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. B, Condensed matter

Comment on ‘‘t/Uexpansion for the Hubbard model’’

Andrzej M. Oleś

Published Feb 1, 1990• 23 citations
Publisher

Abstract

It is shown that a unitary transformation which eliminates the terms which couple states with different numbers of doubly occupied sites in the Hubbard model, proposed recently by MacDonald, Girvin, and Yoshioka, is conceptually equivalent to a canonical perturbation expansion, derived over a decade ago. In the half-filled case, the latter method gives an effective Hamiltonian in which many-body interactions result from even numbers of interconfigurational transitions. It is argued that it is hard to conclude anything about the nature of the ground state from the energies of the N\'eel state and various valence-bond states, as calculated by MacDonald et al., due to the importance of quantum fluctuations in a two-dimensional lattice.

DOI
10.1103/physrevb.41.2562

Related papers

t U expansion for the Hubbard model

Allan H. MacDonald, S. M. Girvin, D. Yoshioka

We describe a unitary transformation which eliminates terms coupling states with differing numbers of doubly occupied sites from the Hamiltonian of the Hubbard model. The $S$ matrix for the transformation, and the transformed Hamiltonian, ${H}^{\ensuremath{'}}$, are generated by an iterative procedure which results in an expansion in powers of the hopping integral $t$ divided by the on-site energy $U$. For a half-filled band and in the space with no doubly occupied sites, ${H}^{\ensuremath{'}}$ is equivalent to a spin Hamiltonian. We discuss the implications of our results for ${H}^{\ensuremath{'}}$ on theories of high-temperature superconductivity.

Physical review. B, Condensed matter•Jun 1, 1988•495 citations

Large-Uexpansions for the Hubbard model atT=0

J. Oitmaa, Jon Henderson

We report results for the ground-state energy of the Hubbard model in the form of perturbation expansions in t/U, i.e., expansions about the strong-coupling or ``atomic'' limit. Expansions to order (t/U${)}^{40}$ are obtained for finite cells with number of sites \ensuremath{\le}10, for the half-filled case, for both the linear chain and two-dimensional (2D) square lattice. Analysis of the series by Pad\'e approximant methods yields, in the 1D case, results consistent with the known exact results. For the largest cell, N=10, on the square lattice there is an indication of a singularity on the real t axis, which may indicate a phase transition at U/t\ensuremath{\simeq}4.

Physical review. B, Condensed matter•Oct 1, 1991•9 citations

Reply to ‘‘Comment on ‘t/Uexpansion for the Hubbard model’’’

Allan H. MacDonald, S. M. Girvin, D. Yoshioka

It is shown that the unitary transformation used as the basis of a t/U expansion for the Hubbard model in an earlier Brief Report is not equivalent to the canonical perturbation expansion (CPE) discussed in the preceding Comment. The CPE is valid only to leading order in t/U. More evidence is offered in support of our suggestion that higher-order terms in the t/U expansion tend to favor spin-liquid states over antiferromagnetically ordered states.

Physical review. B, Condensed matter•Feb 1, 1990•94 citations

A STRONG-COUPLING EXPANSION FOR THE HUBBARD MODEL

N. Dupuis, Stéphane Pairault

We reconsider the strong-coupling expansion for the Hubbard model recently introduced by Sarker and Pairault et al. By introducing slave particles that act as projection operators onto the empty, singly occupied and doubly occupied atomic states, the perturbation theory around the atomic limit distinguishes between processes that do conserve or do not conserve the total number of doubly occupied sites. This allows for a systematic t/U expansion that does not break down at low temperature (t being the intersite hopping amplitude and U the local Coulomb repulsion). The fermionic field becomes a two-component field, which reflects the presence of the two Hubbard bands. The single-particle propagator is naturally expressed as a function of a 2×2 matrix self-energy. Furthermore, by introducing a time- and space-fluctuating spin-quantization axis in the functional integral, we can expand around a "non-degenerate" ground-state where each singly occupied site has a well defined spin direction (which may fluctuate in time). This formalism is used to derive the effective action of charge carriers in the lower Hubbard band to first order in t/U. We recover the action of the t–J model in the spin-hole coherent-state path integral. We also compare our results with those previously obtained by studying fluctuations around the large-U Hartree–Fock saddle point.

arXiv:cond-mat/0008447•International Journal of Modern Physics B•Sep 30, 2000•6 citations

Large-Uexpansions for the Hubbard model atT=0. II

J. Oitmaa, Yang Jie, Jon Henderson +1 more

We investigate the ground state of the Hubbard model in one and two dimensions. Long perturbation expansions in t/U, i.e., about the atomic limit, are computed for the ground-state energy of finite lattices with N\ensuremath{\le}10 for arbitrary electron density. We also extend the series for the one-dimensional infinite lattice by expanding the Bethe-ansatz equations. Pad\'e-approximant analysis of the series shows a distribution of singularities that varies dramatically with the electron density n. We also investigate the effect of inclusion of next-nearest-neighbor hopping. This appears to have little effect at half-filling but is significant away from half-filling.

Physical review. B, Condensed matter•Sep 1, 1993•1 citation

Antiferromagnetism of the two-dimensional Hubbard model at half-filling: the analytic ground state for weak coupling

Michele Cini, Gianluca Stefanucci

We introduce a local formalism to deal with the Hubbard model on an N × N square lattice (for even N ) in terms of eigenstates of number operators, having well defined point symmetry. For U →0, the low-lying shells of the kinetic energy are filled in the ground state. At half-filling, using the 2 N -2 one-body states of the partially occupied shell hf , we build a set of degenerate unperturbed ground states with S z = 0 which are then resolved by the Hubbard interaction Ŵ = U ∑ r r ↑ r ↓ . We study the many-body eigenstates in hf of the kinetic energy with vanishing eigenvalue of the Hubbard repulsion ( W = 0 states). In the S z = 0 sector, this is an N -times-degenerate multiplet. From the singlet component one obtains the ground state of the Hubbard model for U = 0 + , which is unique, in agreement with a theorem of Lieb. The wave function demonstrates an antiferromagnetic order, a lattice step translation being equivalent to a spin flip. We show that the total momentum vanishes, while the point symmetry is s or d for even or odd N /2, respectively.

arXiv:cond-mat/0009058•Journal of Physics Condensed Matter•Jan 25, 2001•2 citations
It is shown that a unitary transformation which eliminates the terms which couple states with different numbers of doubly occupied sites in the Hubbard model, proposed recently by MacDonald, Girvin, and Yoshioka, is conceptually equivalent to a canonical perturbation expansion, derived over a decade ago. In the half-filled case, the latter method gives an effective Hamiltonian in which many-body interactions result from even numbers of interconfigurational transitions. It is argued that it is hard to conclude anything about the nature of the ground state from the energies of the N'eel state and various valence-bond states, as calculated by MacDonald et al., due to the importance of quantum fluctuations in a two-dimensional lattice.
We describe a unitary transformation which eliminates terms coupling states with differing numbers of doubly occupied sites from the Hamiltonian of the Hubbard model. The SSS matrix for the transformation, and the transformed Hamiltonian, ${H}^{\ensuremath{'}}$, are generated by an iterative procedure which results in an expansion in powers of the hopping integral ttt divided by the on-site energy UUU. For a half-filled band and in the space with no doubly occupied sites, ${H}^{\ensuremath{'}}$ is equivalent to a spin Hamiltonian. We discuss the implications of our results for ${H}^{\ensuremath{'}}$ on theories of high-temperature superconductivity.
We report results for the ground-state energy of the Hubbard model in the form of perturbation expansions in t/U, i.e., expansions about the strong-coupling or ``atomic'' limit. Expansions to order (t/U)40{)}^{40})40 are obtained for finite cells with number of sites \ensuremath{\le}10, for the half-filled case, for both the linear chain and two-dimensional (2D) square lattice. Analysis of the series by Pad\'e approximant methods yields, in the 1D case, results consistent with the known exact results. For the largest cell, N=10, on the square lattice there is an indication of a singularity on the real t axis, which may indicate a phase transition at U/t\ensuremath{\simeq}4.
We reconsider the strong-coupling expansion for the Hubbard model recently introduced by Sarker and Pairault et al. By introducing slave particles that act as projection operators onto the empty, singly occupied and doubly occupied atomic states, the perturbation theory around the atomic limit distinguishes between processes that do conserve or do not conserve the total number of doubly occupied sites. This allows for a systematic t/U expansion that does not break down at low temperature (t being the intersite hopping amplitude and U the local Coulomb repulsion). The fermionic field becomes a two-component field, which reflects the presence of the two Hubbard bands. The single-particle propagator is naturally expressed as a function of a 2×2 matrix self-energy. Furthermore, by introducing a time- and space-fluctuating spin-quantization axis in the functional integral, we can expand around a "non-degenerate" ground-state where each singly occupied site has a well defined spin direction (which may fluctuate in time). This formalism is used to derive the effective action of charge carriers in the lower Hubbard band to first order in t/U. We recover the action of the t–J model in the spin-hole coherent-state path integral. We also compare our results with those previously obtained by studying fluctuations around the large-U Hartree–Fock saddle point.
We investigate the ground state of the Hubbard model in one and two dimensions. Long perturbation expansions in t/U, i.e., about the atomic limit, are computed for the ground-state energy of finite lattices with N\ensuremath{\le}10 for arbitrary electron density. We also extend the series for the one-dimensional infinite lattice by expanding the Bethe-ansatz equations. Pad\'e-approximant analysis of the series shows a distribution of singularities that varies dramatically with the electron density n. We also investigate the effect of inclusion of next-nearest-neighbor hopping. This appears to have little effect at half-filling but is significant away from half-filling.
We introduce a local formalism to deal with the Hubbard model on an N × N square lattice (for even N ) in terms of eigenstates of number operators, having well defined point symmetry. For U →0, the low-lying shells of the kinetic energy are filled in the ground state. At half-filling, using the 2 N -2 one-body states of the partially occupied shell hf , we build a set of degenerate unperturbed ground states with S z = 0 which are then resolved by the Hubbard interaction Ŵ = U ∑ r r ↑ r ↓ . We study the many-body eigenstates in hf of the kinetic energy with vanishing eigenvalue of the Hubbard repulsion ( W = 0 states). In the S z = 0 sector, this is an N -times-degenerate multiplet. From the singlet component one obtains the ground state of the Hubbard model for U = 0 + , which is unique, in agreement with a theorem of Lieb. The wave function demonstrates an antiferromagnetic order, a lattice step translation being equivalent to a spin flip. We show that the total momentum vanishes, while the point symmetry is s or d for even or odd N /2, respectively.