Physical review. B, Condensed matter

Extended Hubbard model in two dimensions

Yi Zhang, J. Callaway

Published May 1, 1989• 113 citations

Abstract

The two-dimensional half-filled extended Hubbard model is studied by means of Quantum Monte Carlo simulation. The model includes the on-site electron interaction (U) and nearest-neighbor interaction (V). We study the formation of antiferromagnetic, charge-density-wave, and superconducting states of the model as the parameters of the model are varied.

DOI: 10.1103/physrevb.39.9397

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Physical review. B, Condensed matter

Extended Hubbard model in two dimensions

Yi Zhang, J. Callaway

Published May 1, 1989• 113 citations

Publisher

Abstract

DOI: 10.1103/physrevb.39.9397

Related papers

Half-filled extended Hubbard model on a square lattice: Phase boundaries from determinant quantum Monte Carlo simulations

Sebastião dos Anjos Sousa-Júnior, Natanael C. Costa, Raimundo R. dos Santos

The extended Hubbard model describes fermions on a lattice coupled through on-site, $U$ , and first-neighbor, $V$ , interactions. In the context of high- ${T}_{c}$ cuprates, antiferromagnetic fluctuations may lead to an attractive channel, hence to superconductivity. Despite interest in the two-dimensional version of the model, the current knowledge about the phase diagram is still far from complete. Here, we report on the results of extensive determinant quantum Monte Carlo simulations for this model at half filling, in which we have used the average sign of the product of fermionic determinants as an additional observable to locate critical points. We arrive at a ground state phase diagram in the $U\text{\ensuremath{-}}V$ plane in which the boundaries involving antiferromagnetic, charge-ordered, $s$ - and $d$ -wave superconductivity, and phase-separated phases are quantitatively set with good accuracy. We have also proposed a partial phase diagram, ${T}_{c}(U,V)$ , featuring critical temperatures for the charge-density wave and $s$ -wave superconducting phases.The extended Hubbard model describes fermions on a lattice coupled through on-site, $U$, and first-neighbor, $V$, interactions. In the context of high-${T}_{c}$ cuprates, antiferromagnetic fluctuations may lead to an attractive channel, hence to superconductivity. Despite interest in the two-dimensional version of the model, the current knowledge about the phase diagram is still far from complete. Here, we report on the results of extensive determinant quantum Monte Carlo simulations for this model at half filling, in which we have used the average sign of the product of fermionic determinants as an additional observable to locate critical points. We arrive at a ground state phase diagram in the $U\text{\ensuremath{-}}V$ plane in which the boundaries involving antiferromagnetic, charge-ordered, $s$- and $d$-wave superconductivity, and phase-separated phases are quantitatively set with good accuracy. We have also proposed a partial phase diagram, ${T}_{c}(U,V)$, featuring critical temperatures for the charge-density wave and $s$-wave superconducting phases.

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We consider an extended Hubbard model of interacting fermions on a lattice. The fermion kinetic energy corresponds to a tight-binding Hamiltonian with nearest-neighbor (-t) and next-nearest-neighbor ( ${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$ ) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For ${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$ =0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero ${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$ , we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of ${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$ .We consider an extended Hubbard model of interacting fermions on a lattice. The fermion kinetic energy corresponds to a tight-binding Hamiltonian with nearest-neighbor (-t) and next-nearest-neighbor (${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For ${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$=0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero ${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$, we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of ${\mathrm{t}}^{|\mathrm{I}\mathrm{H}}$.

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Sergei G. Ovchinnikov, Elena I. Shneyder

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Two-dimensional Hubbard model with nearest- and next-nearest-neighbor hopping

H. Q. Lin, J. E. Hirsch

We study the magnetic properties of the two-dimensional Hubbard model with nearest-neighbor (t) and next-nearest-neighbor ( ${t}_{2}$ ) hopping and on-site repulsion U. We first calculate the mean-field phase diagram as a function of band filling and U. Because of the Van Hove singularity in the density of states, we find a ferromagnetic phase extending to zero U for certain band filling. For the half-filled band case, antiferromagnetism sets in at a finite value of U if ${t}_{2}$ \ensuremath{\ne}0. We study the behavior of spin-spin correlation functions for small lattices of up to N=64 atoms using Monte Carlo simulations, as well as exact diagonalization for N=4. Our results show enhanced ferromagnetic correlations in some regions, but apparently no ferromagnetic long-range order. In the half-filled case, our numerical results are consistent with a nonzero critical U. For a non-half-filled band our results suggest that there is no long-range order.We study the magnetic properties of the two-dimensional Hubbard model with nearest-neighbor (t) and next-nearest-neighbor (${t}_{2}$) hopping and on-site repulsion U. We first calculate the mean-field phase diagram as a function of band filling and U. Because of the Van Hove singularity in the density of states, we find a ferromagnetic phase extending to zero U for certain band filling. For the half-filled band case, antiferromagnetism sets in at a finite value of U if ${t}_{2}$\ensuremath{\ne}0. We study the behavior of spin-spin correlation functions for small lattices of up to N=64 atoms using Monte Carlo simulations, as well as exact diagonalization for N=4. Our results show enhanced ferromagnetic correlations in some regions, but apparently no ferromagnetic long-range order. In the half-filled case, our numerical results are consistent with a nonzero critical U. For a non-half-filled band our results suggest that there is no long-range order.

Physical review. B, Condensed matterMar 1, 1987185 citations

The extended Hubbard model describes fermions on a lattice coupled through on-site,

U

, and first-neighbor,

V

, interactions. In the context of high-

{T}_{c}

cuprates, antiferromagnetic fluctuations may lead to an attractive channel, hence to superconductivity. Despite interest in the two-dimensional version of the model, the current knowledge about the phase diagram is still far from complete. Here, we report on the results of extensive determinant quantum Monte Carlo simulations for this model at half filling, in which we have used the average sign of the product of fermionic determinants as an additional observable to locate critical points. We arrive at a ground state phase diagram in the $U\text{\ensuremath{-}}V$ plane in which the boundaries involving antiferromagnetic, charge-ordered,

s

- and

d

-wave superconductivity, and phase-separated phases are quantitatively set with good accuracy. We have also proposed a partial phase diagram,

{T}_{c}(U,V)

, featuring critical temperatures for the charge-density wave and

s

-wave superconducting phases.

{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}

) hopping matrix elements. In addition to the on-site Hubbard interaction (U) we also consider a nearest-neighbor repulsion (V). We obtain the zero-temperature phase diagram of our model within the Hartree-Fock approximation. We consider ground states having charge- and spin-density-wave ordering as well as states with orbital antiferromagnetism or spin nematic order. The latter two states correspond to particle-hole binding with ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ symmetry in the charge and spin channels, respectively. For

{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}

=0, only the charge-density-wave and spin-density-wave states are energetically stable. For nonzero

{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}

, we find that orbital antiferromagnetism (or spin nematic) order is stable over a finite portion of the phase diagram at weak coupling. This region of stability is seen to grow with increasing values of

{\mathrm{t}}^{|\mathrm{I}\mathrm{H}}

U

. In the ground state, the system is found to exhibit antiferromagnetic long-range order for all values of

U

, although greatly reduced from the mean-field-theory predictions.

We design a novel formalism of determinant quantum Monte Carlo method for the half-filled Hubbard model with on-site Hubbard interaction

U

and nearest-neighbor density-density interaction

V

on the square lattice. The formalism is free of sign problem for $|U|\ensuremath{\ge}4|V|$, and is achieved by introducing discrete auxiliary fields on the nearest-neighbor bonds alone. Based on this formalism, we study the ground-state phase diagram of the model systematically using the projector algorithm. Within the sign-problem free parameter space $|U|\ensuremath{\ge}4|V|$, we obtain antiferromagnetism for $U\ensuremath{\ge}4|V|>0$, charge density wave for $U<\ensuremath{-}4V$ and

V>0, s

-wave superconductivity for

U<0

and small negative

V

, and phase separation for

U<0

and a larger negative

V

. We also obtain the single particle gap and spin excitation spectra, and by comparison to mean field results, as well as available literatures, we discuss the possible phase boundary beyond the sign-problem free region. The unbiased numerically exact results can be taken as benchmarks in future studies. Finally, we discuss possible extension for longer-range interactions in the model.

We study the magnetic properties of the two-dimensional Hubbard model with nearest-neighbor (t) and next-nearest-neighbor (

{t}_{2}

) hopping and on-site repulsion U. We first calculate the mean-field phase diagram as a function of band filling and U. Because of the Van Hove singularity in the density of states, we find a ferromagnetic phase extending to zero U for certain band filling. For the half-filled band case, antiferromagnetism sets in at a finite value of U if

{t}_{2}

\ensuremath{\ne}0. We study the behavior of spin-spin correlation functions for small lattices of up to N=64 atoms using Monte Carlo simulations, as well as exact diagonalization for N=4. Our results show enhanced ferromagnetic correlations in some regions, but apparently no ferromagnetic long-range order. In the half-filled case, our numerical results are consistent with a nonzero critical U. For a non-half-filled band our results suggest that there is no long-range order.