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Physical review. B, Condensed matter

Thermodynamics of the infinite-UHubbard model

Jonathan S. Yedidia

Published May 1, 1990• 13 citations
Publisher

Abstract

I observe that the high-temperature expansions of the U=\ensuremath{\infty} Hubbard model on two- and three-dimensional bipartite lattices are consistent with a strong separation of energy scales for spin and translational degrees of freedom at an electron density n near 8/11. The high-temperature expansion of the specific heat is nearly identical with the high-temperature expansion of spinless fermionic holes of density 1-n, while the expansion of the uniform magnetic susceptibility is nearly identical with the susceptibility of free spins. Previous finite-size calculations are also consistent with this result.

DOI
10.1103/physrevb.41.9397

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I observe that the high-temperature expansions of the U=\ensuremath{\infty} Hubbard model on two- and three-dimensional bipartite lattices are consistent with a strong separation of energy scales for spin and translational degrees of freedom at an electron density n near 8/11. The high-temperature expansion of the specific heat is nearly identical with the high-temperature expansion of spinless fermionic holes of density 1-n, while the expansion of the uniform magnetic susceptibility is nearly identical with the susceptibility of free spins. Previous finite-size calculations are also consistent with this result.
High-temperature expansions for the specific heat and susceptibility of the strongly correlated Hubbard model are compared with corresponding quantities for spinless free fermions and free spins. It is found numerically that, when the temperature is high, the ratio of the specific heat of the strongly correlated Hubbard model to that of spinless free fermions approaches a particle-density-dependent constant. For fixed temperature, the susceptibility of the Hubbard model is less (greater) than that of free spins when the particle density is below (above) a certain threshold density of approximately 0.7. The ratio of the susceptibilities for the two systems, however, appears to be finite for any value of the particle density. Given that the free spin system does not have a ferromagnetic state at any finite temperature, it is concluded that the strongly correlated Hubbard model does not have a finite phase transition temperature. This conclusion is consistent with recent high-temperature expansion studies.
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.
A perturbation expansion for the thermodynamic potential is performed to the third order by HTSE method. Entropy, internal energy, specific heat, and magnetic susceptibility of the one-dimensional extended Hubbard model are computed as functions of temperature, electron density, and t, U, W. Thermodynamics can be discussed in two temperature regions. The high T-region corresponds to increase in number of double occupied states. The low T-region corresponds to the distribution of electron spin. The effect of t on thermodynamic properties is obvious.
The selfconsistent high-density expansion for thermodynamic quantities of the quantum spin-polarized fermion lattice gas, derived in the preceding paper, is extended to higher orders. The (1/z)2 terms in the free energy are evaluated by summing two-loop diagrams and the corresponding equation for critical temperature in the classical limit is discussed. Ground state phase diagram of the system for arbitrary electron density is derived in the first and second order of the expansion. Properties of the half-filled, one-dimensional spinless fermion model at T = 0, obtained here in the first three orders in 1/z, are compared with the results of exact calculations and of other approximate methods. This provides a test for the unrenormalized high-density expansion method.
All the three-electron wave functions and energy levels are obtained for the one-dimensional Hubbard model. The second and third virial coefficients for the system are calculated exactly. The virial expansion is converted into an electron density expansion which appears to converge much better than the virial series. The two-particle and three-particle contributions to the specific heat, entropy, intra-atomic correlation function, and magnetic susceptibility of the system are obtained and presented graphically.