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Symmetry-breaking Hartree--Fock solutions in a many-body system

MANUEL DE LLANO

Published Jun 1, 1973• 0 citations
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Abstract

Systems of N(>>1) fermions (or bosons) interacting via either repulsive long-ranged forces (e.g., Coulomb) or short-ranged forces (e.g., repulsive barrier plus attractive Yukawa tail) are clearly of interest in problems from nuclear physics to astrophysics. By comparing expectation values for determinants (or permanents) of plane-wave orbitals in the lowest kinetic-energy state, and Wannier-like orbitals localized in the points of a perfect lattice, the conditions on the coupling and particle density for the existence of energetically-preferred, spatially-periodic, nontrivial (i.e.. nonplane-wave) solutions to the Hartree - Fock equations were deduced. That an electron gas in a crystalline confirmation, the Wigner solid,'' has lower energy than the fluid (plane-waves) configuration at low densities (or all densities, if the charge is increased indefinitely) can be shown to imply the existence of lower-energy, spatially-periodic HF solutions. Analogously, for the short-ranged force case cited before one can show very simply that a determinant (or permanent) of nonover-lapping Wannier orbitals gives lower energy for high enough repulsive barrier in an intermediate range of densities (or all densities if the barrier height is increased indefinitely to the hard core limit) in three dimensions. Lower-energy non-trivial solutions therefore emerge under these conditions, giving a finite energy per particle evenmore » in the hard core limit. These results differ qualitatively from the one-dimensional case first studied by Overhauser and agree with those conjectured for 3-dimensions by several subsequent authors. The inclusion of attractive forces can, moreover, provide a condensed (negative energy) spatiallyperiodic state lower than the trivial state. One can furthermore show that there will exist a whole family of aperiodic singledeterminantal states of higher (but finite) energy than the perfect hep lattice state but lower still than the trivial state. (auth)« less

Related papers

Non-trivial solutions of the Hartree--Fock equations for a very-many- particle system

MANUEL DE LLANO

Work related to proofs for the existence of energetically preferred, non- trivial (mostly periodic-density) solutions to the Hartree -- Fock equations is reviewed. Both long- and short-ranged interacting systems of N-bosons or - fermions (N >> 1) are considered, and stability criteria discussed. Also, some exact results associated with trivial (plane-wave) solutions are surveyed. The main conclusion is that, for any fixed physical density, a sufficiently strong interparticle coupling can induce the appearance of non-trivial HF states. These solutions provide a first step in studying the properties of many-particle systems, which range from astrophysical plasmas and quantum crystals to nuclear matter and helium at very low temperature. (auth)

•Jan 1, 1973•0 citations

Hartree-Fock states in the thermodynamic limit

MANUEL DE LLANO, Angel Ricardo Plastino

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.

Physical Review A•Apr 1, 1976•23 citations

Hartree-Fock states in the thermodynamic limit. III. Low-density clustering effects

L. Döhnert, MANUEL DE LLANO, Angelo Plastino

We present an orthonormal set of single-particle orbital functions explicitly satisfying the Hartree-Fock (HF) equations for occupied states, and which give low-density few-particle clustering effects in a many-fermion system. The initial study given here is in one and three dimensions for attractive $\ensuremath{\delta}$ pair interactions. In one dimension the numerical results compare favorably with the exact (lowest) HF state at zero density. The results turn out to be superior to the classic Overhauser HF orbitals, as well as to some more recent ones.

Physical Review A•Feb 1, 1978•14 citations

Hartree-Fock states in the thermodynamic limit. II. Generalized Overhauser orbitals

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Two infinite families of two-parameter generalized Overhauser orbitals are introduced and shown to satisfy explicitly, for occupied states, the self-consistent Hartree-Fock equations in the thermodynamic limit. For an attractive $\ensuremath{\delta}$ interaction, they give lower Hartree-Fock energy than the usual plane-wave solutions, even for relatively weak coupling and/or low density. The limiting members (possessing an infinite number of harmonics) of both families appear to tend to a "classical static lattice" state, via a second-order transition for one family and via first order for the other. The related density profiles and energy expressions are calculated as functions of the two new parameters. A direct variation with respect to these parameters was done numerically and results are presented graphically.

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Generalized Fermi sea for plane-wave Hartree-Fock theory. II. Semi-realistic interactions in three dimensions

Valdir C. Aguilera-Navarro, Robert Hartman Belehrad, MANUEL DE LLANO +3 more

The possibility of abnormally occupying plane-wave Hartree-Fock determinants is studied analytically for several many-fermion systems interacting with short- and long-ranged pair potentials in three dimensions. Lower energy states are found in some cases, but they always depend sensitively on the shape of the potential as well as on the amount of space exchange present in the interaction.NUCLEAR STRUCTURE Plane-wave Hartree-Fock, nuclear matter.

Physical Review C•Sep 1, 1980•11 citations

Low-density Hartree-Fock solutions for a one-dimensional gas with attractive delta-function interactions

Gonzalo Gutiérrez, Angel Ricardo Plastino, G. Zannoli

New Hartree-Fock (HF) solutions are presented for a one-dimensional gas with attractive delta-function interactions. They are found by recourse to the variational principle and have, as special limiting cases, both the exact, zero-density HF solution and the plane-wave orbitals.

Physical Review A•Mar 1, 1981•0 citations
Systems of N(>>1) fermions (or bosons) interacting via either repulsive long-ranged forces (e.g., Coulomb) or short-ranged forces (e.g., repulsive barrier plus attractive Yukawa tail) are clearly of interest in problems from nuclear physics to astrophysics. By comparing expectation values for determinants (or permanents) of plane-wave orbitals in the lowest kinetic-energy state, and Wannier-like orbitals localized in the points of a perfect lattice, the conditions on the coupling and particle density for the existence of energetically-preferred, spatially-periodic, nontrivial (i.e.. nonplane-wave) solutions to the Hartree - Fock equations were deduced. That an electron gas in a crystalline confirmation, the Wigner solid,'' has lower energy than the fluid (plane-waves) configuration at low densities (or all densities, if the charge is increased indefinitely) can be shown to imply the existence of lower-energy, spatially-periodic HF solutions. Analogously, for the short-ranged force case cited before one can show very simply that a determinant (or permanent) of nonover-lapping Wannier orbitals gives lower energy for high enough repulsive barrier in an intermediate range of densities (or all densities if the barrier height is increased indefinitely to the hard core limit) in three dimensions. Lower-energy non-trivial solutions therefore emerge under these conditions, giving a finite energy per particle evenmore » in the hard core limit. These results differ qualitatively from the one-dimensional case first studied by Overhauser and agree with those conjectured for 3-dimensions by several subsequent authors. The inclusion of attractive forces can, moreover, provide a condensed (negative energy) spatiallyperiodic state lower than the trivial state. One can furthermore show that there will exist a whole family of aperiodic singledeterminantal states of higher (but finite) energy than the perfect hep lattice state but lower still than the trivial state. (auth)« less
Work related to proofs for the existence of energetically preferred, non- trivial (mostly periodic-density) solutions to the Hartree -- Fock equations is reviewed. Both long- and short-ranged interacting systems of N-bosons or - fermions (N >> 1) are considered, and stability criteria discussed. Also, some exact results associated with trivial (plane-wave) solutions are surveyed. The main conclusion is that, for any fixed physical density, a sufficiently strong interparticle coupling can induce the appearance of non-trivial HF states. These solutions provide a first step in studying the properties of many-particle systems, which range from astrophysical plasmas and quantum crystals to nuclear matter and helium at very low temperature. (auth)
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.
We present an orthonormal set of single-particle orbital functions explicitly satisfying the Hartree-Fock (HF) equations for occupied states, and which give low-density few-particle clustering effects in a many-fermion system. The initial study given here is in one and three dimensions for attractive $\ensuremath{\delta}$ pair interactions. In one dimension the numerical results compare favorably with the exact (lowest) HF state at zero density. The results turn out to be superior to the classic Overhauser HF orbitals, as well as to some more recent ones.
Two infinite families of two-parameter generalized Overhauser orbitals are introduced and shown to satisfy explicitly, for occupied states, the self-consistent Hartree-Fock equations in the thermodynamic limit. For an attractive $\ensuremath{\delta}$ interaction, they give lower Hartree-Fock energy than the usual plane-wave solutions, even for relatively weak coupling and/or low density. The limiting members (possessing an infinite number of harmonics) of both families appear to tend to a "classical static lattice" state, via a second-order transition for one family and via first order for the other. The related density profiles and energy expressions are calculated as functions of the two new parameters. A direct variation with respect to these parameters was done numerically and results are presented graphically.