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Journal of Mathematical Physics

Fermi states of Bose systems in three space dimensions. II. The solvable version of the Hubbard model

Piotr Garbaczewski

Published Oct 1, 1985• 2 citations
Publisher

Abstract

An exact spectral solution of the Hubbard problem on the tetrahedron is given. It is demonstrated that it provides at the same time a partial spectral solution for the Bose version of this (Fermi) model. A remarkable relationship of the Hamiltonians HB=HF+(1−P)HB(1−P) is established, where P is a projection in the state space of the interacting Bose system. This partial observation is then extended to the spatial lattice composed of the four-site Hubbard cells, which interact due to the superimposed perturbation of the hopping type, affecting all sites of the lattice; H=∑(i j) ∑σ=± bi j c*iσ cjσ+U ∑i ni+ni− being the corresponding Hamiltonian. It is thus demonstrated that the joint Bose–Fermi spectral problem (previously established in 1+1 dimensions) makes sense in 1+3 dimensions as well.

DOI
10.1063/1.526738

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An exact spectral solution of the Hubbard problem on the tetrahedron is given. It is demonstrated that it provides at the same time a partial spectral solution for the Bose version of this (Fermi) model. A remarkable relationship of the Hamiltonians HB=HF+(1−P)HB(1−P) is established, where P is a projection in the state space of the interacting Bose system. This partial observation is then extended to the spatial lattice composed of the four-site Hubbard cells, which interact due to the superimposed perturbation of the hopping type, affecting all sites of the lattice; H=∑(i j) ∑σ=± bi j c*iσ cjσ+U ∑i ni+ni− being the corresponding Hamiltonian. It is thus demonstrated that the joint Bose–Fermi spectral problem (previously established in 1+1 dimensions) makes sense in 1+3 dimensions as well.
Recently an exact spectral solution was constructed by Sudarshan and Tata for the (NΘ) Fermi version of the Lee model. We demonstrate that it provides a partial solution for the related pure Bose spectral problems. Moreover, the (NΘ) Bose (Bolsterli–Nelson) version of the Lee model is shown to possess Fermi partners, both exhibiting the partial solubility interplay: finding solutions in the Fermi case would presumably be easier than in the original Bose model. Fermi states of the underlying Bose systems in three space dimensions are explicitly identified.
Abstract It is shown that the extended one-dimensional dimer Bose–Hubbard model with multi-body interactions can be solved exactly by using the algebraic Bethe ansatz mainly due to the site-permutation S 2 symmetry. The solution for the model with up to three-particle hopping and three-body on-site interaction is explicitly shown. As an example of the application, lower part of the excitation energy levels and the ground-state entanglement measure of the standard Bose–Hubbard Hamiltonian with the attractive two-body on-site interaction plus the three-body on-site interaction for 100 bosons with variation of the control parameter are calculated by using the exact solution. It is shown that the attractive three-body on-site interaction reinforces the critical point entanglement of the system, which may be helpful for design of an optical lattice for ultracold atoms or a tuneable superconducting quantum interference device with maximal entanglement.
In this continued study of the connection between classical c-number spinor models and their quantized Fermi partners, we elaborate further necessary consequences of the bosonization. The genuine (c-number) path integral representation of tr exp(−iHt) is derived for the Fermi oscillator and simple lattice Fermi models. We find that the underlying Hamiltonian of the Fermi system HF can be equivalently written as PHBP, where HB is the related Bose Hamiltonian, P is an appropriate projection in the state space of the Bose system, and [P, HB]−=0, HF=PHBP. Grassmann algebras are not used. We prove further that both for the massive Thirring model (MT) and the chiral invariant Gross–Neveu model (CN), the Bethe ansatz eigenstates for the Fermi Hamiltonians are exact eigenstates of the Bose MT and CGN Hamiltonians, so introduced that HF=PHBP, [P, HB]−=0. As a consequence, through studying the c-number path integral representation for tr exp(−iHFt), we establish a class of classical (c-number) spinor solutions of the underlying field equations, which at the same time stationary both the c-number Bose and c-number Fermi actions.
A unified algebraic approach to both the hard-core Fermi– and Bose–Hubbard models is extended to both the finite- and infinite-site with periodic condition cases. Excitation energies and the corresponding wavefunctions of both the models with nearest neighbor hopping are exactly derived by using a new and simple algebraic method. It is found that spectra of both the models are determined simply by eigenvalue problem of N × N hopping matrix, where N is the number of sites for finite system or the period of sites for infinite system.
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.
Based on a class of exactly solvable models of interacting Bose and Fermi liquids, we compute the single-particle propagators of these systems exactly for all wavelengths and energies and in any number of spatial dimensions. The field operators are expressed in terms of Bose fields that correspond to displacements of the condensate in the Bose case and displacements of the Fermi sea in the Fermi case. Unlike some of the previous attempts, the present attempt reduces the answer for the spectral function in any dimension in both Fermi and Bose systems to quadratures. It is shown that when only the lowest-order sea-displacement terms are included, is the random-phase approximation in its many guises recovered in the Fermi case, and Bogoliubov's theory in the Bose case. The momentum distribution is evaluated using two different approaches, exact diagonalization and the equation of motion approach; the novelty being, of course, the exact computation of single-particle properties including short-wavelength behavior.