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Journal of Physics A Mathematical and General

Bosons, fermions and spins 1/2: the interplay on lattices of arbitrary dimension

P Garbaczewksi

Published Aug 1, 1987• 1 citation
Publisher

Abstract

The author establishes relationships between Bose, Fermi and spin-1/2 systems on lattices of arbitrary dimension and arbitrary but finite size which are defined to live in the (common) state space of the boson system. The underlying systems display an essential affinity by being described by precisely the same form of Hamiltonians, which are the polynomial functions of canonical operator variables: boson, fermion or spin-1/2 variables, respectively. For large enough, densely populated finite volume lattices it entails the computation of partition and correlation functions for lattice fermions in terms of bosons only. Albeit an approximation, its accuracy increases as the continuum limit is approached.

DOI
10.1088/0305-4470/20/11/040

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The author establishes relationships between Bose, Fermi and spin-1/2 systems on lattices of arbitrary dimension and arbitrary but finite size which are defined to live in the (common) state space of the boson system. The underlying systems display an essential affinity by being described by precisely the same form of Hamiltonians, which are the polynomial functions of canonical operator variables: boson, fermion or spin-1/2 variables, respectively. For large enough, densely populated finite volume lattices it entails the computation of partition and correlation functions for lattice fermions in terms of bosons only. Albeit an approximation, its accuracy increases as the continuum limit is approached.
I consider general interacting systems of quantum particles in one spatial dimension. These consist of bosons or fermions, which can have any number of components, arbitrary spin, or a combination thereof, featuring low-energy two- and multiparticle interactions. The single-particle dispersion can be Galilean (nonrelativistic), relativistic, or have any other form that may be relevant for the continuum limit of lattice theories. Using an algebra of generalized functions, statistical transmutation operators that are genuinely unitary are obtained, putting bosons and fermions in a one-to-one correspondence without the need for a short-distance hard core. In the nonrelativistic case, low-energy interactions for bosons yield, order by order, fermionic dual interactions that correspond to the standard low-energy expansion for fermions. In this way, interacting fermions and bosons are fully equivalent to each other at low energies. While the Bose-Fermi mappings do not depend on microscopic details, the resulting statistical interactions heavily depend on the kinetic energy structure of the respective Hamiltonians. These statistical interactions are obtained explicitly for a variety of models, and regularized and renormalized in the momentum representation, which allows for theoretically and computationally feasible implementations of the dual theories. The mapping is rewritten as a gauge interaction, and one-dimensional anyons are also considered.
For any purely fermionic system on a lattice in arbitrary space dimension we prove that there is a different but well-defined system on the same lattice, consisting of both bona fide fermions and bosons with an interaction depending on a parameter \ensuremath{\lambda}, which characterizes a same-site repulsion between particles. The energy spectrum and the scattering matrix of the former are identical to those in the finite energy sector of the latter, in the limit \ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty} (i.e., hard-core repulsion). This general theorem is applied to a resonance-boson model in a lattice fermion system, whose solution may have relevance to high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductivity.
The well-known Jordan-Wigner transformation maps an arbitrary system of fermions on a one-dimensional lattice to a system of spins. An essential property of this transformation is that it preserves the locality of observables. This transformation has been recently extended to two-dimensional lattices, where it maps an arbitrary system of fermions to a gauge theory, with locality preserved. Here, the authors construct a bosonization map for arbitrary systems of fermions in three dimensions. As an application, the authors design a model of interacting spins on a cubic lattice, which is exactly soluble and is equivalent to a tight-binding model of free fermions with Dirac points.
One of the methods used to extend two-dimensional bosonization to four space-time dimensions involves a transformation to new spatial variables so that only one of them appears kinematically. The problem is then reduced to an Abelian version of two-dimensional bosonization with extra ``internal'' coordinates. On a formal level, putting these internal coordinates on a finite lattice seems to provide a well-defined prescription for calculating correlation functions. However, in the infinite-lattice or continuum limits, certain difficulties appear that require very delicate specification of all of the many limiting procedures involved in the construction.