Physical review. B, Solid state

Phase separation instability in the Hubbard model

Pieter B. Visscher

Published Aug 1, 1974• 144 citations

Abstract

The simple cubic Hubbard model is examined at zero temperature, for the case of strong interactions and average densities near but not at the half-filled band. It is shown that the ferromagnetic state which is usually assumed to be the ground state is unstable against phase separation. A new phase diagram is proposed, featuring a two-phase region in which the equilibrium state is separated into macroscopic antiferromagnetic and ferromagnetic phases (of differing density).

DOI: 10.1103/physrevb.10.943

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Physical review. B, Solid state

Phase separation instability in the Hubbard model

Pieter B. Visscher

Published Aug 1, 1974• 144 citations

Publisher

Abstract

DOI: 10.1103/physrevb.10.943

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Phase separation in the one-dimensional Hubbard model

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The Hartree-Fock ground-state phase diagram of the one-dimensional Hubbard model is calculated in the $μ-U$ 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 $U$ this occurs at hole (or electron) doping of $(3t/π^{2}U)^{1/3}$ from half filling.The Hartree-Fock ground-state phase diagram of the one-dimensional Hubbard model is calculated in the $μ-U$ 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 $U$ this occurs at hole (or electron) doping of $(3t/π^{2}U)^{1/3}$ from half filling.

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Physical review. B, Condensed matterApr 1, 1990116 citations

The Hartree-Fock ground-state phase diagram of the one-dimensional Hubbard model is calculated in the

μ-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

U

this occurs at hole (or electron) doping of

(3t/π^{2}U)^{1/3}

from half filling.

(0<n<1.75)

electron density when the interactions are sufficiently strong. The quasi-long-range order of triplet superconductivity coexists with the ferromagnetic order for a strong Hund coupling region, where the interorbital interaction ${U}^{\ensuremath{'}}\ensuremath{-}J$ is attractive. At quarter-filling

(n=1),

the insulating ferromagnetic state appears, accompanying orbital quasi-long-range order. For low densities

(n<1),

ferromagnetism occurs owing to the ferromagnetic exchange interaction caused by virtual hoppings of electrons, the same as in the quarter-filled system. This comes from separation of the charge and spin-orbital degrees of freedom in the strong-coupling limit. This ferromagnetism is fragile against variation of band structure. For high densities

(n>1),

the phase diagram of the ferromagnetic phase is similar to that obtained in infinite dimensions. In this case, the double exchange mechanism is operative for stabilizing the ferromagnetic order and this long-range order is robust against variation of the band dispersion. A partially polarized state appears in the density region $1.68\ensuremath{\lesssim}n\ensuremath{\lesssim}1.75$ and phase separation occurs for n just below the half-filling

(n=2).

The Hubbard model including nearest-neighbor interaction is studied at

T=0

on a

d

-dimensional hypercubic lattice ($d\ensuremath{\ge}3$) close to half filling. It is shown that the ground state at weak coupling is phase separated, consisting of homogeneous lower-density and antiferromagnetic or charge density wave higher-density regions. Incommensurate phases are unstable. The exact critical doping level and the exact order parameter are renormalized relative to the Hartree results. The renormalization factor at weak coupling turns out to be identical to the result found previously for the same model at half filling.