Physical review. B, Condensed matter

Onsager-reaction-field correction in the half-filled Hubbard model

Peter Kopietz

Published Nov 1, 1993• 15 citations

Abstract

The Onsager-reaction-field correction to Hartree-Fock theory recently derived by Georges and Yedidia [Phys. Rev. B 43, 3475 (1991)] is evaluated analytically for the half-filled positive-U Hubbard model on a d-dimensional hypercubic lattice. It is shown that the Onsager correction is for all d\ensuremath{\ge}3 irrelevant in the weak-coupling limit, while for d=1 and d=2 it strongly renormalizes the Hartree-Fock result, although it does note completely destabilize the spin-density wave. We examine the perturbation theory for the energy change induced by a small staggered magnetization m to all orders in U, and show that one obtains an infinite series in powers of (U/t)ln[${\mathit{m}}^{\mathrm{\ensuremath{-}}1}$ln(${\mathit{m}}^{\mathrm{\ensuremath{-}}1}$)] for d=1, and in powers of (U/t)${\mathrm{ln}}^{2}$[${\mathit{m}}^{\mathrm{\ensuremath{-}}1}$${\mathrm{ln}}^{2}$(${\mathit{m}}^{\mathrm{\ensuremath{-}}1}$)] for d=1. The close similarity between d=1 and d=2 is interpreted as evidence that antiferromagnetism in the weak coupling regime is not stable in d=2.

DOI: 10.1103/physrevb.48.13789

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Physical review. B, Condensed matter

Onsager-reaction-field correction in the half-filled Hubbard model

Peter Kopietz

Published Nov 1, 1993• 15 citations

Publisher

Abstract

{\mathit{m}}^{\mathrm{\ensuremath{-}}1}

ln(

{\mathit{m}}^{\mathrm{\ensuremath{-}}1}

)] for d=1, and in powers of (U/t)

{\mathrm{ln}}^{2}

[

{\mathit{m}}^{\mathrm{\ensuremath{-}}1}$${\mathrm{ln}}^{2}

(

{\mathit{m}}^{\mathrm{\ensuremath{-}}1}

)] for d=1. The close similarity between d=1 and d=2 is interpreted as evidence that antiferromagnetism in the weak coupling regime is not stable in d=2.

DOI: 10.1103/physrevb.48.13789

Related papers

Extended Hubbard model at weak coupling

P. G. J. van Dongen

The half-filled extended Hubbard model (including nearest-neighbor interaction) is considered in perturbation theory at weak coupling on a d-dimensional hypercubic lattice (d\ensuremath{\gtrsim}3). It is shown that the exact critical temperature and the exact order parameter in the weak-coupling limit differ from the corresponding Hartree results by a renormalization factor q of order unity. This renormalization factor q is calculated systematically up to O(1/d) in a 1/d expansion. The results depend sensitively on dimension and on the relative strength of the nearest-neighbor interaction. The renormalization factor in d=3, estimated from the 1/d expansion, is q\ensuremath{\simeq}0.282 for the pure Hubbard model, q\ensuremath{\simeq}0.179 near the spin-density-wave to charge-density-wave transition and q\ensuremath{\simeq}0.830 in the limit of pure nearest-neighbor interaction.The half-filled extended Hubbard model (including nearest-neighbor interaction) is considered in perturbation theory at weak coupling on a d-dimensional hypercubic lattice (d\ensuremath{\gtrsim}3). It is shown that the exact critical temperature and the exact order parameter in the weak-coupling limit differ from the corresponding Hartree results by a renormalization factor q of order unity. This renormalization factor q is calculated systematically up to O(1/d) in a 1/d expansion. The results depend sensitively on dimension and on the relative strength of the nearest-neighbor interaction. The renormalization factor in d=3, estimated from the 1/d expansion, is q\ensuremath{\simeq}0.282 for the pure Hubbard model, q\ensuremath{\simeq}0.179 near the spin-density-wave to charge-density-wave transition and q\ensuremath{\simeq}0.830 in the limit of pure nearest-neighbor interaction.

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A strong-coupling expansion for the antiferromagnetic phase of the Hubbard model is derived in the framework of the slave-boson mean-field approximation. The expansion can be obtained in terms of moments of the density of states of freely hopping electrons on a lattice, which in turn are obtained for hypercubic lattices in arbitrary dimension. The expansion is given for the case of half-filling and for the energy up to fifth order in the ratio of hopping integral t over on-site interaction U, but can straightforwardly be generalized to the non-half-filled case and be extended to higher orders in t/U. For the energy the expansion is found to have an accuracy of better than 1% for U/t\ensuremath{\ge}8. A comparison is given with an earlier perturbation expansion based on the linear-spin-wave approximation and with a similar expansion based on the Hartree-Fock approximation. The case of an infinite number of spatial dimensions is discussed. \textcopyright{} 1996 The American Physical Society.A strong-coupling expansion for the antiferromagnetic phase of the Hubbard model is derived in the framework of the slave-boson mean-field approximation. The expansion can be obtained in terms of moments of the density of states of freely hopping electrons on a lattice, which in turn are obtained for hypercubic lattices in arbitrary dimension. The expansion is given for the case of half-filling and for the energy up to fifth order in the ratio of hopping integral t over on-site interaction U, but can straightforwardly be generalized to the non-half-filled case and be extended to higher orders in t/U. For the energy the expansion is found to have an accuracy of better than 1% for U/t\ensuremath{\ge}8. A comparison is given with an earlier perturbation expansion based on the linear-spin-wave approximation and with a similar expansion based on the Hartree-Fock approximation. The case of an infinite number of spatial dimensions is discussed. \textcopyright{} 1996 The American Physical Society.

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Michael A. Tusch, Yolande H. Szczech, David E. Logan

We present an approach to the magnetic properties of the half-filled Hubbard model, based on an approximate mapping of its low-energy transverse spin excitations on to those of an effective underlying Heisenberg model, but with effective spin interactions which are self-consistently determined and not confined solely to nearest-neighbor couplings. The mapping is exact in strong-coupling and is found to be accurate over a very wide range of interaction strengths, down to weak coupling. At zero temperature, it permits ready evaluation at finite U of the one-loop effects of zero-point spin fluctuations on, e.g., the sublattice magnetization. At finite temperatures, thermodynamic properties of the system in the thermal paramagnet are studied via a physically transparent Onsager reaction field approach, which amounts to a self-consistent treatment of paramagnetic spin correlations. This is central not only in recovering the correct dimensionality dependence of antiferromagnetic long-ranged order, but also for the d=3 case of primary interest here yields a N\'eel temperature in close agreement with known strong- and weak-coupling limits. Spin correlation functions and magnetic susceptibilities also show very good agreement with quantum Monte Carlo calculations over an appreciable temperature range in which the low-lying transverse spin excitations are thermally dominant. \textcopyright{} 1996 The American Physical Society.We present an approach to the magnetic properties of the half-filled Hubbard model, based on an approximate mapping of its low-energy transverse spin excitations on to those of an effective underlying Heisenberg model, but with effective spin interactions which are self-consistently determined and not confined solely to nearest-neighbor couplings. The mapping is exact in strong-coupling and is found to be accurate over a very wide range of interaction strengths, down to weak coupling. At zero temperature, it permits ready evaluation at finite U of the one-loop effects of zero-point spin fluctuations on, e.g., the sublattice magnetization. At finite temperatures, thermodynamic properties of the system in the thermal paramagnet are studied via a physically transparent Onsager reaction field approach, which amounts to a self-consistent treatment of paramagnetic spin correlations. This is central not only in recovering the correct dimensionality dependence of antiferromagnetic long-ranged order, but also for the d=3 case of primary interest here yields a N\'eel temperature in close agreement with known strong- and weak-coupling limits. Spin correlation functions and magnetic susceptibilities also show very good agreement with quantum Monte Carlo calculations over an appreciable temperature range in which the low-lying transverse spin excitations are thermally dominant. \textcopyright{} 1996 The American Physical Society.

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Hiroyuki Kaga, M. Cyrot

We show that the Onsager-field theory for itinerant systems is also quite useful for arbitrary filling of the two-dimensional (2D) and 3D Hubbard models in the weak to intermediate coupling of $U,$ yielding various magnetic properties in agreement with the results of quantum Monte Carlo simulations and the two-particle self-consistent method of Vilk et al. Any small doping into the antiferromagnetic ground state of the half-filled 3D model introduces an incommensurate phase with a significant reduction due to fluctuations.We show that the Onsager-field theory for itinerant systems is also quite useful for arbitrary filling of the two-dimensional (2D) and 3D Hubbard models in the weak to intermediate coupling of $U,$ yielding various magnetic properties in agreement with the results of quantum Monte Carlo simulations and the two-particle self-consistent method of Vilk et al. Any small doping into the antiferromagnetic ground state of the half-filled 3D model introduces an incommensurate phase with a significant reduction due to fluctuations.

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Magnetic phase diagram of the half-filled three-dimensional Hubbard model

Kok-Kwei Pan, Yung‐Li Wang

Magnetic properties of the half-filled Hubbard model on three-dimensional bipartite lattices have been studied by using the linked-cluster series expansion method. The series for the free energy and the staggered susceptibility have been calculated as a function of \ensuremath{\mathrm{B}}t for three-dimensional bipartite lattices for an arbitrary value of the Coulomb interaction U. We obtain a transition from a paramagnetic to an antiferromagnetic U/t- ${\mathrm{T}}_{\mathrm{N}}$ /t phase diagram from the divergence of the staggered susceptibility series. Comparisons with Monte Carlo simulation, Onsager reaction field result of an effective spin Hamiltonian, and large-U Heisenberg high-temperature solutions are made.Magnetic properties of the half-filled Hubbard model on three-dimensional bipartite lattices have been studied by using the linked-cluster series expansion method. The series for the free energy and the staggered susceptibility have been calculated as a function of \ensuremath{\mathrm{B}}t for three-dimensional bipartite lattices for an arbitrary value of the Coulomb interaction U. We obtain a transition from a paramagnetic to an antiferromagnetic U/t-${\mathrm{T}}_{\mathrm{N}}$ /t phase diagram from the divergence of the staggered susceptibility series. Comparisons with Monte Carlo simulation, Onsager reaction field result of an effective spin Hamiltonian, and large-U Heisenberg high-temperature solutions are made.

Physical review. B, Condensed matterFeb 1, 19975 citations

{\mathit{m}}^{\mathrm{\ensuremath{-}}1}

ln(

{\mathit{m}}^{\mathrm{\ensuremath{-}}1}

)] for d=1, and in powers of (U/t)

{\mathrm{ln}}^{2}

[

{\mathit{m}}^{\mathrm{\ensuremath{-}}1}$${\mathrm{ln}}^{2}

(

{\mathit{m}}^{\mathrm{\ensuremath{-}}1}

)] for d=1. The close similarity between d=1 and d=2 is interpreted as evidence that antiferromagnetism in the weak coupling regime is not stable in d=2.

The cluster slave-spin method is employed to investigate systematically the ground state properties of the Hubbard model on a square lattice with doping $\ensuremath{\delta}$ and coupling strength

U

being its parameters. At half-filling, a relation between the staggered magnetization

M

and the antiferromagnetic (AFM) gap ${\mathrm{\ensuremath{\Delta}}}_{\text{AFM}}$ is established in the small

U

limit to compare with that from the Hartree-Fock theory, and a first-order metal-insulator Mott transition in the paramagnetic state is substantiated, which is characterized by discontinuities and hystereses at

{U}_{\text{Mott}}=10t

. The interaction

{U}_{c}

for the crossover in the AFM state, separating the weak- and strong-coupling regimes, is found to remain almost unchanged with large dopings, and smaller than

{U}_{\text{Mott}}

at half-filling because of long range AFM correlations. Finally, an overall phase diagram in the $U\ensuremath{-}\ensuremath{\delta}$ plane is presented, which is composed of four regimes: the AFM insulator at half-filling, the AFM metal with the compressibility $\ensuremath{\kappa}>0$ or $\ensuremath{\kappa}<0$, and the paramagnetic metal, as well as three phase transitions: (i) From the AFM metal to the paramagnetic metal, (ii) between the AFM metal phases with positive and negative $\ensuremath{\kappa}$, and (iii) separating the AFM insulating phase at $\ensuremath{\delta}=0$ from the AFM metal phase for $\ensuremath{\delta}>0$.

We show that the Onsager-field theory for itinerant systems is also quite useful for arbitrary filling of the two-dimensional (2D) and 3D Hubbard models in the weak to intermediate coupling of

U,

yielding various magnetic properties in agreement with the results of quantum Monte Carlo simulations and the two-particle self-consistent method of Vilk et al. Any small doping into the antiferromagnetic ground state of the half-filled 3D model introduces an incommensurate phase with a significant reduction due to fluctuations.

{\mathrm{T}}_{\mathrm{N}}

/t phase diagram from the divergence of the staggered susceptibility series. Comparisons with Monte Carlo simulation, Onsager reaction field result of an effective spin Hamiltonian, and large-U Heisenberg high-temperature solutions are made.