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Physical Review Letters

Ferromagnetism in the two-dimensionalt-Jmodel

W. O. Putikka, Marco U. Luchini, Masao Ogata

Published Oct 12, 1992• 97 citations
Publisher

Abstract

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.

DOI
10.1103/physrevlett.69.2288

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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 \ensuremathχ0{\mathrm{\ensuremath{\chi}}}_{0}\ensuremathχ0​. For J=0 fourteenth-order series have been calculated for F and \ensuremathχ0{\mathrm{\ensuremath{\chi}}}_{0}\ensuremathχ0​ 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 \ensuremathχ0{\mathrm{\ensuremath{\chi}}}_{0}\ensuremathχ0​ 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 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
We present results for the thermodynamics of the two-dimensional t-J model on finite square lattices in the grand canonical ensemble. The specific heat has a double-peak structure independent of J. The susceptibility shows antiferromagnetic behavior above a critical value Jc{\mathit{J}}_{\mathit{c}}Jc​ where for JJc{\mathit{J}}_{\mathit{c}}Jc​ the ground state of the one-hole sector has maximal spin. We give evidence that phase separation observed in the ground state of the t-J model at large J may be suppressed at any finite, nonzero T.
Abstract The instability of the fully polarized ferromagnetic ground state (Nagaoka state) of the Hubbard model on the square lattice is investigated. We use single spin flip variational wave functions including majority spin correlation effects and calculate spin flip energies in the thermodynamic limit. With very local wave functions and with moderate numbers of variational parameters we reproduce the best known estimate for the critical hole density δcr = 0.29 and we obtain an estimate of Ucr = 63 t for the critical coupling which is considerably better than the best estimate of Ucr = 42 t previously known. The simplicity of our wave functions makes the physical origin of the various aspects of the instability particularly transparent.
Using finite-temperature strong-coupling expansions for the SU(NNN) Hubbard model, we calculate the thermodynamic properties of the model in the infinite-UUU limit for arbitrary density $0\ensuremath{\le}\ensuremath{\rho}\ensuremath{\le}1$ and all NNN. We express the ferromagnetic susceptibility of the model as a Curie term plus $\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}$, an excess susceptibility above the Curie behavior. We show that, on a bipartite lattice, graph by graph the contributions to $\mathrm{\ensuremath{\Delta}}\ensuremath{\chi}$ are non-negative in the limit that the hole density $\ensuremath{\delta}=1\ensuremath{-}\ensuremath{\rho}$ goes to zero. By summing the contributions from all graphs consisting of closed loops we find that the low-hole-density ferromagnetic susceptibility diverges exponentially as $exp\mathrm{\ensuremath{\Delta}}/T$ as $T\ensuremath{\rightarrow}0$ in two and higher dimensions. This demonstrates that the Nagaoka-Thouless ferromagnetic state exists as a thermodynamic state of matter at low enough density of holes and sufficiently low temperatures. The constant $\mathrm{\ensuremath{\Delta}}$ scales with the SU(NNN) parameter NNN as 1/N1/N1/N implying that ferromagnetism is gradually weakened with increasing NNN as the characteristic temperature scale for ferromagnetic order goes down.
We present different numerical calculations based on variational quantum Monte Carlo simulations supporting a ferromagnetic ground state for finite and small hole densities in the two-dimensional infinite-U Hubbard model. Moreover, by studying the energies of different total spin sectors, these calculations strongly suggest that the paramagnetic phase is unstable against a phase with a partial polarization for large hole densities delta approximately 0.40 with evidence for a second-order transition to the paramagnetic large doping phase.