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Journal of Physics C Solid State Physics

The stability and thermodynamic behaviour of spin-density waves I. Hartree-Fock theory

M J Morris, J. F. Cornwell

Published Oct 1, 1968• 14 citations
Publisher

Abstract

A detailed investigation has been made of the application of the Hartree-Fock approximation to the stability and thermodynamic behaviour of spin-density waves in a model with a single electronic energy band and a delta-function-type interaction between electrons. It is shown that for a half-filled band the ground state at zero temperature is always a spin-density wave, provided only that the width of the band is sufficiently small. For other bands the conditions for the relative stability of the spin-density wave, ferromagnetic and paramagnetic states are more complicated, and are studied in detail for a special band form. For a half-filled band the energy per atom and specific heat have been calculated as functions of temperature for the spin-density wave, ferromagnetic and paramagnetic states. The temperature dependence of the amplitude for the spin-density wave and ferromagnetic states, and of the entropy for the spin-density wave and paramagnetic states have also been calculated. The formulation of the problem has been made using a Green function method in order to allow the development in a later paper of a higher-order approximation to the correlation problem.

DOI
10.1088/0022-3719/1/5/301

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The stability and thermodynamic behaviour of spin-density waves II. Correlation theory

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The Green function theory formulated in I is generalized to give a higher-order approximation to the correlation problem. The spin-density-wave stability in this approximation is compared in detail with the Hartree-Fock approximation results of I and are found to be in conflict in several respects.

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The stability condition of the Hartree-Fock states, which are constructed out of the plane waves times spin-up or -down eigenfunctions. i.e. the paramagnetic (P), ferromagnetic (F) and unsaturated ferromagnetic (UF) states, is examined in some detail in the case of repulsive δ-function. Yukawa and Coulomb interactions. The spherical Fermi surface leads of course to the minimum energy in each case; this energy is then minimized as the function of polarization n=(n+-n-)/(n++n-,n± being the total number of spin-up and -down electrons. We thus can determine the domains of density where each of the three states is most stable. Then the stability of these states against small deformations of one-particle wave functions is examined in their respective domains. In the case of δ-function potential no instability occurs, while for an electron gas the P state (rs ≪ 5.45) is always unstable and the F state (rs > 5.45) is unstable near the critical density. The variational method in the form of canonical transformation is presented to obtain a new Hartree-Fock solution having helical spin structure in case instability occurs. The case of δ-function interaction is worked out for the P state and it is actually shown that such a solution can exist if and only if the P state becomes unstable. This new solution is of non-perturbative nature in the sense that it can never be obtained by perturbation theory starting from the free ground state.

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J H Samson

The Hartree - Fock ground-state phase diagram of the one-dimensional Hubbard model is calculated, constrained to uniform phases, which have no charge density modulation. The allowed solutions are saturated ferromagnetism (FM), a spiral spin density wave (SSDW) and a double spin density wave (DSDW). The DSDW phase comprises two canted interpenetrating antiferromagnetic sublattices. FM occurs for small filling, SSDW in most of the remainder of the phase diagram, and DSDW in a narrow tongue near quarter (and three-quarter) filling. Itinerant electrons lift the degeneracy with respect to canting angle in the DSDW. The Hartree - Fock states are metallic except at multiples of a quarter filling. Near half filling the uniform SSDW phase is unstable against phase separation into a half-filled antiferromagnetic phase and a hole-rich SSDW phase. The dependence of the ground-state wave number on chemical potential is conjectured to be a staircase. Comparison is made with higher-dimensional Hubbard models and the Heisenberg model.

arXiv:cond-mat/9511116•Journal of Physics Condensed Matter•Jan 29, 1996•1 citation
A detailed investigation has been made of the application of the Hartree-Fock approximation to the stability and thermodynamic behaviour of spin-density waves in a model with a single electronic energy band and a delta-function-type interaction between electrons. It is shown that for a half-filled band the ground state at zero temperature is always a spin-density wave, provided only that the width of the band is sufficiently small. For other bands the conditions for the relative stability of the spin-density wave, ferromagnetic and paramagnetic states are more complicated, and are studied in detail for a special band form. For a half-filled band the energy per atom and specific heat have been calculated as functions of temperature for the spin-density wave, ferromagnetic and paramagnetic states. The temperature dependence of the amplitude for the spin-density wave and ferromagnetic states, and of the entropy for the spin-density wave and paramagnetic states have also been calculated. The formulation of the problem has been made using a Green function method in order to allow the development in a later paper of a higher-order approximation to the correlation problem.
The stability condition of the Hartree-Fock states, which are constructed out of the plane waves times spin-up or -down eigenfunctions. i.e. the paramagnetic (P), ferromagnetic (F) and unsaturated ferromagnetic (UF) states, is examined in some detail in the case of repulsive δ-function. Yukawa and Coulomb interactions. The spherical Fermi surface leads of course to the minimum energy in each case; this energy is then minimized as the function of polarization n=(n+-n-)/(n++n-,n± being the total number of spin-up and -down electrons. We thus can determine the domains of density where each of the three states is most stable. Then the stability of these states against small deformations of one-particle wave functions is examined in their respective domains. In the case of δ-function potential no instability occurs, while for an electron gas the P state (rs ≪ 5.45) is always unstable and the F state (rs > 5.45) is unstable near the critical density. The variational method in the form of canonical transformation is presented to obtain a new Hartree-Fock solution having helical spin structure in case instability occurs. The case of δ-function interaction is worked out for the P state and it is actually shown that such a solution can exist if and only if the P state becomes unstable. This new solution is of non-perturbative nature in the sense that it can never be obtained by perturbation theory starting from the free ground state.
We calculate the Hartree-Fock energy of a density-wave in a spin polarized two-dimensional electron gas using a short-range repulsive interaction. We find that the stable ground state for a short-range potential is always either the paramagnetic state or the uniform ferromagnetic state. The energy of a density-wave state is, however, reduced by a factor proportional to (1 - zeta^2), where zeta is the polarization of the electron gas. Since this situation occurs in the most unfavorable conditions (short-range repulsive interaction) it is therefore conceivable that by including higher order many-body corrections to the interaction a density-wave ground state is indeed found to be stable.
The Hartree-Fock ground state phase diagram of the one-dimensional Hubbard model is calculated, constrained to uniform phases, which have no charge density modulation. The allowed solutions are saturated ferromagnetism (FM), a spiral spin density wave (SSDW) and a double spin density wave} (DSDW). The DSDW phase comprises two canted interpenetrating antiferromagnetic sublattices. FM occurs for small filling, SSDW in most of the remainder of the phase diagram, and DSDW in a narrow tongue near quarter (and three-quarter) filling. Itinerant electrons lift the degeneracy with respect to canting angle in the DSDW. The Hartree-Fock states are metallic except at multiples of a quarter filling. Near half filling the uniform SSDW phase is unstable against phase separation into a half-filled antiferromagnetic phase and a hole-rich SSDW phase. The dependence of the ground state wave number on chemical potential is conjectured to be a staircase. Comparison is made with higher dimensional Hubbard models and the J1−J2J_{1}-J_{2}J1​−J2​ Heisenberg model.
The Hartree - Fock ground-state phase diagram of the one-dimensional Hubbard model is calculated, constrained to uniform phases, which have no charge density modulation. The allowed solutions are saturated ferromagnetism (FM), a spiral spin density wave (SSDW) and a double spin density wave (DSDW). The DSDW phase comprises two canted interpenetrating antiferromagnetic sublattices. FM occurs for small filling, SSDW in most of the remainder of the phase diagram, and DSDW in a narrow tongue near quarter (and three-quarter) filling. Itinerant electrons lift the degeneracy with respect to canting angle in the DSDW. The Hartree - Fock states are metallic except at multiples of a quarter filling. Near half filling the uniform SSDW phase is unstable against phase separation into a half-filled antiferromagnetic phase and a hole-rich SSDW phase. The dependence of the ground-state wave number on chemical potential is conjectured to be a staircase. Comparison is made with higher-dimensional Hubbard models and the Heisenberg model.