Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:cond-mat/0508750 · Physical Review B

Many-body density matrices on a two-dimensional square lattice: Noninteracting and strongly interacting spinless fermions

Siew Ann Cheong, Christopher L. Henley

Published Oct 31, 2006• 4 citations
arXivHTMLPDF

Abstract

The reduced density matrix of an interacting system can be used as the basis for a truncation scheme, or in an unbiased method to discover the strongest kind of correlation in the ground state. In this paper, we investigate the structure of the many-body fermion density matrix of a small cluster in a square lattice. The cluster density matrix is evaluated numerically over a set of finite systems, subject to nonsquare periodic boundary conditions given by the lattice vectors ${\mathbf{R}}_{1}\ensuremath{\equiv}({R}_{1x},{R}_{1y})$ and ${\mathbf{R}}_{2}\ensuremath{\equiv}({R}_{2x},{R}_{2y})$. We then approximate the infinite-system cluster density-matrix spectrum by averaging the finite-system cluster density matrix (i) over degeneracies in the ground state, and orientations of the system relative to the cluster, to ensure it has the proper point-group symmetry; and (ii) over various twist boundary conditions to reduce finite size effects. We then compare the eigenvalue structure of the averaged cluster density matrix for noninteracting and strongly interacting spinless fermions, as a function of the filling fraction $\overline{n}$, and discuss whether it can be approximated as being built up from a truncated set of single-particle operators.

DOI
10.1103/physrevb.74.165121

Related papers

Operator-based truncation scheme based on the many-body fermion density matrix

Siew Ann Cheong, Christopher L. Henley

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.

arXiv:cond-mat/0307172•Physical Review B•Feb 27, 2004•28 citations

Density-matrix functional theory of the Hubbard model: An exact numerical study

Román López-Sandoval, G. M. Pastor

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 ${N}_{a},$ band filling ${N}_{e},$ 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 ${N}_{a},$ ${N}_{e},$ 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 $(U>0)$ and attractive $(U<0)$ interactions are contrasted.

arXiv:cond-mat/9910264•Physical review. B, Condensed matter•Jan 15, 2000•45 citations

Density-matrix based numerical methods for discovering order and correlations in interacting systems

Christopher L. Henley, Hitesh J. Changlani

We review recently introduced numerical methods for the unbiased detection of the order parameter and/or dominant correlations in many-body interacting systems, by using reduced density matrices. Most of the paper is devoted to the ‘quasi-degenerate density matrix’ (QDDM) which is rooted in Anderson's observation that the degenerate symmetry-broken states valid in the thermodynamic limit are manifested in finite systems as a set of low-energy ‘quasi-degenerate’ states (in addition to the ground state). This method, its original form due to Furukawa et al (2006 Phys. Rev. Lett. 96 047211 ), is given a number of improvements here, above all the extension from two-fold symmetry breaking to arbitrary cases. This is applied to two test cases: (1) interacting spinless hardcore bosons on the triangular lattice and (2) a spin-1/2 antiferromagnetic system at the percolation threshold. In addition, we survey a different method called the ‘correlation density matrix’, which detects (possibly long-range) correlations only from the ground state, but using the reduced density matrix from a cluster consisting of two spatially separated regions.

arXiv:1407.4189•Journal of Statistical Mechanics Theory and Experiment•Nov 6, 2014•9 citations

Expansion of the Density Matrix of an N-Fermion System in Terms of the Correlation Densities of Fermion Clusters

Tai‐ichi Shibuya, Oktay Si̇nanoğlu

The N-fermion density operator D = |Ψ〈〉Ψ| is decomposed into the densities of clusters of correlating fermions. The m-body cluster densities γ̂(m) are ``orthogonal'' to the one-body ``Fock-Dirac''-type densities ρ1, i.e., Γr1(ρ1γ̂1,2,(m)…,m)=0. The reduced-correlation-density matrices obtained differ from the conventional reduced-density-matrices and are particularly convenient for the treatment of fermion-fermion correlations.

Journal of Mathematical Physics•Jun 1, 1969•4 citations

Model derived reduced density matrix restrictions for correlated fermions

Joel H. Sebold, Jerome K. Percus

We illustrate the importance of restrictions in improving ground state energy lower bounds of a model of correlated electrons on a lattice. A reduced density matrix (RDM) formalism is employed. The restrictions are derived from closely related and exactly solved models. Such conditions raise the estimates without resorting to increasing the size of the physical space, thus improving computational efficiency. Our main motivation for this work is the problematic picture of Hohenberg–Kohn–Sham density functional theory for strongly correlated fermions. We find that using small cluster representations, errors can be reduced by more than 50% depending on the nature of the model and parameter regime studied. We obtain results for one- and two-dimensional lattices at half filling in the thermodynamic limit, although the method could be easily adapted to finite molecular structures as well.

The Journal of Chemical Physics•May 1, 1996•6 citations

Many body density of states of a system of non interacting spinless fermions

R. Lefèvre, Krissia Zawadzki, Grégoire Ithier

Abstract The modeling of out-of-equilibrium many-body quantum systems requires to go beyond low-energy physics and single or few bodies densities of states. Many-body localization, presence or lack of thermalization and quantum chaos are examples of phenomena in which states at different energy scales, including highly excited ones, contribute to dynamics and therefore affect the system’s properties. Quantifying these contributions requires the many-body density of states (MBDoS), a function whose calculation becomes challenging even for non-interacting identical particles due to the difficulty to enumerate accessible states while enforcing the exchange symmetry. In the present work, we introduce a new approach to evaluate the MBDoS in the general case of non-interacting systems of identical quantum particles. The starting point of our method is the principal component analysis of a filling matrix F describing how N particles can be distributed into L single-particle energy levels. We show that the many body spectrum can be expanded as a weighted sum of singular vectors of the filling matrix. The weighting coefficients only involve renormalized energies obtained from the single body spectrum. We illustrate our method in two classes of problems that are mapped into spinless fermions : (i) non-interacting electrons in a homogeneous tight-binding model in 1D and 2D, and (ii) interacting spins in a chain under a transverse field.

arXiv:2208.02236•New Journal of Physics•May 25, 2023•3 citations
The reduced density matrix of an interacting system can be used as the basis for a truncation scheme, or in an unbiased method to discover the strongest kind of correlation in the ground state. In this paper, we investigate the structure of the many-body fermion density matrix of a small cluster in a square lattice. The cluster density matrix is evaluated numerically over a set of finite systems, subject to nonsquare periodic boundary conditions given by the lattice vectors R1\ensuremath≡(R1x,R1y){\mathbf{R}}_{1}\ensuremath{\equiv}({R}_{1x},{R}_{1y})R1​\ensuremath≡(R1x​,R1y​) and R2\ensuremath≡(R2x,R2y){\mathbf{R}}_{2}\ensuremath{\equiv}({R}_{2x},{R}_{2y})R2​\ensuremath≡(R2x​,R2y​). We then approximate the infinite-system cluster density-matrix spectrum by averaging the finite-system cluster density matrix (i) over degeneracies in the ground state, and orientations of the system relative to the cluster, to ensure it has the proper point-group symmetry; and (ii) over various twist boundary conditions to reduce finite size effects. We then compare the eigenvalue structure of the averaged cluster density matrix for noninteracting and strongly interacting spinless fermions, as a function of the filling fraction n‾\overline{n}n, and discuss whether it can be approximated as being built up from a truncated set of single-particle 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.
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.
We review recently introduced numerical methods for the unbiased detection of the order parameter and/or dominant correlations in many-body interacting systems, by using reduced density matrices. Most of the paper is devoted to the ‘quasi-degenerate density matrix’ (QDDM) which is rooted in Anderson's observation that the degenerate symmetry-broken states valid in the thermodynamic limit are manifested in finite systems as a set of low-energy ‘quasi-degenerate’ states (in addition to the ground state). This method, its original form due to Furukawa et al (2006 Phys. Rev. Lett. 96 047211 ), is given a number of improvements here, above all the extension from two-fold symmetry breaking to arbitrary cases. This is applied to two test cases: (1) interacting spinless hardcore bosons on the triangular lattice and (2) a spin-1/2 antiferromagnetic system at the percolation threshold. In addition, we survey a different method called the ‘correlation density matrix’, which detects (possibly long-range) correlations only from the ground state, but using the reduced density matrix from a cluster consisting of two spatially separated regions.
We illustrate the importance of restrictions in improving ground state energy lower bounds of a model of correlated electrons on a lattice. A reduced density matrix (RDM) formalism is employed. The restrictions are derived from closely related and exactly solved models. Such conditions raise the estimates without resorting to increasing the size of the physical space, thus improving computational efficiency. Our main motivation for this work is the problematic picture of Hohenberg–Kohn–Sham density functional theory for strongly correlated fermions. We find that using small cluster representations, errors can be reduced by more than 50% depending on the nature of the model and parameter regime studied. We obtain results for one- and two-dimensional lattices at half filling in the thermodynamic limit, although the method could be easily adapted to finite molecular structures as well.
Abstract The modeling of out-of-equilibrium many-body quantum systems requires to go beyond low-energy physics and single or few bodies densities of states. Many-body localization, presence or lack of thermalization and quantum chaos are examples of phenomena in which states at different energy scales, including highly excited ones, contribute to dynamics and therefore affect the system’s properties. Quantifying these contributions requires the many-body density of states (MBDoS), a function whose calculation becomes challenging even for non-interacting identical particles due to the difficulty to enumerate accessible states while enforcing the exchange symmetry. In the present work, we introduce a new approach to evaluate the MBDoS in the general case of non-interacting systems of identical quantum particles. The starting point of our method is the principal component analysis of a filling matrix F describing how N particles can be distributed into L single-particle energy levels. We show that the many body spectrum can be expanded as a weighted sum of singular vectors of the filling matrix. The weighting coefficients only involve renormalized energies obtained from the single body spectrum. We illustrate our method in two classes of problems that are mapped into spinless fermions : (i) non-interacting electrons in a homogeneous tight-binding model in 1D and 2D, and (ii) interacting spins in a chain under a transverse field.