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Journal of Mathematical Physics

Dynamics near Equilibrium of Systems Described in Thermal Hartree-Fock Approximation

B. U. Felderhof

Published Jun 1, 1969• 6 citations
Publisher

Abstract

The dynamics of systems described in Hartree-Fock approximation is studied near a stationary point of the free energy. It is shown that the second-order free-energy functional is a constant for the linearized self-consistent equation of motion. This leads to the stability criterion derived by Mermin. A simple collision model is constructed and is shown to satisfy the H theorem. It exhibits the effect of critical slowing down. The formalism is shown to be applicable to superconductors.

DOI
10.1063/1.1664928

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The dynamics of systems described in Hartree-Fock approximation is studied near a stationary point of the free energy. It is shown that the second-order free-energy functional is a constant for the linearized self-consistent equation of motion. This leads to the stability criterion derived by Mermin. A simple collision model is constructed and is shown to satisfy the H theorem. It exhibits the effect of critical slowing down. The formalism is shown to be applicable to superconductors.
In the time-dependent Hartree-Fock approximation, the fermions are assumed to interact only through the mean field and the collisions between particles are neglected. We formulate an extended time-dependent Hartree-Fock approximation which incorporates particle collisions due to the residual interaction, with the usual time-dependent Hartree-Fock approximation as the collisionless limit. It is obtained by a truncation of the Martin-Schwinger hierarchy at the second level and by using a simple representation of the one-body Green's function in terms of time-dependent occupation numbers. The final set of coupled equations consists of a modified time-dependent Hartree-Fock equation and a master equation for the occupation numbers. These results are physically transparent and describe properly the physics of the collision process. They may also be simple enough to be of practical use to study heavy-ion collisions or the dynamics of other fermion systems. Furthermore, as the configuration-space analog of the quantum Boltzmann equation, many important results concerning statistical dynamics are obtained. Concepts such as entropy, temperature, and local and thermal equilibrium can be quantitatively introduced. The well-known HHH theorem that entropy never decreases can be readily recovered. With the collision term explicitly exhibited, the macroscopic equations (equations of continuity, momentum flux, and energy) and their associated conservation theorems can also be derived. Analytic solutions for the master equation for simple cases lead to new "level crossing" formulas having characteristics distinctly different from the Landau-Zener level-crossing formula and illuminate the salient features as to how a nonequilibrium fermion system approaches thermal equilibrium.NUCLEAR REACTIONS Extension of time-dependent Hartree-Fock approximation. Collisions between particles. Master equation for occupation probabilities. Entropy, temperature, and thermal equilibrium. HHH theorem. Analytic solution of the master equation. New types of level crossing formula.
In this paper, we develop a self-consistent Hartree-Fock approach to theoretically study the far-from-equilibrium quantum dynamics of interacting fermions, and apply this approach to explore the onset of many-body localization (MBL) in these systems. We investigate the dynamics of a state with a nonequilibrium density profile; we find that at weak disorder the density profile equilibrates rapidly, whereas for strong disorder it remains frozen on the accessible timescales. We analyze this behavior in terms of the Hartree-Fock self-energy. At weak disorder, the self-energy fluctuates strongly and can be interpreted as a self-consistent noise process. By contrast, at strong disorder the self-energy evolves with a few coherent oscillations which cannot delocalize the system. Accordingly, the nonequilibrium site-resolved spectral function shows a broad spectrum at weak disorder and sharp spikes at strong disorder. Our Hartree-Fock theory incorporates spatial fluctuations and rare-region effects. As a consequence, we find subdiffusive relaxation in random systems; but, when the system is subjected to weak quasiperiodic potentials, the subdiffusive response ceases to exist, as rare region effects are absent in this case. This self-consistent Hartree-Fock approach can be regarded as a relatively simple theory that captures much of the MBL phenomenology.
Applying the master equation to an Ising model, equations of motion are obtained for the average of a single spin and for the average of a pair of spins. These equations relax to equilibrium values that predict a self-consistency relationship among the internal field, temperature, and exchange energy. The critical temperature obtained is that of the Bethe approximation, and is extended to include a static external field in the magnetization direction. With this, the relaxation of an Ising model near the Curie temperature has been investigated in a self-consistent manner. The results are similar to those of Suzuki and Kubo, but in the Bethe approximation instead of the Bragg-Williams approximation, and hence predict the relaxation of the correlation as well. In particular we studied a nearly Ising spin system that is a system with relatively small isotropic transverse coupling. An oscillating, linear, and external magnetic field applied transverse to the strongly coupled direction of the spins leads to lowering of the critical temperature.
Some model Hamiltonians are proposed for quantum-mechanical many-body systems with pair forces. In the case of an infinite system in thermal equilibrium, they lead to temperature-domain propagator expansions which are expressible by closed, formally exact equations. The expansions are identical with infinite subclasses of terms from the propagator expansion for the true many-body problem. The two principal models introduced correspond, respectively, to ring and ladder summations from the true propagator expansion, but augmented by infinite classes of self-energy corrections. The latter are expected to yield damping of single-particle excitations. The eigenvalues of the ring and ladder model Hamiltonians are real, and they are bounded from below if the pair potential obeys certain conditions. The models are formulated for fermions, bosons, and distinguishable particles. In addition to the ring and ladder models, two simpler types are discussed, one of which yields the Hartree-Fock approximation to the true problem. A novel feature of all the model Hamiltonians (except the Hartree-Fock) is that they contain an infinite number of parameters whose phases are fixed by random choices. Explicit closed expressions are obtained for the Helmholtz free energy of all the models in the classical limit.
A two-parameter class of single-particle orbitals, giving rise to long-range order in the local (spatial and/or spin) density, is shown to satisfy the full Hartree-Fock (HF) equations for occupied states in the thermodynamic limit. For a $\ensuremath{\delta}$ interparticle potential, these states are stabler (have lower HF energy) than the usual plane-wave (or trivial) HF solutions, for sufficiently strong coupling and/or high density. Minimization of the energy with respect to the (new) free parameters leads to sometimes gradual (second-order transition) and sometimes abrupt (first-order transition) onset of order, accompanied by a "bifurcation" of the new energy state from the old. The existence of even lower-energy, nontrivial HF states is also mentioned. We discuss the relevance to neutron and nuclear matter, to the Pauling "close-packed spheron" model of nuclei, as well as to the electron-gas problem.