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arXiv:cond-mat/9609004 · Physical review. B, Condensed matter

One-dimensional pair hopping and attractive Hubbard models: A comparative study

Mathias van den Bossche, Michel Caffarel

Published Dec 15, 1996• 16 citations
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Abstract

The low-energy physics of the one-dimensional pair hopping (PH) and attractive Hubbard models are expected to be similar. Based on numerical calculations on small chains, several authors have recently challenged this idea and predicted the existence of a phase transition at half filling and finite positive coupling for the pair-hopping model. We reexamine the controversy by making systematic comparisons between numerical results obtained for the PH and attractive Hubbard models. To do so, we have calculated the Luttinger parameters (spin and charge velocities, stiffnesses, etc.) of the two models using both the density matrix renormalization-group method for large systems and Lancz\'os calculations with twisted boundary conditions for smaller systems. Although most of our results confirm that both models are very similar we have found some important differences in the spin properties for the small sizes considered by previous numerical studies (6-12 sites). However, we show that these differences disappear at larger sizes (14-42 sites) when sufficiently accurate eigenstates are considered. Accordingly, our results strongly suggest that the ground-state phase transition previously found for small systems is a finite size artifact. Interpreting our results within the framework of the Luttinger liquid theory, we discuss the origin of the apparent contradiction between the predictions of the perturbative renormalization-group approach and numerical calculations at small sizes.

DOI
10.1103/physrevb.54.17414

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The low-energy physics of the one-dimensional pair hopping (PH) and attractive Hubbard models are expected to be similar. Based on numerical calculations on small chains, several authors have recently challenged this idea and predicted the existence of a phase transition at half filling and finite positive coupling for the pair-hopping model. We reexamine the controversy by making systematic comparisons between numerical results obtained for the PH and attractive Hubbard models. To do so, we have calculated the Luttinger parameters (spin and charge velocities, stiffnesses, etc.) of the two models using both the density matrix renormalization-group method for large systems and Lancz'os calculations with twisted boundary conditions for smaller systems. Although most of our results confirm that both models are very similar we have found some important differences in the spin properties for the small sizes considered by previous numerical studies (6-12 sites). However, we show that these differences disappear at larger sizes (14-42 sites) when sufficiently accurate eigenstates are considered. Accordingly, our results strongly suggest that the ground-state phase transition previously found for small systems is a finite size artifact. Interpreting our results within the framework of the Luttinger liquid theory, we discuss the origin of the apparent contradiction between the predictions of the perturbative renormalization-group approach and numerical calculations at small sizes.
We use the density-matrix renormalization group to calculate the phase diagram of the one-dimensional extended Hubbard chain at half filling. We find that for one specific U,U,U, the critical value of Vc{V}_{c}Vc​ for the charge-density-wave versus spin-density-wave phase transition is smaller than that predicted from the Monte Carlo simulation. Our results are consistent with the strong- and weak-coupling limit results, whereas the previous results from the Monte Carlo simulations deviate from the weak-coupling limit result. As UUU increases, we find that the first-order phase transition develops gradually. The tricritical point is estimated to occur at Uc=4−−6,{U}_{c}=4--6,Uc​=4−−6, not Uc=3{U}_{c}=3Uc​=3 as previously predicted from the Monte Carlo simulation. Due to the topological difference, we observe that with the periodic boundary condition, the phase transition is more abrupt for finite chains of length N=4MN=4MN=4M than for chains of N=4M+2,N=4M+2,N=4M+2, where MMM is an integer, for the same UUU and V.V.V. The difference will diminish in the thermodynamic limit.
The phase diagram of a one-dimensional tight-binding model with a pair-hopping term (amplitude V) has been the subject of some controversy. Using two-loop renormalization-group equations and the density matrix renormalization group with lengths L\ensuremath{\le}60, we argue that no spin-gap transition occurs at half-filling for positive V, contrary to recent claims. However, we point out that away from half-filling, a phase-separation transition occurs at finite V. This transition and the spin-gap transition occurring at half-filling and negative V are analyzed numerically.
We revisit the one-dimensional attractive Hubbard model by using the Bethe-ansatz-based density-functional theory and density-matrix renormalization method. The ground-state properties of this model are discussed in details for different fillings and different confining conditions in weak-to-intermediate coupling regime. We investigate the ground-state energy, energy gap, and pair-binding energy and compare them with those calculated from the canonical Bardeen-Cooper-Schrieffer approximation. We find that the Bethe-ansatz-based density-functional theory is computationally easy and yields an accurate description of the ground-state properties for weak-to-intermediate interaction strength, different fillings, and confinements. In order to characterize the quantum phase transition in the presence of a harmonic confinement, we calculate the thermodynamic stiffness, the density-functional fidelity, and fidelity susceptibility, respectively. It is shown that with the increase in the number of particles or attractive interaction strength, the system can be driven from the Luther-Emery-type phase to the composite phase of Luther-Emery-type in the wings and insulatinglike in the center.
We investigate numerically the ground state phase diagram of the one-dimensional extended Hubbard model, including an on-site interaction U and a nearest-neighbor interaction V. We focus on the ground state phases of the model in the $V\ensuremath{\gg}U$ region, where previous studies have suggested the possibility of dominant superconducting pairing fluctuations before the system phase separates at a critical value V=VPS.{V=V}_{\mathrm{PS}}.V=VPS​. Using quantum Monte Carlo methods on lattices much larger than in previous Lancz\"os diagonalization studies, we determine the boundary of phase separation, the Luttinger-liquid correlation exponent ${K}_{\ensuremath{\rho}},$ and other correlation functions in this region. We find that phase separation occurs for V significantly smaller than previously reported. In addition, for negative U, we find that a uniform state reenters from phase separation as the electron density is increased towards half filling. For V<VPS,V<{V}_{\mathrm{PS}},V<VPS​, our results show that superconducting fluctuations are not dominant. The system behaves asymptotically as a Luttinger liquid with ${K}_{\ensuremath{\rho}}<1,$ but we also find strong low-energy (but gapped) charge-density fluctuations at a momentum not expected for a standard Luttinger liquid.
Using density matrix renormalization group calculations, we compare results obtained for the $t\ensuremath{-}J,$ one-band Hubbard, and three-band Hubbard models of a two-leg CuO ladder. Spin and charge gaps, pair binding energies, and effective pair hoppings are calculated for a wide range of parameters. All three models have an insulating state at a filling corresponding to one hole per Cu site. For physically relevant parameters their spin gaps are similar in size but they exhibit quite different charge gaps. We find that the binding energy of a pair of doped holes is of the order of the undoped ladder spin gap for all three models. The main difference between the models is the size of the effective pair hopping, which is significantly larger in the three-band model with parameters appropriate for CuO materials than in the other two models.
We have studied the two-dimensional Hubbard model on a square lattice with nearest-neighbor hopping. We first discuss the properties of the model within the mean-field approximation: Because of the form of the band structure, some peculiar features are found. We then discuss the simulation algorithm used and compare simulation results with exact results for 6-site chains to test the reliability of the approach. We present results for thermodynamic properties and correlation functions for lattices up to 8\ifmmode\times\else\texttimes\fi{}8 in spatial size. The system is found to be an antiferromagnetic insulator for all values of the coupling constant at zero temperature in the half-filled-band case, but the long-range order is much smaller than predicted by mean-field theory. We perform a finite-size-scaling analysis to determine the character of the transition at zero coupling. For non-half-filled-band cases, our results suggest that the system is always paramagnetic, in contradiction with Hartree-Fock predictions. The system does not show tendency to ferromagnetism nor triplet superconductivity in the parameter range studied. We also discuss some properties of the attractive Hubbard model in the half-filled-band case.