Bosonisation as the Hubbard Stratonovich Transformation

The problem of strongly correlated electrons in one dimension attracted the attention of condensed matter physicists since early 50’s. After the seminal paper of Tomonaga [8] who suggested the first soluble model in 1950, there were essential achievements reflected in papers by Luttinger [6] (1963) and Mattis and Lieb [7] (1963). Considerable contribution to understanding of the generic properties of the 1D electron liquid has been done by Dzyaloshinskii and Larkin [1] (1973) and Efetov and Larkin [2] (1976). Despite the fact that the main features of the 1D electron liquid were captured and described by the end of 70’s, the investigators felt dissatisfied with the rigour of the theoretical description. The most famous example is the paper by Haldane [4] (1981) where the author developed the fundamentals of the modern bosonisation technique, known as the operator approach. This paper became famous because the author has rigourously shown how to construct the Fermi creation/anihilation operators out of Bose ones. The most recent example of dissatisfaction is the review due to von Delft and Schoeller [9] (1998) who revised the approach to the bosonisation and came up with what they called constructive bosonisation. There exists another way to bosonise the problem. It was suggested 25 years ago by [3] and found its further elaboration in the paper by Lee and Chen [5] in 1988. This approach (sometimes called ”functional bosonisation”), or more precisely its pieces, can be found in variety of papers. The main objective of this lecture is to assemble these pieces in a clear picture. The aim is purely methodological which is the reason why the number of complications (like backscattering, presence of lattice and spin degrees of freedom) are kept at the lowest possible level. Any generalisations can be easily incorporated into the scheme. The final formulation of the bosonised problem will look distinct from what can be found elsewhere. We believe that the approach present in this lecture is the ”shortest shortcut” from the original fermionic problem of interacting electrons to the problem of non-interacting bosons.