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

Physical Review A

Boson correlation energies and density matrices from reduced Hamiltonian interpolation

David A. Mazziotti, Dudley R. Herschbach

Published Sep 1, 2000• 16 citations
Publisher

Abstract

The ground-state energies of interacting bosons are computed beyond the mean-field approximation by a new method known as reduced Hamiltonian interpolation (RHI) [D. A. Mazziotti and D. R. Herschbach, Phys. Rev. Lett. 83, 5185 (1999)]. In the RHI, the N-particle Hamiltonian is represented through a sequence of p-particle expanded and reduced Hamiltonians that give upper and lower bounds on the true energy. Assimilating ideas from N representability and dimensional interpolation, the RHI technique interpolates over the number p of quasiparticles to determine the N-particle energy with close upper and lower bounds. With the Hellmann-Feynman theorem, we extend the RHI to compute the two-particle reduced density matrix (2RDM) as well as the energy. We examine both the computational advantages of RHI in comparison with traditional methods and the possibility of extending the RHI to treat fermion correlation. Applied to bosons with harmonic interactions as well as a two-level system, the RHI technique yields more than 99% of the correlation energy and an accurate correlation correction for the elements of the 2RDM.

DOI
10.1103/physreva.62.043603

Related papers

Boson Correlation Energies from Reduced Hamiltonian Interpolation

David A. Mazziotti, Dudley R. Herschbach

The ground-state energies of interacting bosons are computed beyond the mean-field approximation by a new method which we call reduced Hamiltonian interpolation (RHI). In this interpolation the $N$-particle Hamiltonian is represented through a sequence of $p$-particle expanded and reduced Hamiltonians that give upper and lower bounds on the true energy. A synthesis of ideas from $N$-representability and dimensional interpolation, the RHI interpolates over the number $p$ of quasiparticles (equivalent to spatial dimension) to calculate the $N$-particle energy as the mean of close upper and lower bounds. Application to bosons with harmonic interactions yields more than $99%$ of the correlation energy.

Physical Review Letters•Dec 20, 1999•19 citations

Uncertainty relations and reduced density matrices: Mapping many-body quantum mechanics onto four particles

David A. Mazziotti, Robert Erdahl

For the description of ground-state correlation phenomena an accurate mapping of many-body quantum mechanics onto four particles is developed. The energy for a quantum system with no more than two-particle interactions may be expressed in terms of a two-particle reduced density matrix (2-RDM), but variational optimization of the 2-RDM requires that it corresponds to an N-particle wave function. We derive N-representability conditions on the 2-RDM that guarantee the validity of the uncertainty relations for all operators with two-particle interactions. One of these conditions is shown to be necessary and sufficient to make the RDM solutions of the dispersion condition equivalent to those from the contracted Schr\"odinger equation (CSE) [Mazziotti, Phys. Rev. A 57, 4219 (1998)]. In general, the CSE is a stronger N-representability condition than the dispersion condition because the CSE implies the dispersion condition as well as additional N-representability constraints from the Hellmann-Feynman theorem. Energy minimization subject to the representability constraints is performed for a boson model with 10, 30, and 75 particles. Even when traditional wave-function methods fail at large perturbations, the present method yields correlation energies within 2%.

Physical Review A•Mar 21, 2001•187 citations

Simple renormalization-group approximation of the ground-state properties of interacting bosonic systems

Roman Werpachowski, Jerzy Kijowski

We present a simple renormalization-group method of investigating ground-state properties of interacting bosonic systems. Our method reduces the number of particles in a system, which makes numerical calculations possible for large systems. It is conceptually simple and easy to implement, and allows investigation of properties unavailable through mean-field approximations, such as one- and two-particle reduced density matrices of the ground state. As an example, we model a weakly interacting one-dimensional Bose gas in a harmonic trap. Compared to the mean-field Gross-Pitaevskii approximation, our method provides a more accurate description of the ground-state one-particle density matrix. We have also obtained the Hall-Post lower bounds for the ground-state energy of the gas. All results have been obtained by straightforward numerical diagonalization of the Hamiltonian matrix.

arXiv:cond-mat/0510436•Physical Review A•Jun 21, 2007•0 citations

Variational reduced-density-matrix calculation of the one-dimensional Hubbard model

Jeff R. Hammond, David A. Mazziotti

Variational reduced-density-matrix theory is applied to calculating the ground-state energy and two-electron reduced density matrices (2-RDMs) of the one-dimensional Hubbard model for a range of interaction strengths. The 2-RDM is constrained to represent an $N$-particle wave function by two sets of $N$-representability conditions, known as the 2- and partial 3-positivity conditions. Variational optimization of the energy with the 2-RDM constrained by $N$-representability conditions is performed using a first-order semidefinite-programming algorithm that was developed for treating atoms and molecules [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Accurate energies for a broad range of interaction strengths indicate that the variational 2-RDM method is a valuable tool for studying strongly correlated electrons.

Physical Review A•Jun 13, 2006•41 citations

Determination of reduced density matrices in the doubly occupied configuration interaction space: A Hellmann–Feynman theorem approach

Adán Garros

In this work, the Hellmann-Feynman theorem is extended within the doubly occupied configuration interaction space to enable practical calculations of reduced density matrices and expected values. This approach is straightforward, employing finite energy differences, yet remains reliable and accurate even with approximate energies from successive approximation methods. The method's validity is rigorously tested against the Richardson-Gaudin-Kitaev and reduced Bardeen-Cooper-Schrieffer models using approximate excitation energies procured from the Hermitian operator method within the same space, effectively proving the approach's reliability with median error rates for reduced density matrix calculations around 0.1%. These results highlight the procedure's potential as a practical tool for computing reduced density matrices and expected values, particularly valuable as an ad hoc method in scenarios where only system energies are easily available.

The Journal of Chemical Physics•Oct 1, 2024•1 citation

Boson correlation energies via variational minimization with the two-particle reduced density matrix: Exact N -representability conditions for harmonic interactions

Gergely Gidofalvi, David A. Mazziotti

A many-body theory for interacting bosons is developed within the framework of minimizing the ground-state energy with respect to the two-particle reduced-density matrix (2-RDM) subject to $N$-representability conditions. The $N$-representability conditions, which ensure that the 2-RDM may be derived from an $N$-particle wave function, are imposed through a hierarchy of positivity conditions where the $p$-positivity conditions restrict the metric matrices for $p∕2$-body operators to be positive semidefinite. Using two-positivity, we minimize the ground-state energies of $5--10 000$ harmonically interacting bosons in a harmonic external potential. The energies and 2-RDMs obtained are in agreement with the exact solution except for round-off errors, which implies that for this class of boson interactions two-positivity conditions alone yield exact results for any interaction strength. The ground-state energies obtained at strong interactions are more accurate than many-body perturbative techniques by many orders of magnitude.

Physical Review A•Apr 16, 2004•42 citations
The ground-state energies of interacting bosons are computed beyond the mean-field approximation by a new method known as reduced Hamiltonian interpolation (RHI) [D. A. Mazziotti and D. R. Herschbach, Phys. Rev. Lett. 83, 5185 (1999)]. In the RHI, the N-particle Hamiltonian is represented through a sequence of p-particle expanded and reduced Hamiltonians that give upper and lower bounds on the true energy. Assimilating ideas from N representability and dimensional interpolation, the RHI technique interpolates over the number p of quasiparticles to determine the N-particle energy with close upper and lower bounds. With the Hellmann-Feynman theorem, we extend the RHI to compute the two-particle reduced density matrix (2RDM) as well as the energy. We examine both the computational advantages of RHI in comparison with traditional methods and the possibility of extending the RHI to treat fermion correlation. Applied to bosons with harmonic interactions as well as a two-level system, the RHI technique yields more than 99% of the correlation energy and an accurate correlation correction for the elements of the 2RDM.
The ground-state energies of interacting bosons are computed beyond the mean-field approximation by a new method which we call reduced Hamiltonian interpolation (RHI). In this interpolation the NNN-particle Hamiltonian is represented through a sequence of ppp-particle expanded and reduced Hamiltonians that give upper and lower bounds on the true energy. A synthesis of ideas from NNN-representability and dimensional interpolation, the RHI interpolates over the number ppp of quasiparticles (equivalent to spatial dimension) to calculate the NNN-particle energy as the mean of close upper and lower bounds. Application to bosons with harmonic interactions yields more than 9999%99 of the correlation energy.
For the description of ground-state correlation phenomena an accurate mapping of many-body quantum mechanics onto four particles is developed. The energy for a quantum system with no more than two-particle interactions may be expressed in terms of a two-particle reduced density matrix (2-RDM), but variational optimization of the 2-RDM requires that it corresponds to an N-particle wave function. We derive N-representability conditions on the 2-RDM that guarantee the validity of the uncertainty relations for all operators with two-particle interactions. One of these conditions is shown to be necessary and sufficient to make the RDM solutions of the dispersion condition equivalent to those from the contracted Schr\"odinger equation (CSE) [Mazziotti, Phys. Rev. A 57, 4219 (1998)]. In general, the CSE is a stronger N-representability condition than the dispersion condition because the CSE implies the dispersion condition as well as additional N-representability constraints from the Hellmann-Feynman theorem. Energy minimization subject to the representability constraints is performed for a boson model with 10, 30, and 75 particles. Even when traditional wave-function methods fail at large perturbations, the present method yields correlation energies within 2%.
We present a simple renormalization-group method of investigating ground-state properties of interacting bosonic systems. Our method reduces the number of particles in a system, which makes numerical calculations possible for large systems. It is conceptually simple and easy to implement, and allows investigation of properties unavailable through mean-field approximations, such as one- and two-particle reduced density matrices of the ground state. As an example, we model a weakly interacting one-dimensional Bose gas in a harmonic trap. Compared to the mean-field Gross-Pitaevskii approximation, our method provides a more accurate description of the ground-state one-particle density matrix. We have also obtained the Hall-Post lower bounds for the ground-state energy of the gas. All results have been obtained by straightforward numerical diagonalization of the Hamiltonian matrix.
Variational reduced-density-matrix theory is applied to calculating the ground-state energy and two-electron reduced density matrices (2-RDMs) of the one-dimensional Hubbard model for a range of interaction strengths. The 2-RDM is constrained to represent an NNN-particle wave function by two sets of NNN-representability conditions, known as the 2- and partial 3-positivity conditions. Variational optimization of the energy with the 2-RDM constrained by NNN-representability conditions is performed using a first-order semidefinite-programming algorithm that was developed for treating atoms and molecules [D. A. Mazziotti, Phys. Rev. Lett. 93, 213001 (2004)]. Accurate energies for a broad range of interaction strengths indicate that the variational 2-RDM method is a valuable tool for studying strongly correlated electrons.
In this work, the Hellmann-Feynman theorem is extended within the doubly occupied configuration interaction space to enable practical calculations of reduced density matrices and expected values. This approach is straightforward, employing finite energy differences, yet remains reliable and accurate even with approximate energies from successive approximation methods. The method's validity is rigorously tested against the Richardson-Gaudin-Kitaev and reduced Bardeen-Cooper-Schrieffer models using approximate excitation energies procured from the Hermitian operator method within the same space, effectively proving the approach's reliability with median error rates for reduced density matrix calculations around 0.1%. These results highlight the procedure's potential as a practical tool for computing reduced density matrices and expected values, particularly valuable as an ad hoc method in scenarios where only system energies are easily available.
A many-body theory for interacting bosons is developed within the framework of minimizing the ground-state energy with respect to the two-particle reduced-density matrix (2-RDM) subject to NNN-representability conditions. The NNN-representability conditions, which ensure that the 2-RDM may be derived from an NNN-particle wave function, are imposed through a hierarchy of positivity conditions where the ppp-positivity conditions restrict the metric matrices for p∕2p∕2p∕2-body operators to be positive semidefinite. Using two-positivity, we minimize the ground-state energies of 5−−100005--10 0005−−10000 harmonically interacting bosons in a harmonic external potential. The energies and 2-RDMs obtained are in agreement with the exact solution except for round-off errors, which implies that for this class of boson interactions two-positivity conditions alone yield exact results for any interaction strength. The ground-state energies obtained at strong interactions are more accurate than many-body perturbative techniques by many orders of magnitude.