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/9802171 · Physical review. B, Condensed matter

Ground-state energy of the Hubbard model: Cluster-perturbative results

Maciej M. Maśka

Published Apr 15, 1998• 7 citations
arXivHTMLPDF

Abstract

When electron correlations are important it is often necessary to use numerical methods to solve the Hamiltonian for a finite system (cluster) ``exactly.'' Unfortunately, such methods are restricted to small systems. We propose to combine the ``exact'' numerical diagonalization for small clusters with the perturbative calculations to take into account the intracluster as well as intercluster interactions.

DOI
10.1103/physrevb.57.8755

Related papers

Cluster perturbation theory for Hubbard models

David Sénéchal, Danny Pérez, Dany Plouffe

Cluster perturbation theory is a technique for calculating the spectral weight of Hubbard models of strongly correlated electrons, which combines exact diagonalizations on small clusters with strong-coupling perturbation theory at leading order. It is exact in both the strong- and weak-coupling limits and provides a good approximation to the spectral function at any wave vector. Following the paper by S\'en\'echal et al. [Phys. Rev. Lett. $84,$ 522 (2000)], we provide a more complete description and derivation of the method. We illustrate some of its capabilities, in particular regarding the effect of doping, the calculation of ground state energy and double occupancy, the disappearance of the Fermi surface in the $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model, and so on. The method is applicable to any model with on-site repulsion only.

arXiv:cond-mat/0205044•Physical review. B, Condensed matter•Aug 30, 2002•238 citations

Correlation between perturbation and variation methods in the study of strongly correlated electron systems

Robinson O. Okanigbuan, JO Idiodi

The ground–state wave function and energy are calculated for two electrons subject to a one-band Hubbard Hamiltonian on a one dimensional lattice containing N electronic sites, N = 2,3,4,5,6, and a 3 x 3 cluster of the square lattice, using perturbation and variational methods. The results from these two approximation methods are then compared with the result from exact calculational method. Journal of the Nigerian Association of Mathematical Physics Vol. 10 2006: pp. 583-594

Journal of the Nigerian Association of Mathematical Physics•May 22, 2008•0 citations

Cluster-based mean-field and perturbative description of strongly correlated fermion systems: Application to the one- and two-dimensional Hubbard model

Carlos A. Jiménez-Hoyos, Gustavo E. Scuseria

We introduce a mean-field and a perturbative approach, based on clusters, to describe the ground state of fermionic strongly correlated systems. In the cluster mean-field approach, the ground-state wave function is written as a simple tensor product over optimized cluster states. The optimization of the single-particle basis where the cluster mean field is expressed is crucial in order to obtain high-quality results. The mean-field nature of the Ansatz allows us to formulate a perturbative approach to account for intercluster correlations; other traditional many-body strategies can be easily devised in terms of the cluster states. We present benchmark calculations on the half-filled 1D and (square) 2D Hubbard model, as well as the lightly doped regime in 2D, using cluster mean-field and second-order perturbation theory. Our results indicate that, with sufficiently large clusters or to second-order in perturbation theory, a cluster-based approach can provide an accurate description of the Hubbard model in the considered regimes. Several avenues to improve upon the results presented in this work are discussed.

arXiv:1505.05909•Physical Review B•Aug 3, 2015•49 citations

Variational cluster approximation and exact diagonalization for calculation of Hubbard models

Alexander E. Lukyanov, Vyacheslav D. Neverov, Yaroslav Zhumagulov +1 more

The numerical method was developed that combines the variational cluster approximation and the exact diagonalization techniques to solve Hubbard models of various dimensionality, number of orbitals, and lattice geometry. The developed method was tested on the one-band Hubbard model; antiferromagnetic ordering and the superconducting phase were observed, in accordance with known results obtained by other techniques. The method may be successfully used for studying properties of high-temperature superconductors.

Journal of Physics Conference Series•Mar 1, 2019•0 citations

Extracting Infinite System Properties from Finite Size Clusters: `Phase Randomization/Boundary Condition Averaging'

J. Tinka Gammel, D. K. Campbell, Jr Loh

When electron-electron correlations are important, it is often necessary to use exact numerical methods, such as Lanczos diagonalization, to study the full many-body Hamiltonian. Unfortunately, such exact diagonalization methods are restricted to small system sizes. We show that if the Hubbard $U$ term is replaced by a ``periodic Hubbard" term, the full many body Hamiltonian may be numerically exactly solved, even for very large systems (even $>$100 sites), though only for low fillings. However, for half-filled systems and large $U$ this approach is not only no longer exact, it no longer improves extrapolation to larger systems. We discuss how generalized ``randomized variable averaging'' (RVA) or ``phase randomization'' schemes can be reliably employed to improve extrapolation to large system sizes in this regime. This general approach can be combined with any many-body method and is thus of broad interest and applicability.

arXiv:cond-mat/9209026•Sep 19, 1992•0 citations

Hubbard Hamiltonian in the dimer representation Large-U case

M. Matlak, J. Aksamit, B. Grabiec +1 more

The nonperturbative approach, presented in this paper, can, in principle, be applied to clusters of any size, as well as, to another type of model Hamiltonians.

arXiv:cond-mat/0204377•Annalen der Physik•Jun 18, 2003•5 citations
When electron correlations are important it is often necessary to use numerical methods to solve the Hamiltonian for a finite system (cluster) exactly.'' Unfortunately, such methods are restricted to small systems. We propose to combine the exact'' numerical diagonalization for small clusters with the perturbative calculations to take into account the intracluster as well as intercluster interactions.
Cluster perturbation theory is a technique for calculating the spectral weight of Hubbard models of strongly correlated electrons, which combines exact diagonalizations on small clusters with strong-coupling perturbation theory at leading order. It is exact in both the strong- and weak-coupling limits and provides a good approximation to the spectral function at any wave vector. Following the paper by S\'en\'echal et al. [Phys. Rev. Lett. 84,84,84, 522 (2000)], we provide a more complete description and derivation of the method. We illustrate some of its capabilities, in particular regarding the effect of doping, the calculation of ground state energy and double occupancy, the disappearance of the Fermi surface in the $t\ensuremath{-}{t}^{\ensuremath{'}}$ Hubbard model, and so on. The method is applicable to any model with on-site repulsion only.
We introduce a mean-field and a perturbative approach, based on clusters, to describe the ground state of fermionic strongly correlated systems. In the cluster mean-field approach, the ground-state wave function is written as a simple tensor product over optimized cluster states. The optimization of the single-particle basis where the cluster mean field is expressed is crucial in order to obtain high-quality results. The mean-field nature of the Ansatz allows us to formulate a perturbative approach to account for intercluster correlations; other traditional many-body strategies can be easily devised in terms of the cluster states. We present benchmark calculations on the half-filled 1D and (square) 2D Hubbard model, as well as the lightly doped regime in 2D, using cluster mean-field and second-order perturbation theory. Our results indicate that, with sufficiently large clusters or to second-order in perturbation theory, a cluster-based approach can provide an accurate description of the Hubbard model in the considered regimes. Several avenues to improve upon the results presented in this work are discussed.
The numerical method was developed that combines the variational cluster approximation and the exact diagonalization techniques to solve Hubbard models of various dimensionality, number of orbitals, and lattice geometry. The developed method was tested on the one-band Hubbard model; antiferromagnetic ordering and the superconducting phase were observed, in accordance with known results obtained by other techniques. The method may be successfully used for studying properties of high-temperature superconductors.
When electron-electron correlations are important, it is often necessary to use exact numerical methods, such as Lanczos diagonalization, to study the full many-body Hamiltonian. Unfortunately, such exact diagonalization methods are restricted to small system sizes. We show that if the Hubbard UUU term is replaced by a ``periodic Hubbard" term, the full many body Hamiltonian may be numerically exactly solved, even for very large systems (even $>$100 sites), though only for low fillings. However, for half-filled systems and large UUU this approach is not only no longer exact, it no longer improves extrapolation to larger systems. We discuss how generalized ``randomized variable averaging'' (RVA) or ``phase randomization'' schemes can be reliably employed to improve extrapolation to large system sizes in this regime. This general approach can be combined with any many-body method and is thus of broad interest and applicability.