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Complete spectrum of the infinite-U Hubbard ring using group theory

Alessandro Soncini, Willem Van den Heuvel

Published Jan 1, 2014• 0 citations
OpenAlex

Abstract

We present a full analytical solution of the multiconfigurational strongly-correlated mixed-valence problem corresponding to the N-Hubbard ring filled with N − 1 elec-trons, and infinite on-site repulsion. While the eigenvalues and the eigenstates of the model are known already, analytical determination of their degeneracy is presented here for the first time. The full solution, including degeneracy count, is achieved for each spin configuration by mapping the Hubbard model into a set of Hückel-annulene problems for rings of variable size. The number and size of these effective Hückel an-nulenes, both crucial to obtain Hubbard states and their degeneracy, are determined by solving a well-known combinatorial enumeration problem, the necklace problem for N − 1 beads and two colors, within each subgroup of the CN−1 permutation group. Symmetry-adapted solution of the necklace enumeration problem is finally achieved by means of the subduction of coset representation technique [S. Fujita, Theor. Chem. Acta 76, 247 (1989)], which provides a general and elegant strategy to solve the one-hole infinite-U Hubbard problem, including degeneracy count, for any ring size. The proposed group theoretical strategy to solve the infinite-U Hub-bard problem for N−1 electrons, is easily generalized to the case of arbitrary electron count L, by analyzing the permutation group CL and all its subgroups.

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We present a full analytical solution of the multiconfigurational strongly-correlated mixed-valence problem corresponding to the N-Hubbard ring filled with N − 1 elec-trons, and infinite on-site repulsion. While the eigenvalues and the eigenstates of the model are known already, analytical determination of their degeneracy is presented here for the first time. The full solution, including degeneracy count, is achieved for each spin configuration by mapping the Hubbard model into a set of Hückel-annulene problems for rings of variable size. The number and size of these effective Hückel an-nulenes, both crucial to obtain Hubbard states and their degeneracy, are determined by solving a well-known combinatorial enumeration problem, the necklace problem for N − 1 beads and two colors, within each subgroup of the CN−1 permutation group. Symmetry-adapted solution of the necklace enumeration problem is finally achieved by means of the subduction of coset representation technique [S. Fujita, Theor. Chem. Acta 76, 247 (1989)], which provides a general and elegant strategy to solve the one-hole infinite-U Hubbard problem, including degeneracy count, for any ring size. The proposed group theoretical strategy to solve the infinite-U Hub-bard problem for N−1 electrons, is easily generalized to the case of arbitrary electron count L, by analyzing the permutation group CL and all its subgroups.
We present a full analytical solution of the multiconfigurational strongly correlated mixed-valence problem corresponding to the N-Hubbard ring filled with N-1 electrons, and infinite on-site repulsion. While the eigenvalues and the eigenstates of the model are known already, analytical determination of their degeneracy is presented here for the first time. The full solution, including degeneracy count, is achieved for each spin configuration by mapping the Hubbard model into a set of Hückel-annulene problems for rings of variable size. The number and size of these effective Hückel annulenes, both crucial to obtain Hubbard states and their degeneracy, are determined by solving a well-known combinatorial enumeration problem, the necklace problem for N-1 beads and two colors, within each subgroup of the CN-1 permutation group. Symmetry-adapted solution of the necklace enumeration problem is finally achieved by means of the subduction of coset representation technique [S. Fujita, Theor. Chim. Acta 76, 247 (1989)], which provides a general and elegant strategy to solve the one-hole infinite-U Hubbard problem, including degeneracy count, for any ring size. The proposed group theoretical strategy to solve the infinite-U Hubbard problem for N-1 electrons is easily generalized to the case of arbitrary electron count L, by analyzing the permutation group CL and all its subgroups.
The t - J model with constant t and J between any pair of sites is studied by exploiting the symmetry of the Hamiltonian with respect to site permutations. For a given number of electrons and a given total spin the exchange term simply yields an additive constant. Therefore the real problem is to diagonalize the ` t model', or equivalently the infinite U Hubbard Hamiltonian. Using extensively the properties of the permutation group, we are able to find explicitly both the energy eigenvalues and eigenstates, labelled according to spin quantum numbers and Young diagrams. As a corollary we also obtain the degenerate ground states of the finite U Hubbard model with infinite-range hopping -t>0 .
The two-orbital degenerate Hubbard model proves to be a powerful tool in the investigation of several 3 d and 4 d transition metal oxides where orbital degeneracy is known to play a crucial role. We present here a finite-size cluster study of this model where an exact numerical diagonalization procedure is used, based on the implementation of the symmetries generated by the spin, the pairing and the orbital pseudospin operators. The technique is then applied to the solution of the model on a four-site ring, and an explicit comparison is presented between the behavior of the spin, charge and orbital gaps as the on-site Coulomb repulsion is varied.
A Hubbard-like model with SU(4) symmetry for electrons with twofold orbital degeneracy is studied extensively. Exact solution in one dimension is derived by means of Bethe ansatz, where the sites are supposed to be occupied by at most two electrons. The features of ground state and excited states for repulsive coupling are shown. For finite NNN number of electrons, the configurations of quantum numbers are given explicitly and the spectra of excitations are obtained by solving Bethe-ansatz equation numerically. For infinite N,N,N, the ground state and various kinds of low-lying excitations are obtained on the basis of thermodynamics limit.
The one-dimensional attractive Hubbard model (U 0) is discussed for the chains of N nodes and the same number of electrons, where N -1 of them have the same spin projection, assuming periodic boundary conditions and the half-lling case.Based on the analysis of the eigenvalue problem we provided the general analytical expression for the eigenvalues, for any number N .This formula implies the existence of two elementary particles with mutually dependent momenta on the ring with N sites the same number of electrons including N -1 of the same spin projection.
An interacting spin-fermion model is exactly solved on an open chain. In a certain representation, it is the nearest-neighbor Hubbard model in the limit of infinite UUU (local interaction). Exact solution of its complete energy eigenspectrum is accomplished by introducing a unitary transformation which maps the original problem to a tight-binding model of the fermions only. Physically, the exact solution implies the absence of Nagaoka ferromagnetism in the ground state for arbitrary electron densities. The present method solves a class of very general models exactly. Few more problems are discussed as an application of this unitary transform method.