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Journal of Physics Condensed Matter

A model for divalent-atom systems-one dimension

John P. Hernandez, L.W. Martin

Published Mar 5, 1990• 0 citations
Publisher

Abstract

A one-dimensional system, with L sites and periodic boundary conditions, is considered. N divalent atoms occupy sites. The atoms have two orbitals, electron hopping is allowed between nearest neighbour atoms, and intra-atomic electron correlation is taken into account. The ground and first excited many-body states are estimated using a basis, appropriate to weak interatomic coupling, in which multiple nearest neighbour resonating valence bond excitations, appropriately symmetrised, are included. Ordered and disordered systems are considered, for large L.

DOI
10.1088/0953-8984/2/9/014

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A one-dimensional system, with L sites and periodic boundary conditions, is considered. N divalent atoms occupy sites. The atoms have two orbitals, electron hopping is allowed between nearest neighbour atoms, and intra-atomic electron correlation is taken into account. The ground and first excited many-body states are estimated using a basis, appropriate to weak interatomic coupling, in which multiple nearest neighbour resonating valence bond excitations, appropriately symmetrised, are included. Ordered and disordered systems are considered, for large L.
A one-dimensional-lattice model consisting of a single narrow band and excluded double occupancy of the sites is considered. The electrons on nearest-neighbor sites interact via a charge interaction and spin exchange. In a previous paper we discussed the exact solution of the model for special values of the coupling constants [related to SU(3) invariance]. Here we construct an asymptotically exact solution in the low-electron-density limit for coupling parameters deviating from SU(3) symmetry. The results are compared to those obtained via multiplicative renormalization for the model in the continuum limit and used to discuss the phase diagram.
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.
A fully variational resonance-valence-bond (RVB) study is performed for two-, three- and four-legged isotropic spin-1/2 Heisenberg ladders under periodic boundary conditions along the rungs and periodic/twisted boundary conditions along the chains. By using the full lattice groups of these systems very accurate results have been obtained for the ground-state bulk limit properties in the case of the dimer RVB basis. The results reported here represent the best one can do within the dimer RVB ansatz and should be useful when comparing with results obtained from an a priori fixed structure of the wavefunction. Further results are presented for some finite-sized systems, extending beyond the dimer RVB approach by including RVB functions containing spin pairings between non-nearest-neighbour sites. These results indicate that the three-legged ladder with periodic boundary conditions along the rungs falls into the same universality class of ladders possessing short-range RVB wavefunctions as the two- and four-legged ladders.
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.