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Physical Review B

Many-body density matrices for free fermions

Siew Ann Cheong, Christopher L. Henley

Published Feb 27, 2004• 163 citations
Publisher

Abstract

Building upon an analytical technique introduced by Chung and Peschel [Phys. Rev. B 64, 064412 (2001)], we calculated the many-body density matrix ${\ensuremath{\rho}}_{B}$ of a finite block of B sites within an infinite system of free spinless fermions in arbitrary dimensions. In terms of the block Green function matrix G (whose elements are ${G}_{\ifmmode \bar{\imath}\else \={\i}\fi{}j}=〈{c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j}〉,$ where ${c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}$ and ${c}_{j}$ are fermion creation and annihilation operators acting on sites i and j within the block, respectively), the density matrix can be written as ${\ensuremath{\rho}}_{B}=\mathrm{det}(1\ensuremath{-}G)\mathrm{exp}({\ensuremath{\sum}}_{\mathrm{ij}}[\mathrm{ln}G(1\ensuremath{-}{G)}^{\ensuremath{-}1}{]}_{\mathrm{ij}}{c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j}).$ Our results suggest that Hilbert space truncation schemes should retain the states created by a subset of the ${c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}$'s (in any combination), rather than selecting eigenvectors of ${\ensuremath{\rho}}_{B}$ independently based on the eigenvalue.

DOI
10.1103/physrevb.69.075111

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Building upon an analytical technique introduced by Chung and Peschel [Phys. Rev. B 64, 064412 (2001)], we calculated the many-body density matrix \ensuremathρB{\ensuremath{\rho}}_{B}\ensuremathρB​ of a finite block of B sites within an infinite system of free spinless fermions in arbitrary dimensions. In terms of the block Green function matrix G (whose elements are G\ifmmodeıˉ\else\iˉ\fij=〈ci\ifmmode†\else\textdagger\ficj〉,{G}_{\ifmmode \bar{\imath}\else \={\i}\fi{}j}=〈{c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j}〉,G\ifmmodeˉ\else\iˉ​\fij​=〈ci\ifmmode†\else\textdagger\fi​cj​〉, where ci\ifmmode†\else\textdagger\fi{c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}ci\ifmmode†\else\textdagger\fi​ and cj{c}_{j}cj​ are fermion creation and annihilation operators acting on sites i and j within the block, respectively), the density matrix can be written as \ensuremathρB=det(1\ensuremath−G)exp(\ensuremath∑ij[lnG(1\ensuremath−G)\ensuremath−1]ijci\ifmmode†\else\textdagger\ficj).{\ensuremath{\rho}}_{B}=\mathrm{det}(1\ensuremath{-}G)\mathrm{exp}({\ensuremath{\sum}}_{\mathrm{ij}}[\mathrm{ln}G(1\ensuremath{-}{G)}^{\ensuremath{-}1}{]}_{\mathrm{ij}}{c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{c}_{j}).\ensuremathρB​=det(1\ensuremath−G)exp(\ensuremath∑ij​[lnG(1\ensuremath−G)\ensuremath−1]ij​ci\ifmmode†\else\textdagger\fi​cj​). Our results suggest that Hilbert space truncation schemes should retain the states created by a subset of the ci\ifmmode†\else\textdagger\fi{c}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}ci\ifmmode†\else\textdagger\fi​'s (in any combination), rather than selecting eigenvectors of \ensuremathρB{\ensuremath{\rho}}_{B}\ensuremathρB​ independently based on the eigenvalue.
Building upon an analytical technique introduced by Chung and Peschel [M. Chung and I. Peschel, Phys. Rev. B 64, art. 064412 (2001)], we calculated the density matrix rho_B of a finite block of B sites within an infinite system of free spinless fermions. In terms of the block Green function matrix G (whose elements are G_ij = < c_i^+ c_j>, where c_i^+ and c_j are fermion creation and annihilation operators acting on sites i and j within the block respectively), the density matrix can be written as rho_B = det(1 - G) exp[ sum_ij (log G(1 - G^{-1})_ij c_i^+ c_j]. Implications of such a result to Hilbert space truncation for real-space renormalization schemes is discussed.
A density-functional theory for many-body lattice models is considered in which the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ is the basic variable. Eigenvalue equations are derived for solving Levy's constrained search of the interaction energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is expressed as the sum of Hartree-Fock energy ${E}_{\mathrm{HF}}[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ and the correlation energy ${E}_{C}[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ Exact results are obtained for ${E}_{C}({\ensuremath{\gamma}}_{12})$ of the Hubbard model on various periodic lattices, where ${\ensuremath{\gamma}}_{\mathrm{ij}}={\ensuremath{\gamma}}_{12}$ for all nearest neighbors i and j. The functional dependence of ${E}_{C}({\ensuremath{\gamma}}_{12})$ is analyzed by varying the number of sites Na,{N}_{a},Na​, band filling Ne,{N}_{e},Ne​, and lattice structure. The infinite one-dimensional chain and one-, two-, or three-dimensional finite clusters with periodic boundary conditions are considered. The properties of ${E}_{C}({\ensuremath{\gamma}}_{12})$ are discussed in the limits of weak $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{0})$ and strong $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$ electronic correlations, and in the crossover region $({\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}<~{\ensuremath{\gamma}}_{12}<~{\ensuremath{\gamma}}_{12}^{0}).$ Using an appropriate scaling we observe that ${\ensuremath{\varepsilon}}_{C}{(g}_{12}{)=E}_{C}{/E}_{\mathrm{HF}}$ has a pseudo-universal behavior as a function of ${g}_{12}=({\ensuremath{\gamma}}_{12}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})/({\ensuremath{\gamma}}_{12}^{0}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}).$ The fact that ${\ensuremath{\varepsilon}}_{C}{(g}_{12})$ depends weakly on Na,{N}_{a},Na​, Ne,{N}_{e},Ne​, and lattice structure suggests that the correlation energy of extended systems could be obtained quite accurately from finite-cluster calculations. Finally, the behaviors of ${E}_{C}({\ensuremath{\gamma}}_{12})$ for repulsive (U>0)(U>0)(U>0) and attractive (U<0)(U<0)(U<0) interactions are contrasted.
The Liouville resolvent method is an unconventional technique used for finding a Green function for a Hamiltonian. Implementation of the method entails the calculation of commutators of a second-quantized Hamiltonian operator with particular generalized stepping operators that are elements of a Hilbert space and that represent transitions between many-particle states. These commutators produce linear combinations of stepping operators, so the results can be arrayed as matrix elements of the Liouville operator L̂ in the Hilbert space of stepping operators. The resulting L̂ matrix is usually of infinite order, and in principle its eigenvalues and eigenvectors can be used to construct the Green function from the L̂ resolvent matrix. Approximations are usually necessary, at least in the form of truncation of the L̂ matrix, and if one produces a sequence of such matrices of increasing order and calculates the eigenvalues and eigenvectors of these matrices, a sequence of approximations for the L̂ resolvent matrix can be produced. This sequence is mathematically guaranteed to converge to the exact result for the L̂ resolvent matrix (except at its singularities). The accuracy of an approximation depends on the order of the matrix at which the sequence is truncated. Application of the method to a Hamiltonian representing interactions between fermions and bosons involves complications arising from the large number of terms generated by the commutation properties of boson operators. This dissertation describes the method and its use in the study of fermion-boson couplings. Approximations to second order in stepping operators are calculated for simplified Froehlich and Lee models. Limited thermodynamic results are obtained from the Lee model. Exact energy eigenvalues are obtained by operator algebra for simplified Froehlich, Lee and Dirac models. These exact solutions comprise the main contribution of this research and will prove to be valuable starting points for further research. Suggestions are made for further research.
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.