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Journal of Physics A Mathematical and General

Functional bosonisation of the Tomonaga-Luttinger model

Derek K. K. Lee, Yang Chen

Published Nov 21, 1988• 53 citations
Publisher

Abstract

A new bosonisation procedure based on an exact method of functional integration is applied to the one-dimensional Tomonaga (1950)-Luttinger (1963) model with forward scattering. This method circumvents the conventional procedure where the fermion field operator is represented by (bosonic) charge and spin density fields. Fictitious auxiliary fields are introduced to deal with the interaction term so that the fermionic degrees of freedom can be integrated out exactly. In addition to the usual description of the system in terms of charge and spin fluctuations, an effective action in terms of the auxiliary fields can be used. The evaluation of the Green function is facilitated if the latter route is taken. The authors' results for the spinless and spin-1/2 cases agree with previous work.

DOI
10.1088/0305-4470/21/22/018

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A new bosonisation procedure based on an exact method of functional integration is applied to the one-dimensional Tomonaga (1950)-Luttinger (1963) model with forward scattering. This method circumvents the conventional procedure where the fermion field operator is represented by (bosonic) charge and spin density fields. Fictitious auxiliary fields are introduced to deal with the interaction term so that the fermionic degrees of freedom can be integrated out exactly. In addition to the usual description of the system in terms of charge and spin fluctuations, an effective action in terms of the auxiliary fields can be used. The evaluation of the Green function is facilitated if the latter route is taken. The authors' results for the spinless and spin-1/2 cases agree with previous work.
Using the functional integral bosonization technique, the spinless Tomonaga - Luttinger model with impurity forward scattering is solved exactly. Explicit analytical results are given for the one- and two-particle spectral functions in terms of a convolution integral representation. Sharp structures in the frequency spectra of an ordered system are smoothed out by the disorder. The existence of a charge - density instability is predicted if the critical exponent is less than -0.5. There is no influence of the disorder on the superconducting spectral function .
We develop a new formulation of the functional renormalization group (RG) for interacting fermions. Our approach unifies the purely fermionic formulation based on the Grassmannian functional integral, which has been used in recent years by many authors, with the traditional Wilsonian RG approach to quantum systems pioneered by Hertz [Phys. Rev. B 14, 1165 (1976)], which attempts to describe the infrared behavior of the system in terms of an effective bosonic theory associated with the soft modes of the underlying fermionic problem. In our approach, we decouple the interaction by means of a suitable Hubbard-Stratonovich transformation (following the Hertz approach), but do not eliminate the fermions; instead, we derive an exact hierarchy of RG flow equations for the irreducible vertices of the resulting coupled field theory involving both fermionic and bosonic fields. The freedom of choosing a momentum transfer cutoff for the bosonic soft modes in addition to the usual band cutoff for the fermions opens the possibility of new RG schemes. In particular, we show how the exact solution of the Tomonaga-Luttinger model (i.e., one-dimensional fermions with linear energy dispersion and interactions involving only small momentum transfers) emerges from the functional RG if one works with a momentum transfer cutoff. Then the Ward identities associated with the local particle conservation at each Fermi point are valid at every stage of the RG flow and provide a solution of an infinite hierarchy of flow equations for the irreducible vertices. The RG flow equation for the irreducible single-particle self-energy can then be closed and can be reduced to a linear integrodifferential equation, the solution of which yields the result familiar from bosonization. We suggest new truncation schemes of the exact hierarchy of flow equations, which might be useful even outside the weak coupling regime.
Correlation functions of one-dimensional Tomonaga-Luttinger models with sine-Gordon-like extensions are computed using bosonisation and scaling techniques. The author shows that by expanding the full correlation functions instead of considering only their first cumulants, exponents for their asymptotic decay, logarithmic corrections as well as renormalised amplitudes can be obtained. Applied to the Hubbard model, the method produces for the first time correlation functions consistent with its symmetries, exact spectrum and numerical simulations.
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.
The theoretical description of interacting fermions in one spatial dimension is simplified by the fact that the low energy spectrum of noniteracting fermions is identical to the one of a harmonic chain. This fermion-boson transmutation allows to describe interacting fermions as a system of coupled oscillators. Tomonaga's model of interacting nonrelativistic fermions on a ring is presented and first discussed in low order perturbation theory. After introducing the concept of two independent species of right and left moving fermions the exact solution of the Tomonaga-Luttinger model is discussed in detail. The momentum distribution and spectral functions are calculated using the method of the bosonization of the field operator. The general Luttinger liquid phenomenology is shortly discussed.