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Physical Review A

Further comment on “Eigenvalue spectrum of the independent-fermion kinetic-energy kernel”

Jerome K. Percus

Published Sep 1, 1999• 6 citations
Publisher

Abstract

When a fermion-system ground state is examined in the grand ensemble, particle density is determined uniquely by local chemical potential except when an additional fermion is in the process of appearing. Conversely, the chemical potential is uniquely determined by the density profile except when there are an integer number of particles, but is then necessarily linear in a two-state mixing parameter. This linear dependence has immediate consequences, one of which has elicited prior commentary.

DOI
10.1103/physreva.60.2601

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Comment on the paper “Ground state instability in systems of strongly interacting fermions”

VICTOR A. KHODEL

Journal of Experimental and Theoretical Physics Letters•Feb 1, 1999•0 citations
When a fermion-system ground state is examined in the grand ensemble, particle density is determined uniquely by local chemical potential except when an additional fermion is in the process of appearing. Conversely, the chemical potential is uniquely determined by the density profile except when there are an integer number of particles, but is then necessarily linear in a two-state mixing parameter. This linear dependence has immediate consequences, one of which has elicited prior commentary.
This thesis encompasses a number of problems related to the number fluctuations from the ground state of ideal particles in different statistical ensembles. In the microcanonical ensemble most of these problems may be solved using number theory. Given an energy E, the well-known problem of finding the number of ways of distributing N bosons over the excited levels of a one-dimensional harmonic spectrum, for instance, is equivalent to the number of restricted partitions of E. As a result, the number fluctuation from the ground state in the microcanonical ensemble for this system may be found analytically. When the particles are fermions instead of bosons, however, it is difficult to calculate the exact ground state number fluctuation because the fermionic ground state consists of many levels. By breaking up the energy spectrum into particle and hole sectors, and mapping the problem onto the classic number partitioning theory, we formulate a method of calculating the particle number fluctuation from the ground state in the microcanonical ensemble for fermions. The same quantity is calculated for particles interacting via an inverse-square pairwise interaction in one dimension. In the canonical ensemble, an analytical formula for the ground state number fluctuation is obtained by using the mapping of this system onto a system of noninteracting particles obeying the Haldane-Wu exclusion statistics. In the microcanonical ensemble, however, the result can be obtained only for a limited set of values of the interacting strength parameter. Usually, for a discrete set of a mean-field single-article quantum spectrum and in the micro canonical ensemble, there are many combinations of exciting particles from the ground state. The spectrum given by the logarithms of the prime number sequence, however, is a counterexample to this rule. Here, as a consequence of the fundamental theorem of arithmetic, there is a one-to-one correspondence between the microstate and the macrostate, resulting in the vanishing of number fluctuation for all excitations. The use of the canonical or grand canonical ensembles, on the other hand, gives a substantial number fluctuation from the ground state. For a related spectrum, that given by the logarithms of an integer n, the microcanonical number fluctuation is non-zero but the application of the other ensembles is still not valid. These two spectra are examples of systems where canonical and grand canonical ensembles averagings yield answers different from the microcanonical result. Some models in physics may be used to obtain formulae known in the theory of number partition. For the same problem of N ideal bosons in a one-dimensional harmonic oscillator potential mentioned earlier, it is well known that the asymptotic (N --> ∞) density of states is identical to the Hardy-Ramanujan formula for the number of partitions of an integer n. The same statistical mechanics technique for the density of states of bosons in a power-law spectrum yields the partitioning formula for the number of partitions of n into a sum of 8th powers of a set of integers. By considering only the particle sector of the fermionic spectrum, a formula for the number of distinct partitions of n is obtained. For the s = 1 case and for finite N, the Erdos-Lehner formula for the restricted partitions, and a new formula for the distinct and restricted partitions are derived. As a diversion, we discuss the microcanonical entropy which may be uniquely defined in terms of the macrostate, or equivalently the many-body degeneracy of the state, at a given energy. The many-body degeneracy factor, however, is exceedingly difficult to calculate in general. It is thus desirable to find a different way to calculate the micro canonical entropy. It has been recently suggested that the microcanonical entropy may be accurately reproduced by including a logarithmic correction to the canonical entropy. This claim is readily tested using some of the models mentioned above, where the many-body degeneracy may be determined exactly. In addition, we also consider a system of N distinguishable particles in a d-dimensional harmonic energy spectrum. In this case the many-body degeneracy factor can be obtained analytically in a closed form.
That the external potential ${v}_{\mathit{ext}}(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})$ of a system of electrons is determined uniquely by the ground-state density is one of the central statements of the first Hohenberg-Kohn theorem. It is known that the validity of this statement extends to densities $n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})$ with noninteger particle number [i.e., $n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})$ integrates to a number that is not an integer] if the functional derivative of ${T}_{s}[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})]+U[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})]+{E}_{xc}[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})]$ exists or (without relying on the existence of functional derivatives) if the ground-state energy is a strictly convex function of the particle number. In the present article, a proof that relies neither on the existence of the above functional derivative nor on the strict convexity of the ground-state energy is presented. The fact that the density determines the external potential leads to a noncrossing theorem for ground-state densities. The noncrossing theorem produces knowledge as to what the integer-particle-number ground-state densities of a system cannot be. The noncrossing theorem produces inequalities that the functional derivatives of the exchange-correlation energy functional ${E}_{xc}[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})]$ and the noninteracting kinetic energy functional ${T}_{s}[n(\stackrel{P\vec}{r}\phantom{\rule{0.1em}{0ex}})]$ must fulfill.
The constrained minimization independent-fermion kinetic-energy kernel, ${\mathrm{\ensuremath{\delta}}}^{2}$Ts{\mathit{T}}_{\mathit{s}}Ts​[\ensuremath{\rho}]/\ensuremath{\delta}\ensuremath{\rho}(r)\ensuremath{\delta}\ensuremath{\rho}(${\mathbf{r}}^{\ensuremath{'}}$), has a zero mode for all \ensuremath{\rho}(r), while it is non-negative for \ensuremath{\rho}(r) noninteracting v representable. \textcopyright{} 1996 The American Physical Society.
Spurious zero-dispersion continua in full (2,q)(2,q)(2,q)-positivity: A linear hierarchy-depth lower bound from an exactly solvable counting operator
Variational two-particle reduced-density-matrix methods impose hierarchies of semidefinite constraints. We ask whether a fixed level of such a hierarchy can recover even an elementary many-fermion spectrum, using the dispersion test, in which a value is accepted as an eigenvalue when the minimized dispersion vanishes. For an exactly solvable operator that counts fermions in one half of the spin orbitals, whose spectrum consists of integers, we determine the complete set of accepted values at every level of the full hierarchy of particle-hole semidefinite constraints. Besides the integers, this set contains an entire interval of noninteger values unless the hierarchy level grows linearly with the particle number, and we give the exact threshold. The proof constructs explicit pseudo reduced density matrices that are diagonal in the occupation basis, obtained by continuing the occupation probabilities of a fixed-particle-number state to a noninteger particle number. Classical semidefinite constraints on occupation correlations, supplemented by nonnegative occupation-pattern probabilities, suffice to establish fermionic positivity. For this counting model the two relaxations have the same dispersion zero set, not necessarily the same feasible states, and a converse argument shows that no other false eigenvalues occur. The limitation is specific to the hierarchy: two-body inequalities that encode the integer spectrum would remove every false value but are not generated at a fixed level. We also verify that the full semidefinite constraints and their complete projection onto two-particle reduced-density-matrix space are equivalent in finite dimensions, so that the result applies equally to the polar formulation.