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Journal of the Physical Society of Japan

Superconducting Correlation of the One-Dimensionalt-JModel

Masatoshi Imada

Published Nov 15, 1990• 18 citations
Publisher

Abstract

The pairing susceptibility of the one-dimensional t-J model is examined numerically by the quantum transfer matrix method. The temperature dependence of the pairing susceptibility χ s shows logarithmic behavior at low temperatures as χ s ∼αlog T for T being the temperature. The coefficient α is determined as a function of J / t and the fillingness for J and t being the exchange interaction and the transfer. Beyond that of noninteracting fermions, α grows rapidly with the increase of | J |.

DOI
10.1143/jpsj.59.4121

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The pairing susceptibility of the one-dimensional t-J model is examined numerically by the quantum transfer matrix method. The temperature dependence of the pairing susceptibility χ s shows logarithmic behavior at low temperatures as χ s ∼αlog T for T being the temperature. The coefficient α is determined as a function of J / t and the fillingness for J and t being the exchange interaction and the transfer. Beyond that of noninteracting fermions, α grows rapidly with the increase of | J |.
The equal-time pairing correlation function of the two-dimensional t-J model on a square lattice is studied using a high-temperature expansion method. The sum of the pairing correlation, its spatial dependence, and the correlation length are obtained as functions of temperature down to T≃0.2tT \simeq 0.2 tT≃0.2t. By comparison of single-particle contributions in the correlation functions, we find an effective attractive interaction between quasi-particles in dx2−y2d_{x^2-y^2}dx2−y2​-wave pairings. It is shown that d-wave correlation grows rapidly at low temperatures for 0.5 < n < 0.9, with n being the electron density. The temperature for this growth is roughly scaled by J/2. This is in sharp contrast to the Hubbard model in a weak or intermediate coupling region, where there is no numerical evidence of superconductivity.
Numerical results obtained from Monte Carlo simulation in the ground state and at finite temperatures as well as the exact diagonalization and the transfer matrix method are reported. Efficiency of the Monte Carlo method in the ground state is examined. Spin, charge and superconducting correlations are investigated for the Hubbard and the t - J model in one and two dimensions. The momentum distribution in the one-dimensional Hubbard model shows fermi-liquid-like behavior at least in two orders of magnitude smaller energy scale than the band width. The short-range incommensurate spin correlation is observed and analyzed in the doped Hubbard and the t - J model both in one and two dimensions. In the ground state, the superconducting correlation shows the absence of system size dependence in two dimensions at the filling smaller than 0.8 and in one dimension at any filling. The binding energy of two fermions and the spin and charge distortion around an itinerant fermion are also discussed in the t - J and the coupled spin-fermion model.
The physics of the one-dimensional t-J model may be determined at J/t\ensuremath{\rightarrow}0 (i.e., the U/t\ensuremath{\rightarrow}\ensuremath{\infty} limit of the repulsive Hubbard model) and at J/t=2 with the use of the Bethe ansatz. To get a full understanding of the charge and spin correlation functions for all values of J/t, one has to resort to numerical methods. We have used two numerical methods (the world-line quantum Monte Carlo algorithm and the quantum transfer-matrix algorithm) to get information on charge and spin static structure factors, pairing correlations, and critical exponents. The (1+1)-dimensional classical system corresponding to the one-dimensional t-J model, on which the Monte Carlo technique is used, is identified to a fifteen-vertex model. We show that the t-J model undergoes phase separation at large values of J/t. Before phase separation, both critical exponents and pairing correlations in the extended s-wave channel favor the onset of superconductivity. For low values of J/t, the model shows a U/t\ensuremath{\rightarrow}\ensuremath{\infty} Hubbard-like character.
Supercomputing correlations in the two dimensional t-J model at zero temperature are evaluated using numerical techniques. At the fermionic density 〈n〉\ensuremath{\sim}1/2, strong signals of dx2{\mathit{d}}_{\mathit{x}}^{2}dx2​-y2{\mathit{y}}^{2}y2 superconductivity were observed in the ground state. These conclusions are based on a study of static pairing correlations, the Meissner effect, and flux quantization as indicators of superconductivity. A tentative phase diagram of the two dimensional t-j model is presented.
We have calculated the spin correlation length of the half-filled one-dimensional Hubbard model down to the temperature of T =0.01×(transfer integral) by the Bethe-Ansatz method for the quantum transfer matrix. Its temperature dependence shows the crossover from (2 log π T ) -1 at high temperatures to a T -1 with the log corrections at low temperatures, the low- T behavior being consistent with the prediction by the conformal field theory. When the Coulomb repulsion U is small, we have found at low temperatures another crossover in the proportional constant a from the value for the free electron system to that for the antiferromagnetic Heisenberg model. This crossover is due to the disappearance of charge fluctuation. A similar crossover has been observed for the linear coefficient of specific heat, C / T .
We have calculated the spin correlation length of the half-filled one-dimensional Hubbard model down to the temperature of T =0.01×(transfer integral) by the Bethe-Ansatz method for the quantum transfer matrix. Its temperature dependence shows the crossover from (2 log π T ) -1 at high temperatures to a T -1 with the log corrections at low temperatures, the low- T behavior being consistent with the prediction by the conformal field theory. When the Coulomb repulsion U is small, we have found at low temperatures another crossover in the proportional constant a from the value for the free electron system to that for the antiferromagnetic Heisenberg model. This crossover is due to the disappearance of charge fluctuation. A similar crossover has been observed for the linear coefficient of specific heat, C / T .