and stripe states. Stripes involve also charge order and can be unidirectional or bidirectional, with horizontal and/or vertical orientation. We present complete phase diagrams in the plane spanned by electron density and temperature, for a moderate Hubbard interaction and various choices of the next-nearest-neighbor hopping amplitude.
The Hartree-Fock ground-state phase diagram of the one-dimensional Hubbard model is calculated in the μ−Uμ-Uμ−U plane, restricted to phases with no charge density modulation, extending the results presented in cond-mat/9511116. This allows antiferromagnetism, saturated ferromagnetism, spiral spin density waves and a collinear structure with unit cell ↑↑↓↓\uparrow \uparrow \downarrow \downarrow↑↑↓↓. The spiral phase is unstable against phase separation near quarter-, half- and three-quarter-filling. For large UUU this occurs at hole (or electron) doping of (3t/π2U)1/3(3t/π^{2}U)^{1/3}(3t/π2U)1/3 from half filling.