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Journal of Statistical Mechanics Theory and Experiment

Rigorous analysis of the Tomonaga model by means of Ward identities and the renormalization group

G. Benfatto, Vieri Mastropietro

Published Apr 12, 2005• 0 citations
Publisher

Abstract

The Tomonaga model describes interacting 1D fermions with a linear dispersion relation and a bandwidth cut-off, which destroys local gauge invariance and makes the model not solvable. We rigorously obtain its Schwinger functions by combining renormalization group analysis with Ward identities and a set of ‘correction identities’, which relate the correction terms to formal Ward identities (due to cut-offs) with some Schwinger functions. Contrary to previous cases, the use of the Luttinger model exact solution is completely avoided. Therefore this should be considered the first proof of what has been so far a conjecture: 1D interacting fermions can be constructed on the basis of a non-perturbative analysis independent of any exact solutions (of models which could be shown to have essentially the same beta function) and entirely based on a functional integral approach. The same method allows us to construct essentially all 1D Luttinger liquid models.

DOI
10.1088/1742-5468/2005/04/l04001

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The Tomonaga model describes interacting 1D fermions with a linear dispersion relation and a bandwidth cut-off, which destroys local gauge invariance and makes the model not solvable. We rigorously obtain its Schwinger functions by combining renormalization group analysis with Ward identities and a set of ‘correction identities’, which relate the correction terms to formal Ward identities (due to cut-offs) with some Schwinger functions. Contrary to previous cases, the use of the Luttinger model exact solution is completely avoided. Therefore this should be considered the first proof of what has been so far a conjecture: 1D interacting fermions can be constructed on the basis of a non-perturbative analysis independent of any exact solutions (of models which could be shown to have essentially the same beta function) and entirely based on a functional integral approach. The same method allows us to construct essentially all 1D Luttinger liquid models.
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.
We examine the spin-charge separation effects which are manifest in the Tomonaga-Luttinger model (TLM) in the framework of the field theoretical renormalization group (RG) theory. Initially, it is shown how far one can go with the perturbative RG method. Following that, we take into full account the Ward identities [Phys. Rev. 78, 172 (1950)]. This allows us to calculate the spin and charge velocities exactly. Next, we construct the Callan-Symanzik equation (CSE) [Phys. Rev. D 5, 3202 (1972); Commun. Math. Phys. 23, 49 (1971)] for the renormalized fermionic one-particle Green's function. Finally, making use of those perturbative results for the TLM and taking the two characteristic velocities as basic ingredients, we solve this CSE in qualitative good agreement with the exact result.
The phenomenon of Spin-Charge separation in non-Fermi liquids is well understood only in certain solvable d = 1 fermionic systems. In this paper we furnish the first example of asymptotic Spin-Charge separation in a d = 1 non solvable model. This goal is achieved using Renormalization Group approach combined with Ward-Identities and Schwinger-Dyson equations, corrected by the presence of a bandwidth cut-offs. Such methods, contrary to bosonization, could be in principle applied also to lattice or higher dimensional systems. 1 Introduction and Main results In recent years the properties of non-Fermi liquids have been extensively investigated, both from experimental and theoretical point of view. In particular, one of the most spectacular feature appearing in non-Fermi liquids is the phenomenon of Spin-Charge (SC) separation, which is surely relevant for the physics
The Luttinger model owes its solvability to a number of peculiar features, like its linear relativistic dispersion relation, which are absent in more realistic fermionic systems. Nevertheless according to the Luttinger liquid conjecture a number of relations between exponents and other physical quantities, which are valid in the Luttinger model, are believed to be true in a wide class of systems, including tight binding or jellium one-dimensional fermionic systems. Recently a rigorous proof of several Luttinger liquid relations in nonsolvable models has been achieved; it is based on exact Renormalization Group methods coming from Constructive Quantum Field Theory and its main steps will be reviewed below.
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.
The Luttinger model owes its solvability to a number of peculiar features, like its linear relativistic dispersion relation, which are absent in more realistic fermionic systems. Nevertheless according to the Luttinger liquid conjecture a number of relations between exponents and other physical quantities, which are valid in the Luttinger model, are believed to be true in a wide class of systems, including tight binding or jellium one-dimensional fermionic systems. Recently a rigorous proof of several Luttinger liquid relations in nonsolvable models has been achieved; it is based on exact Renormalization Group methods coming from Constructive Quantum Field Theory and its main steps will be reviewed below.