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Journal of Physics C Solid State Physics

Bosonisation of the one-dimensional two-fermion model: boson representation

Marian Apostol

Published Nov 10, 1983• 10 citations
Publisher

Abstract

Jordan's theory of the boson representation of the fermion fields in one dimension is reviewed and generalised to the one-dimensional two-fermion model (TFM). It is shown that an additional constraint (Jordan's commutator) on the fermion fields has been overlooked so far by the theory of the TFM. This supplementary condition requires a particular cut-off procedure which has been little considered in recent times.

DOI
10.1088/0022-3719/16/31/010

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Jordan's theory of the boson representation of the fermion fields in one dimension is reviewed and generalised to the one-dimensional two-fermion model (TFM). It is shown that an additional constraint (Jordan's commutator) on the fermion fields has been overlooked so far by the theory of the TFM. This supplementary condition requires a particular cut-off procedure which has been little considered in recent times.
The Jordan boson representation and point-splitting regularisation are applied to the one-dimensional two-fermion model (TFM) with momentum-transfer cut-off and zero-mode terms included. It is shown that the correlation functions of the Tomonago-Luttinger model obtained in this way coincide with those given by diagram summation. The energy gaps entering the well known Luther-Emery-Peschel back-scattering and Umklapp scattering solutions acquire a physical dependence on the momentum-transfer cut-off. First-order perturbation calculations of the charge-density function give different results for the spin-parallel 'back-scattering' contribution to the bosonised TFM and corresponding Fermi gas model, thus indicating that these models are not equivalent.
The model one-dimensional electron gas, defined by a particular ultraviolet cut-off procedure, is shown to decompose into two parts, a Tomonaga-Luttinger model and a model equivalent to the two-dimensional Coulomb gas. For a particular value of one of its coupling constants, this latter model is shown to be equivalent to a model of free fermions. These results, familiar features of the boson representation of the one-dimensional electron gas, are here derived entirely within the context of fermions. The cut-off procedure used differs from those normally considered in the literature. The significance of these differences is discussed.
Bosonisation is a technique for studying one-dimensional fermion systems by the means of transforming them to a boson system. S. Coleman showed that the correlation functions for the Sine Gordon field, a massless boson with a cosine interaction term, coincide with those of the massive Thirring model, which is a Dirac fermion with self-interactions. An explicit map can be found between these two systems, and it can in fact be generalised to an entire class of fermions in which the boson-fermion correspondence is a well-defined operator equality in Fock space. More generally, it is a widely used technique to transform a model into another whose nature simplifies the solution or allows for obtaining more information that appears more clearly in the new representation. Another example is the Jordan Wigner transformation, which associates quantum spin chains to one-dimensional fermions. Some aspects about the statistical properties of the chain can be inferred from the fermionic representation, and more properties can be studied by applying bosonisation and examine the system in this new representation.
The equivalence of the one-dimensional two-fermion model, on one hand, with the two-dimensional Coulomb gas and sine-Gordon models on the other is worked out by employing the Jordan boson representation and the point-splitting regularization. The distinction is made throughout between the bandwidth cut-off and momentum-transfer cut-off. It is shown that the coupling constants, both in the weak-coupling regime and near the Luther-Emery exact solution, scale with the momentum-transfer cut-off. The relationship between the bosonized form of the two-fermion model and the corresponding Fermi gas model is briefly discussed.
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.