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arXiv:cond-mat/9401012 · Physical review. B, Condensed matter

Bosonization of current-current interactions

Dmitri V. Khveshchenko

Published Jun 15, 1994• 34 citations
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Abstract

We discuss a generalization of the conventional bosonization procedure to the case of current-current interactions which have a natural representation in terms of current instead of fermion-number-density operators. A consistent bosonization procedure requires a geometrical quantization of the Hamiltonian action of ${\mathit{W}}_{\mathrm{\ensuremath{\infty}}}$ on its coadjoint orbits. An integrable example of a nontrivial realization of this symmetry is presented by the Calogero-Sutherland model. For an illustrative nonintegrable example we consider transverse gauge interactions and calculate the fermion Green function.

DOI
10.1103/physrevb.49.16893

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We discuss a generalization of the conventional bosonization procedure to the case of current-current interactions which have a natural representation in terms of current instead of fermion-number-density operators. A consistent bosonization procedure requires a geometrical quantization of the Hamiltonian action of W\ensuremath∞{\mathit{W}}_{\mathrm{\ensuremath{\infty}}}W\ensuremath∞​ on its coadjoint orbits. An integrable example of a nontrivial realization of this symmetry is presented by the Calogero-Sutherland model. For an illustrative nonintegrable example we consider transverse gauge interactions and calculate the fermion Green function.
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We propose an approach to treating (1 + 1)-dimensional fermionic systems based on the idea of algebraic bosonization. This amounts to decomposing the elementary low-lying excitations around the Fermi surface in terms of basic building blocks carrying a representation of the [Formula: see text] algebra, which is the dynamical symmetry of the Fermi quantum incompressible fluid. This symmetry simply expresses the local particle number current conservation at the Fermi surface. The general approach is illustrated in detail through two examples: the Heisenberg and Calogero–Sutherland models, which allow comparison with the exact Bethe ansatz solution.
Chapter 31 discussed the generic O(N) non-linear σ-model. We have noticed that the abelian case N = 2 is special because the RG β-function vanishes in two dimensions. The corresponding O(2) invariant spin model is specially interesting: it provides an example of the celebrated Kosterlitz–Thouless phase transition and will be examined in Chapter 33. However, a thorough discussion of this model requires a new technique: we have to establish relations, special to two dimensions, between fermion and boson local field theories, by a method called bosonization. This chapter illustrates the derivation involves several steps with the help of various other two-dimensional models which are physically interesting in their own right. It first studies the free massless boson and fermion fields. It evaluates the determinant of the covariant fermion derivative in the presence of an external gauge field. The result is at the basis of the bosonization technique. The chapter then discusses the sine–Gordon (SG) model. It solves the Schwinger model, QED with massless fermions. Finally, it demonstrates the equivalence between the SG model and two fermion models with current-current interaction: the Thirring model and another model with two species of fermions.
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