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Journal of the Physical Society of Japan

Kosterlitz–Thouless Transition in Two-Dimensional Hubbard Model Evidenced from Quantum Monte Carlo Calculations of Susceptibilities

Takashi Yanagisawa

Published Jun 10, 2010• 7 citations
Publisher

Abstract

Susceptibilities in the static limit ω→0 are evaluated to show that the superconducting Kosterlitz–Thouless transition exists in the two-dimensional Hubbard model for optimal band parameters. By using quantum Monte Carlo methods, we have examined the size dependence of the spin susceptibility χ( Q ) at half filling and the pair susceptibilities with d - and s -wave symmetries for the repulsive and attractive interactions, respectively. The numerical results for systems of up to 14×14 sites show that the s -wave pair susceptibility scales as χ pair ∼ L 2 for the attractive case, in agreement with previous quantum Monte Carlo studies. The scaling χ pair ∼ L 2 also holds for the d -wave pair susceptibility for the repulsive Hubbard model if we adjust the band parameter t '. Therefore, it is likely that the superconducting Kosterlitz–Thouless transition exists in the two-dimensional repulsive Hubbard model.

DOI
10.1143/jpsj.79.063708

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We study the superconducting Kosterlitz-Thouless (KT) transition of the attractive Hubbard model on a two-dimensional triangular lattice using auxiliary field quantum Monte Carlo method for system sizes up to $14\ifmmode\times\else\texttimes\fi{}14$ sites. Combining three methods to analyze the numerical data, we find, for the attractive interaction of $U=\ensuremath{-}4t$, that the transition temperature is broadly maximized around the band filling of 1.2<n<1.61.2<n<1.61.2<n<1.6, while it is found to be rather low in the range of n<1n<1n<1 and n>1.8n>1.8n>1.8. We discuss that this band filling dependence may be explained in terms of the band filling dependence of the superfluid density and the effective mass. Our analysis also teaches us that the estimation of the KT transition temperature requires some caution when the density of states near the Fermi level is small and the calculation is restricted to small system sizes.
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Based on extensions of the grand-canonical quantum Monte Carlo algorithm to incorporate magnetic fields, we provide numerical data confirming the existence of a Kosterlitz-Thouless transition in the attractive Hubbard model. Here, we calculate the temperature derivative of the superfluid density, [\ensuremath{\partial}\ensuremath{\beta}Ds{\mathit{D}}_{\mathit{s}}Ds​(\ensuremath{\beta})]/\ensuremath{\partial}\ensuremath{\beta}, to pin down the transition. The latter quantity is obtained from the difference in internal energy of systems which differ by a phase twist \ensuremath{\pi}/2 in the boundary condition along one lattice direction. Away from half-band filling, [\ensuremath{\partial}\ensuremath{\beta}Ds{\mathit{D}}_{\mathit{s}}Ds​(\ensuremath{\beta})]/\ensuremath{\partial}\ensuremath{\beta} shows a response which increases with lattice size at the transition temperature. In contrast, such a signal is not observed at half-band filling.
In this study we have computed the pair correlation functions in the two-dimensional Hubbard model using a quantum Monte Carlo method. We employ a new diagonalization algorithm in the quantum Monte Carlo method which is free from the negative sign problem. We show that the d-wave pairing correlation function is indeed enhanced slightly for the positive on-site Coulomb interaction U when doping away from the half-filling. When the system size becomes large, the pair correlation function P d increases for U > 0 compared to the non-interacting case, while P d is suppressed for U > 0 when the system size is small. The enhancement ratio P d [ U ]/ P d [ U = 0] will give a criterion on the existence of superconductivity. The ratio P d [ U ]/ P d [ U = 0] increases almost linearly ∝ L when the system size L × L is increased. This increase is a good indication of the existence of a superconducting phase in the two-dimensional Hubbard model. There is, however, no enhancement of pair correlation functions in the half-filled case, which indicates the absence of superconductivity without hole doping.
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