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Physical review. B, Condensed matter

Exact many-body solution of the periodic-clustert-t’-Jmodel for cubic systems: Ground-state properties

James K. Freericks, L. M. Falicov

Published Sep 15, 1990• 20 citations
Publisher

Abstract

The t-t'-J model (strongly interacting limit of a particular Hubbard model) is solved exactly on small clusters of eight sites with periodic boundary conditions for the simple-, body-centered-, and face-centered-cubic lattices and for the two-dimensional square lattice. The symmetry, k vector, and spin of the ground state are studied as functions of crystalline environment, interaction strength, and electron concentration. Phase diagrams are presented for stable solutions, and regions of parameter space that exhibit ferromagnetism and heavy-fermionic behavior are identified.

DOI
10.1103/physrevb.42.4960

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Nigerian Journal of Physics•Oct 18, 2006•0 citations
The t-t'-J model (strongly interacting limit of a particular Hubbard model) is solved exactly on small clusters of eight sites with periodic boundary conditions for the simple-, body-centered-, and face-centered-cubic lattices and for the two-dimensional square lattice. The symmetry, k vector, and spin of the ground state are studied as functions of crystalline environment, interaction strength, and electron concentration. Phase diagrams are presented for stable solutions, and regions of parameter space that exhibit ferromagnetism and heavy-fermionic behavior are identified.
An exact solution of an eight-site crystal model with periodic boundary conditions, a {001} thin film with face-centered-cubic crystal structure, is presented for the case of a heavy-fermion system. The object is to study modifications of the many-body electronic structure in the presence of surfaces. The model consists of (a) a single, fully symmetric orbital per site, with nearest-neighbor and second-nearest-neighbor hopping; (b) an infinite Coulomb repulsion between electrons on the same site; (c) antiferromagnetic superexchange interactions; and (d) a nearly half-filled band ((7/8 electron per site). Two structures are investigated: a two-layer and a four-layer fcc slab. The nature of the change from bulk to surface is studied by introducing a variable coupling between periodically extended slabs. The many-body spectrum, the one-electron occupancy, and the change in properties from bulk to surface are obtained and discussed.
This thesis addresses the question of a strongly interacting many-body problem: the Hubbard model on one- and two-dimensional lattices. The technique employed is the small-cluster method wherein the full many-body Hamiltonian for finite clusters with periodic boundary conditions (PBC's) is solved exactly. We first apply this technique to an extended one-dimensional chain and a two-dimensional square-lattice Hubbard model. We then construct a two-dimensional overlayer/substrate Hubbard model with first- and second-nearest-neighbor hopping, on-site and nearest-neighbor interactions, and overlayer-site orbital energy. The ground-state properties are investigated and discussed in a many-body picture.
A density-functional theory for many-body lattice models is considered in which the single-particle density matrix ${\ensuremath{\gamma}}_{\mathrm{ij}}$ is the basic variable. Eigenvalue equations are derived for solving Levy's constrained search of the interaction energy functional $W[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ $W[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ is expressed as the sum of Hartree-Fock energy ${E}_{\mathrm{HF}}[{\ensuremath{\gamma}}_{\mathrm{ij}}]$ and the correlation energy ${E}_{C}[{\ensuremath{\gamma}}_{\mathrm{ij}}].$ Exact results are obtained for ${E}_{C}({\ensuremath{\gamma}}_{12})$ of the Hubbard model on various periodic lattices, where ${\ensuremath{\gamma}}_{\mathrm{ij}}={\ensuremath{\gamma}}_{12}$ for all nearest neighbors i and j. The functional dependence of ${E}_{C}({\ensuremath{\gamma}}_{12})$ is analyzed by varying the number of sites Na,{N}_{a},Na​, band filling Ne,{N}_{e},Ne​, and lattice structure. The infinite one-dimensional chain and one-, two-, or three-dimensional finite clusters with periodic boundary conditions are considered. The properties of ${E}_{C}({\ensuremath{\gamma}}_{12})$ are discussed in the limits of weak $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{0})$ and strong $({\ensuremath{\gamma}}_{12}\ensuremath{\simeq}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})$ electronic correlations, and in the crossover region $({\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}<~{\ensuremath{\gamma}}_{12}<~{\ensuremath{\gamma}}_{12}^{0}).$ Using an appropriate scaling we observe that ${\ensuremath{\varepsilon}}_{C}{(g}_{12}{)=E}_{C}{/E}_{\mathrm{HF}}$ has a pseudo-universal behavior as a function of ${g}_{12}=({\ensuremath{\gamma}}_{12}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}})/({\ensuremath{\gamma}}_{12}^{0}\ensuremath{-}{\ensuremath{\gamma}}_{12}^{\ensuremath{\infty}}).$ The fact that ${\ensuremath{\varepsilon}}_{C}{(g}_{12})$ depends weakly on Na,{N}_{a},Na​, Ne,{N}_{e},Ne​, and lattice structure suggests that the correlation energy of extended systems could be obtained quite accurately from finite-cluster calculations. Finally, the behaviors of ${E}_{C}({\ensuremath{\gamma}}_{12})$ for repulsive (U>0)(U>0)(U>0) and attractive (U<0)(U<0)(U<0) interactions are contrasted.
The ground state of the two-dimensional (2D) t-J model is studied by exact diagonalization for seven different unit cells with N=10, 16, 18, 20, 26, 32, and 36 sites on a square lattice. In the limit of low electron densities (up to six electrons) we calculate the lowest eigenvalues and classify the ground state according to wave vector K, total spin S, and point-group symmetry both in the limit J=0 (U=\ensuremath{\infty} Hubbard model) and for finite J>0. The transition to a completely phase-separated state for J>JPS{\mathit{J}}_{\mathrm{PS}}JPS​(N) is examined by calculating the free energy as a function of the electron concentration. We find that JPS{\mathit{J}}_{\mathrm{PS}}JPS​(N) increases with the lattice size N. An extrapolation of our finite-size results to the thermodynamic limit gives JPS{\mathit{J}}_{\mathrm{PS}}JPS​(\ensuremath{\infty})/t=4.1 (\ifmmode\pm\else\textpm\fi{}0.1).
The ground-state and thermodynamic properties of a simple Hubbard Hamiltonian are studied for an eight-site cubic cluster as functions of the number of electrons and the strength of the electron interaction. The ground-state energy and spin are determined, and the excitation spectrum is computed. Thermodynamic properties including the specific heat, the magnetic susceptibility, and spin correlation functions are calculated. Computations are performed both by explicit diagonalization of the Hamiltonian and by Monte Carlo simulation.
We study the one-band Hubbard model in a three dimensional simple cubic lattice, with periodic boundary conditions, by means of a variational analytic approach. Ground state energies and pairing correlations depend implicitly on the interaction strength (U/41). It is shown that for two electrons, the interaction is always repulsive in the ground state for any positive value of the on-site Coulomb interaction U. Implications of this result for superconductivity are also discussed. Nigerian Journal of Physics Vol. 16(2) 2004: 167-171