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arXiv:hep-lat/9205024 · Physical review. B, Condensed matter

Grand-canonical partition function of a two-dimensional Hubbard model

Ian M. Barbour, Elyakum G. Klepfish

Published Jul 1, 1992• 6 citations
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Abstract

We present a technique for a numerical analysis of the phase structure of the two-dimensional Hubbard model as a function of the hole chemical potential. The grand-canonical partition function for the model is obtained via Monte Carlo simulations. The dependence of the hole occupation number on the chemical potential and the temperature is evaluated. These calculations, together with a study of the Yang-Lee zeros of the grand-canonical partition function, show evidence of a phase transition at zero temperature and particle density below half filling. The binding energy of a pair of holes is calculated in the low-temperature regime and the possibility for pairing is explored.

DOI
10.1103/physrevb.46.469

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We present a technique for a numerical analysis of the phase structure of the two-dimensional Hubbard model as a function of the hole chemical potential. The grand-canonical partition function for the model is obtained via Monte Carlo simulations. The dependence of the hole occupation number on the chemical potential and the temperature is evaluated. These calculations, together with a study of the Yang-Lee zeros of the grand-canonical partition function, show evidence of a phase transition at zero temperature and particle density below half filling. The binding energy of a pair of holes is calculated in the low-temperature regime and the possibility for pairing is explored.
We report on a numerical study of the two-dimensional Hubbard model and describe two new algorithms for the simulation of many-electron systems. These algorithms allow one to carry out simulations within the grand canonical ensemble at significantly lower temperatures than had previously been obtained and to calculate ground-state properties with fixed numbers of electrons. We present results for the two-dimensional Hubbard model with half- and quarter-filled bands. Our results support the existence of long-range antiferromagnetic order in the ground state at half-filling and its absence at quarter-filling. Results for the magnetic susceptibility and the momentum occupation along with an upper bound to the spin-wave spectrum are given. We find evidence for an attractive effective d-wave pairing interaction near half-filling but have not found evidence for a phase transition to a superconducting state.
We present a new numerical method for the study of binding energies of particles in fermionic systems. Working with an imaginary chemical potential we can obtain results in the canonical ensemble by simple modifications of standard numerical techniques. We applied the technique to the two-dimensional Hubbard model observing binding of holes at half-filling on lattices of 4\ifmmode\times\else\texttimes\fi{}4 sites. For U/t=4 we estimate that the binding energy is \ensuremath{\Delta}=-0.10\ifmmode\pm\else\textpm\fi{}0.02.
A finite-size scaling is applied to the Yang-Lee zeros of the grand canonical partition function for the two-dimensional Hubbard model in the complex chemical potential plane. The logarithmic scaling of the imaginary part of the zeros with the system size indicates a singular dependence of the carrier density on the chemical potential. Our analysis points to a second-order phase transition with critical exponent 1/\ensuremath{\delta}=1/2\ifmmode\pm\else\textpm\fi{}1/3, which leads to a divergence of the electronic susceptibility. We interpret these results as reflecting the singular behavior of the density of states in the quasiparticle spectrum. \textcopyright{} 1996 The American Physical Society.
We present a method for obtaining grand canonical expectation values as a continuous function of the chemical potential in quantum Monte Carlo simulations. We show how canonical ensemble expectation values can be extracted in such simulations. In order to obtain results for all particle sectors, a modest number of grand canonical simulations must be performed, each at a different chemical potential. With the canonical ensemble results in hand, grand canonical expectation values can be constructed for arbitrary dopings. Results are presented from the application of this method to the two-dimensional Hubbard model.
The electronic momentum distribution n(k) of the two-dimensional (2D) Hubbard model is studied for different values of the coupling U/t, electronic density 〈n〉, and temperature, using quantum Monte Carlo techniques. A detailed analysis of the data on 8\ifmmode\times\else\texttimes\fi{}8 clusters shows that along the line from (0,\ensuremath{\pi}) to (\ensuremath{\pi},0) a minimum in n(k) at momenta k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2) appears as the system is doped away from half-filling. In the range of couplings studied, the depth of this minimum is the largest at density 〈n〉\ensuremath{\approxeq}0.9, and it increases with decreasing temperature. Its existence is correlated with the strength of the antiferromagnetic correlations in the system. The apparent absence of structure at k=(\ensuremath{\pi}/2,\ensuremath{\pi}/2) in previous numerical studies of this model is explained. The results discussed here, together with the recent experimental data for the cuprates discussed by Aebi et al. [Phys. Rev. Lett. 72, 2757 (1994)], reopens the possibility that hole pockets exist in the doped 2D Hubbard model, although more evidence is needed to fully demonstrate their existence.