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Quantum Monte Carlo simulations of the one-dimensional extended Hubbard model

W. R. Somsky, James E. Gubernatis

Published Jan 1, 1989• 0 citations
OpenAlex

Abstract

We report preliminary results of an investigation of the thermodynamic properties of the extended Hubbard model in one- dimension, calculated with the world-line Monte Carlo method described by Hirsch et al. With strictly continuous world-lines, we are able to measure the expectation of operators that conserve fermion number locally, such as the energy and (spatial) occupation number. By permitting the world-lines to be broken'' stochastically, we may also measure the expectation of operators that conserve fermion number only globally, such as the single-particle Green's function. For a 32 site lattice we present preliminary calculations of the average electron occupancy as a function of wavenumber when U = 4, V = 0 and {beta} = 16. For a half-filled band we find no indications of a Fermi surface. Slightly away from half-filling, we find Fermi-surface-like behavior similar to that found in other numerical investigations. 8 refs., 3 figs.

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As a continuation of a previous paper [J. Phys. Soc. Jpn. 56 (1987) 1490], the variational Monte-Carlo method is extended to include the antiferromagnetic long-range order. The theory is based on the Gutzwiller-type correlation factor and its effect is exactly taken into account by the Monte-Carlo procedure. An application is made to the half-filled-band case of one-dimensional lattice (50 sites), two-dimensional square lattice (up to 20×20 sites) and three-dimensional simple cubic lattice (6×6×6 sites). The result is qualitatively different from previous studies relying on the random-phase-type "Gutzwiller approximation." The variational energy for two and three dimensions is favorably compared with Hirsch's quantum Monte-Carlo data.

Journal of the Physical Society of Japan•Oct 15, 1987•153 citations
We report preliminary results of an investigation of the thermodynamic properties of the extended Hubbard model in one- dimension, calculated with the world-line Monte Carlo method described by Hirsch et al. With strictly continuous world-lines, we are able to measure the expectation of operators that conserve fermion number locally, such as the energy and (spatial) occupation number. By permitting the world-lines to be broken'' stochastically, we may also measure the expectation of operators that conserve fermion number only globally, such as the single-particle Green's function. For a 32 site lattice we present preliminary calculations of the average electron occupancy as a function of wavenumber when U = 4, V = 0 and {beta} = 16. For a half-filled band we find no indications of a Fermi surface. Slightly away from half-filling, we find Fermi-surface-like behavior similar to that found in other numerical investigations. 8 refs., 3 figs.
Ground-state properties of the Hubbard model on a two-dimensional square lattice are studied by the auxiliary-field quantum Monte Carlo method. Accurate results for energy, double occupancy, effective hopping, magnetization, and momentum distribution are calculated for interaction strengths of U/tU/tU/t from 2 to 8, for a range of densities including half-filling and n=0.3,0.5,0.6,0.75n=0.3,0.5,0.6,\phantom{\rule{0.16em}{0ex}}0.75n=0.3,0.5,0.6,0.75, and 0.8750.8750.875. At half-filling, the results are numerically exact. Away from half-filling, the constrained path Monte Carlo method is employed to control the sign problem. Our results are obtained with several advances in the computational algorithm, which are described in detail. We discuss the advantages of generalized Hartree-Fock trial wave functions and its connection to pairing wave functions, as well as the interplay with different forms of Hubbard-Stratonovich decompositions. We study the use of different twist angle sets when applying the twist averaged boundary conditions. We propose the use of quasirandom sequences, which improves the convergence to the thermodynamic limit over pseudorandom and other sequences. With it and a careful finite size scaling analysis, we are able to obtain accurate values of ground-state properties in the thermodynamic limit. Detailed results for finite-sized systems up to $16\ifmmode\times\else\texttimes\fi{}16$ are also provided for benchmark purposes.
Quantum Monte Carlo methods are used to study the physical properties of the two-dimensional Hubbard model for various band fillings. Results for the energy, magnetic moment, magnetic structure factor, and integrated optical spectral weight are presented. A comparison of the half-filled (〈n〉=1) and quarter-filled (〈n〉=0.5) bands is made using results for the momentum distribution 〈nk{n}_{k}nk​〉, the compressibility K, and the one-electron self-energy \ensuremath{\Sigma}(k,i${\ensuremath{\omega}}_{n}$), which shows that a gap exists in the one-electron excitations at half-filling but is absent for the one-quarter-filled case. At a filling of 〈n〉=0.87, the locus of k values for which 〈nk{n}_{k}nk​〉=0.5 is found to be close to the free Fermi surface for this filling.
We discuss a new method to perform numerical simulations of one-dimensional systems with fermion and boson degrees of freedom. The method is based on a direct-space, imaginary-time representation of the fermion field. It is fast so that systems having up to 100 sites can easily be simulated. In addition, the method provides an intuitive physical "picture" of the ground state of a one-dimensional many-body system. We discuss in detail how to implement the method and how to compute various physical quantities. In particular, we show how to extend the method to study averages of off-diagonal quantities in an occupation-number representation. To assess the accuracy of our procedure, we apply it to free fermions in one dimension and compare with exact results. We then study a model of spinless interacting fermions and obtain the expected phase structure and behavior of correlation functions. We also consider the extended Hubbard model at various points in its phase diagram and study the behavior of spin-density, charge-density, and pairing correlation functions. We then study the Gross-Neveu model and show how the behavior depends on the number of fermion flavors. Finally, we consider an electron-phonon model and study its behavior both in the one-particle polaron sector and in the half-filled-band case. Along the way we show pictures of the ground-state configurations that give physical insight into the properties of the systems, like charge-density-wave, spin-density-wave, and superconducting states, "fractional charges," and solitons. We conclude by comparing our method with other methods and discuss the possibility of extending it to higher dimensions.
Recent computer simulations performed on the one-dimensional Hubbard model with a new algorithm developed by Sorella et al. confirm the absence of a Fermi surface in this model at half filling. However, these same simulations suggest the existence of a Fermi surface at zero temperature away from half filling, contrary to expectations based on renormalization-group analysis. The results found here, using a more standard algorithm and a simple low-temperature extrapolation, appear consistent with the above-mentioned numerical results in the half-filled and quarter-filled band cases. However, it is argued that the discontinuity at the Fermi level found in all these simulations is likely to come from the zero-temperature finite-size dependence of the quasiparticle weight Z, which is also discussed here.
The Quantum Monte-Carlo simulations of the two-dimensional Hubbard model are presented for the half filling. The method based on the direct-space proposed by Suzuki and al., and Hirsch and al. was used. The states generated by this method are basis states in occupation number representation built with Wannier states localised on each site of the square array. The configurations of fermions can be observed on the real 2D array. An antiferromagnetic factor is defined and calculated for each temperature. The curves of energy, specific heat, conducivity and antiferromagnetic factor are presented for different values of the repulsive coulombian on site interaction U. There is a metal-insulator transition at low temperature for small values of U. This transition corresponds with a paramagnetic-ferromagnetic first order transition. Indeed, for these interaction values, the energy curves show a gap which is a characteristic of a first order transition. An hysteresis phenomenon appears on the conductivity curves. There is a behaviour change for U/t=3.5. For the values U>3.5 there is ferromagnetic-paramagnetic change without observable effect on the energy and the specific heat. The metal-insulator transition does not exist any more, the conductivity stays very small. Isotherms of the physical quantities versus U/t show a transition which seems to be the metal-insulator Mott transition. These results allow to draw a phase diagram with two first order transition lines.
As a continuation of a previous paper [J. Phys. Soc. Jpn. 56 (1987) 1490], the variational Monte-Carlo method is extended to include the antiferromagnetic long-range order. The theory is based on the Gutzwiller-type correlation factor and its effect is exactly taken into account by the Monte-Carlo procedure. An application is made to the half-filled-band case of one-dimensional lattice (50 sites), two-dimensional square lattice (up to 20×20 sites) and three-dimensional simple cubic lattice (6×6×6 sites). The result is qualitatively different from previous studies relying on the random-phase-type "Gutzwiller approximation." The variational energy for two and three dimensions is favorably compared with Hirsch's quantum Monte-Carlo data.