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Journal of Mathematical Physics

Fermi states of Bose systems in three space dimensions

Piotr Garbaczewski

Published Mar 1, 1985• 2 citations
Publisher

Abstract

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.

DOI
10.1063/1.526636

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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.
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.
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.
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.
We generalize the notion of partial dynamical symmetry (PDS) to a system of interacting bosons and fermions. In a PDS, selected states of the Hamiltonian are solvable and preserve the symmetry exactly, while other states are mixed. As a first example of such novel symmetry construction, spectral features of the odd-mass nucleus 195Pt^{195}\mathrm{Pt}195Pt are analyzed.
We demonstrate that Bose systems can be used to simulate properties of their Fermi partners in thermal bath, provided the spectral property HB =PHBP +(1−P)HB(1−P), HF =PHBP can be proved in the state space of the Bose system. The approximation accuracy can be made arbitrarily good by varying the (free) coupling parameter λ∈(0,∞), and permits studying of fermionic partition or correlation functions in a finite volume by means of the standard techniques (Bose:path integral, Monte Carlo, etc).