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Physical Review Letters

Aspects of the phase diagram of the two-dimensionalt-Jmodel

W. O. Putikka, Marco U. Luchini, Thomas Maurice Rice

Published Jan 27, 1992• 172 citations
Publisher

Abstract

The phase diagram of the 2D t-J model is investigated using high-temperature expansions. Series for the Helmholtz free energy, the inverse compressibility, the chemical potential, and the uniform spin susceptibility through tenth order are calculated and analyzed. A region of phase separation is found at T=0 for J/t lying above a line extending from J/t=3.8 at zero filling to J/t=1.2 at half filling. For very small J/t near half filling where the Nagaoka effect is possible, we find a region of divergent uniform magnetic susceptibility at T=0.

DOI
10.1103/physrevlett.68.538

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The phase diagram of the 2D t-J model is investigated using high-temperature expansions. Series for the Helmholtz free energy, the inverse compressibility, the chemical potential, and the uniform spin susceptibility through tenth order are calculated and analyzed. A region of phase separation is found at T=0 for J/t lying above a line extending from J/t=3.8 at zero filling to J/t=1.2 at half filling. For very small J/t near half filling where the Nagaoka effect is possible, we find a region of divergent uniform magnetic susceptibility at T=0.
The phase diagram of the 2D t-J model has been investigated using high temperature expansions. Series for the Helmholtz free energy, the inverse compressibility, the chemical potential and the uniform spin susceptibility through tenth order on a square lattice have been calculated using the finite cluster method.1 The series are analytically continued beyond their radius of convergence by Padé and integral approximants. The most accurate extrapolations can be made for the Helmholtz free energy where for J/t≈0.3 and n≈0.9 we can reach T~t/5. We can test the accuracy of the continuation methods by comparing with the 1D results for the boundary of phase separation of Ogata et al.2 In 2D a region of phase separation was found at T=0 for J/t lying above a line extending from J/t=3.8 at zero filling to J/t=1.2 at half filling. No phase separation was found at very small J/t contrary to the earlier suggestion of Emery et al3 which was based on results from exact diagonalisation on 4×4 clusters but in agreement with Quantum Monte Carlo (QMC) calculations on the Hubbard model.4 For very small J/t near half filling where the Nagaoka effect is possible, we find a region of divergent uniform magnetic susceptibility at T=0. However the divergence is very weak when compared with the exponential behaviour expected from a 2D ferromagnet. This might imply a substantially reduced moment which is consistent with the recent QMC estimates of Zhang et al.5
The occurrance of ferromagnetism for the t-J model on a square lattice is investigated using high-temperature series for the Helmholtz free energy F and the uniform magnetic susceptibility ${\mathrm{\ensuremath{\chi}}}_{0}$. For J=0 fourteenth-order series have been calculated for F and ${\mathrm{\ensuremath{\chi}}}_{0}$ and we estimate F(T=0) to find the ground-state energy. A region of fully polarized ferromagnetism is found for J0. A ferrimagnetic region is observed for larger J/t where ${\mathrm{\ensuremath{\chi}}}_{0}$ is divergent, but F(T=0) is below that of spinless fermions. Our results do not support Nagaoka's theorem at a thermodynamic density of holes.
The phase diagram of the one-dimensional t-J model is investigated by analyzing the results of exact diagonalization and the exact solutions at J/t=0 and 2. Phase separation takes place above a critical value of J around Jc{\mathit{J}}_{\mathit{c}}Jc​/t=2.5--3.5 depending on the electron density. In the small-J region, Tomonaga-Luttinger liquid theory holds and its correlation exponents are calculated as a function of J/t and the electron density. Superconducting correlations become dominant in a region between the solvable case (J/t=2) and phase separation. A spin-gap region is also found at low density.
Abstract We investigated the two-dimensional Hubbard model using many-body perturbation theory, focusing on the systematic computation of thermodynamic properties. The contributions of generalized Hugenholtz (Feynman) diagrams were evaluated symbolically and reformulated using divided differences, enabling efficient and precise calculations. We derived the coefficients of the grand potential expansion up to the eighth order in both the interaction potential (U-expansion) and the inverse temperature (β-expansion). Additionally, we presented results for magnetic susceptibility and heat capacity at half-filling, providing critical insights into the thermodynamic behavior of the model in this regime.