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The Ground State Of Two-Dimensional Hubbard-Like Models

Allman, Eric

Published Jan 1, 2002• 0 citations
Publisher

Abstract

We present results from a constrained path Monte Carlo (CPMC) study of a modified two-dimensional Hubbard model. We include more general forms of the band structure and electron interaction in order to examine their effects on ground-state properties, such as electron pairing correlations. Both next-nearest neighbor hopping, t', and third-nearest neighbor hopping, t″, are introduced in the Hamiltonian. A nearest neighbor interaction of strength V is also included. We carry out CPMC calculations on system sizes up to 16 x 16, at various electron fillings, to investigate the ground state of the model for different values of these parameters. For realistic systems, these calculations indicate that Hubbard-like models are not capable of showing enhanced pairing. The modified Hamiltonian also presents an opportunity to more closely examine the accuracy and robustness of the CPMC algorithm. Results of further benchmark calculations involving CPMC are presented. These benchmarks confirm that CPMC is able to show enhanced pairing in systems where such behavior is known to exist.

DOI
10.21220/s2-d114-2g03

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We present results from a constrained path Monte Carlo (CPMC) study of a modified two-dimensional Hubbard model. We include more general forms of the band structure and electron interaction in order to examine their effects on ground-state properties, such as electron pairing correlations. Both next-nearest neighbor hopping, t', and third-nearest neighbor hopping, t″, are introduced in the Hamiltonian. A nearest neighbor interaction of strength V is also included. We carry out CPMC calculations on system sizes up to 16 x 16, at various electron fillings, to investigate the ground state of the model for different values of these parameters. For realistic systems, these calculations indicate that Hubbard-like models are not capable of showing enhanced pairing. The modified Hamiltonian also presents an opportunity to more closely examine the accuracy and robustness of the CPMC algorithm. Results of further benchmark calculations involving CPMC are presented. These benchmarks confirm that CPMC is able to show enhanced pairing in systems where such behavior is known to exist.
We simulated the three-band Hubbard model using the constrained-path Monte Carlo (CPMC) method in search for a possible superconducting ground state. The CPMC is a ground-state method that is free of the exponential scaling of computing time with system size. We calculated the binding energy of a pair of holes for systems up to $6\ifmmode\times\else\texttimes\fi{}4$ unit cells. We also studied the pairing correlation functions versus distance for both the ddd-wave and extended sss-wave channels in systems up to $6\ifmmode\times\else\texttimes\fi{}6$. We found that holes bind for a wide range of parameters and that the binding increased as the system size is increased. However, the pairing correlation functions decay quickly with distance. For the extended sss channel, we found that as the Coulomb interaction Ud{U}_{d}Ud​ on the Cu sites is increased, the long-range part of the correlation functions is suppressed and fluctuates around zero. For the ${d}_{{x}^{2}\ensuremath{-}{y}^{2}}$ channel, we found that the correlations decay rapidly with distance towards a small positive value. However, this value becomes smaller as the interaction Ud{U}_{d}Ud​ or the system size is increased.
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.
To clarify the effects of next-nearest-neighbor electron hopping ${t}^{\ensuremath{'}}$ on the spin, charge, and pairing correlations, we studied the two-dimensional $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}U$ Hubbard model using the constrained-path Monte Carlo technique. We found that a negative ${t}^{\ensuremath{'}}$ suppresses the spin fluctuations near the Brillouin zone point $M=(\ensuremath{\pi},\ensuremath{\pi})$ and the charge fluctuations near the point $X=(\ensuremath{\pi},0),$ while a positive ${t}^{\ensuremath{'}}$ has the opposite effect. For a negative ${t}^{\ensuremath{'}},$ we also found that the long-range vertex contributions to the pairing correlations are heavily suppressed, but the local contributions are enhanced. Over the entire hole doping range studied, the d-wave long-range vertex contributions to pairing dominated those of the extended s-wave vertex contributions. In general, we did not find any tendency for these correlations to grow with increasing system size. We also did a limited number of simulations with an attractive nearest neighbor density-density interaction added to the $t\ensuremath{-}{t}^{\ensuremath{'}}\ensuremath{-}U$ model. It enhanced the long-range vertex contributions to the pairing correlations only for weak values of U.
We benchmark constrained-path Monte Carlo (CPMC) on the triangular-lattice Hubbard model for several fillings and UUU values and show that symmetry-adapted trial wave functions substantially improve quantitative accuracy. Away from half-filling, simple free-electron-based trials that preserve the ground state symmetry yield energy deviations ≲1%\lesssim 1\%≲1% from exact diagonalization and density matrix renormalization group results. At half-filling, strong frustration in the intermediate to large UUU regimes necessitates symmetry-projected trials to reach comparable accuracy, where both free-electron and symmetry-broken Hartree-Fock trials incur substantial constraint bias. Since the computational cost of CPMC with symmetry projection scales polynomially with system size, our results motivate its use as a practical route for studying competing ground states in strongly correlated, frustrated systems.
The emergence of superconductivity (SC) in lattice models, such as the attractive Hubbard one, has renewed interest since the realization of cold-atom experiments. However, reducing the temperature in these experiments is a bottleneck; therefore, investigating how to increase the energy scale for SC is crucial to cold atoms. In view of this, we examine the effects of next-nearest-neighbor hoppings (t′) on the pairing properties of the attractive Hubbard model in a square lattice. To this end, we analyze the model through unbiased Quantum Monte Carlo simulations for fixed density n=0.87, and perform finite-size scaling analysis to the thermodynamic limit. As our main result, we notice that the existence of further hopping channels leads to an enhancement of the pairing correlations, which, in turn, increases the ground-state order parameter Δ. Finally, at finite temperatures, for t′/t≠0, this enhancement of pairing correlations leads to an increase in the critical temperature Tc. That is, the fine-tuning of second-neighbor hoppings increases the energy scales for SC, and may be a route by which cold-atom experiments can achieve such a phase and to help us further understand the nature of this phenomenon.
We study the magnetic properties of the two-dimensional Hubbard model with nearest-neighbor (t) and next-nearest-neighbor (t2{t}_{2}t2​) 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 t2{t}_{2}t2​\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.