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arXiv:cond-mat/0112270

Bosonisation as the Hubbard-Stratonovich Transformation

I. Yurkevich

Published Dec 14, 2001• 1 citation
arXivHTMLPDF

Abstract

The purpose of this Lecture is to demonstrate how the well-known results on Luttinger Liquid can be obtained with the use of the coherent state integral.
DOI
10.48550/arxiv.cond-mat/0112270

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The present work consists of two parts. The first part deals with hyperbolic Hubbard-Stratonovich transformations. Such transformations are used to derive non-linear sigma models that describe the low energy behaviour of disordered electron systems. For a long time the mathematical status of hyperbolic Hubbard-Stratonovich transformations of Pruisken-Schäfer type remained unclear. Only recently the two special cases of pseudounitary and pseudoorthogonal symmetry were proven . In this thesis we prove the transformation for a general (essentially semisimple) symmetry group. The proof is descriptive and shows explicitly the connection to the standard Gaussian integrals. In the second part we develop a novel method to bosonise granular fermionic systems. The method is related to the method of coherent states. In particular it is not based on the well known bosonisation of (1+1)-dimensional systems. One approach is to use the colour-flavour transformation to transform the Grassmann path integral representation of a grand canonical partition function in a way that allows to eliminate the Grassmann variables. The result is a path integral in generalised coherent states with special boundary conditions.