There are few exact results for the Hubbard model on bipartite lattices of spatial dimension
d>1. Nevertheless, the Hubbard model with transfer integral
t and onsite repulsion
U on bipartite lattices with
Na sites, such as the square, honeycomb, cubic, body-centered cubic, face-centered cubic, and diamond lattices, provides the simplest toy model for describing electronic correlations in many condensed-matter systems and is therefore a quantum problem of considerable physical interest. Seven exact theorems that provide new physical insight into the model are established. Overall, the exact framework based on physical spins and physical
η-spins for the Hubbard model on bipartite lattices of spatial dimension
d>1 introduced in this paper offers a robust foundation for future studies of the model, as well as of the condensed-matter materials, such as cuprate superconductors, graphene and graphene-derived systems, and other quantum systems, that it describes.
We present a new model describing strongly correlated electrons on a general d-dimensional lattice. It is an extended Hubbard model and it contains the t-J model as a special case. The model naturally describes local electron pairs, which can move coherently at arbitrary momentum. By using an \ensuremath{\eta}-pairing mechanism we can construct eigenstates of the Hamiltonian with off-diagonal long-range order. In the attractive case the exact ground state is superconducting in any number of dimensions. On a one-dimensional lattice, the model is exactly solvable by Bethe ansatz.
We construct a solvable Lindblad model in arbitrary dimensions, in which the Liouvillian can be mapped to a BCS-Hubbard model featuring an imaginary Hubbard interaction. The Hilbert space of the system can be divided into multiple sectors, each characterized by an on-site invariant configuration. The model exhibits bistable steady states in all spatial dimensions, which is guaranteed by the fermion-number parity. Notably, the Liouvillian gap exhibits a Zeno transition, below which the Liouvillian gap is linear with respect to the dissipation. We also uncover a generic dimension-dependent gap behavior: In one dimension, the gap originates from multiple sectors with spectral crossing; in higher dimensions, a single sector determines the gap.
We investigate the problem of constructing solvable models of nonrelativistic spin-1/2 interacting fermions on an infinite lattice. The extended Hubbard model, which includes the direct and exchange intersite interaction, is considered. The hopping term in the Hamiltonian is not restricted to nearest-neighbor sites (the ``overjumping'' model), but the short-range character of the standard Hubbard model is retained. The solutions in one- and two-particle sectors are obtained in an explicit form by using the theory of Weierstrass elliptic functions. The possibilities of further development of models of that type are also considered.
We show that the exact ground state for a class of extended Hubbard models including bond-charge, exchange, and pair-hopping terms, is the Yang ``\ensuremath{\eta}-paired'' state for any nonvanishing positive value of the pair-hopping amplitude, at least when the on-site Coulomb interaction is attractive enough and the remaining physical parameters satisfy a single constraint. The ground state is thus rigorously superconducting. Our result holds on a bipartite lattice in any dimension, at any band filling, and for arbitrary electron hopping. \textcopyright{} 1996 The American Physical Society.