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Reports on Mathematical Physics

Relation between the range of an elementary 3-particle 2-body operator and the convex structure of the set of fermion 3-representable 2-density operators

Hubert Grudziński

Published Jun 1, 2009• 0 citations
Publisher
DOI
10.1016/s0034-4877(09)90013-1

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In an earlier work [S. A. Cheong and C. L. Henley, preceding paper], we derived an exact formula for the many-body density matrix ${\ensuremath{\rho}}_{B}$ of a block of B sites cut out from an infinite chain of noninteracting spinless fermions, and found that the many-particle eigenvalues and eigenstates of ${\ensuremath{\rho}}_{B}$ can all be constructed out of the one-particle eigenvalues and one-particle eigenstates, respectively. In this paper we improved upon this understanding, and developed a statistical-mechanical analogy between the density-matrix eigenstates and the many-body states of a system of noninteracting fermions. Each density-matrix eigenstate corresponds to a particular set of occupation of single-particle pseudoenergy levels, and the density-matrix eigenstate with the largest weight, having the structure of a Fermi sea ground state, unambiguously defines a pseudo-Fermi level. Based on this analogy, we outlined the main ideas behind an operator-based truncation of the density-matrix eigenstates, where single-particle pseudoenergy levels far away from the pseudo-Fermi level are removed as degrees of freedom. We report numerical evidence for scaling behaviors in the single-particle pseudoenergy spectrum for different block sizes B and different filling fractions $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}.$ With the aid of these scaling relations, which tell us that the block size B plays the role of an inverse temperature in the statistical-mechanical description of the density-matrix eigenstates and eigenvalues, we looked into the performance of our operator-based truncation scheme in minimizing the discarded density-matrix weight and the error in calculating the dispersion relation for elementary excitations. This performance was compared against that of the traditional density-matrix-based truncation scheme, as well as against an operator-based plane-wave truncation scheme, and found to be very satisfactory.

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In [J. Math. Phys. 13, 1608–1621 (1972)], Erdahl10.1063/1.1665885 considered the convex structure of the set of N-representable 2-body reduced density matrices in the case of fermions. Some of these results have a straightforward extension to the m-body setting and to the more general quantum marginal problem. We describe these extensions, but cannot resolve a problem in the proof of Erdahl's claim that every extreme point is exposed in finite dimensions. Nevertheless, we can show that when 2m ⩾ N every extreme point of the set of N-representable m-body reduced density matrices has a unique pre-image in both the symmetric and anti-symmetric setting. Moreover, this extends to the quantum marginal setting for a pair of complementary m-body and (N − m)-body reduced density matrices.
Abstract N‐representability conditions for a two‐particle density operator implied by positive‐semidefiniteness of the projection operator PN+1(ϕ1 Λ ΨN) are derived and discussed. The operator PN+1(ϕ1 Λ ΨN) projects onto an (N + 1)‐particle antisymmetric function ϕ1 Λ ΨN, the Grassmann product of a one‐particle factor ϕ1 and an N‐particle factor ΨN. The polar subcone 𝒫2N(g, q) to the set of N‐representable two‐particle density operators 𝒫2N which corresponds to these conditions is found. It is shown that its extreme rays belong to two orbits for the action of the unitary group of transformations in one‐particle Hilbert space. The facial structure of the convex set 𝒫2N exposed by elements of 𝒫2N(g, q) is analyzed. An example of the operator that changes the structure of its bottom eigenspace when the number of fermions N surpasses a certain value is noted. A new approach to the diagonal conditions for N‐representability is found. It consists of the decomposition of the N‐particle antisymmetric identity operator onto the mutually orthogonal projection operators.
In an earlier work [S. A. Cheong and C. L. Henley, preceding paper], we derived an exact formula for the many-body density matrix ${\ensuremath{\rho}}_{B}$ of a block of B sites cut out from an infinite chain of noninteracting spinless fermions, and found that the many-particle eigenvalues and eigenstates of ${\ensuremath{\rho}}_{B}$ can all be constructed out of the one-particle eigenvalues and one-particle eigenstates, respectively. In this paper we improved upon this understanding, and developed a statistical-mechanical analogy between the density-matrix eigenstates and the many-body states of a system of noninteracting fermions. Each density-matrix eigenstate corresponds to a particular set of occupation of single-particle pseudoenergy levels, and the density-matrix eigenstate with the largest weight, having the structure of a Fermi sea ground state, unambiguously defines a pseudo-Fermi level. Based on this analogy, we outlined the main ideas behind an operator-based truncation of the density-matrix eigenstates, where single-particle pseudoenergy levels far away from the pseudo-Fermi level are removed as degrees of freedom. We report numerical evidence for scaling behaviors in the single-particle pseudoenergy spectrum for different block sizes B and different filling fractions $\mathrm{n\ifmmode \bar{}\else \={}\fi{}}.$ With the aid of these scaling relations, which tell us that the block size B plays the role of an inverse temperature in the statistical-mechanical description of the density-matrix eigenstates and eigenvalues, we looked into the performance of our operator-based truncation scheme in minimizing the discarded density-matrix weight and the error in calculating the dispersion relation for elementary excitations. This performance was compared against that of the traditional density-matrix-based truncation scheme, as well as against an operator-based plane-wave truncation scheme, and found to be very satisfactory.