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Journal of Physics C Solid State Physics

On the vertex function method in the Tomonaga-Luttinger model

Marian Apostol

Published Feb 10, 1983• 0 citations
Publisher

Abstract

The Ward identities relating the three-point vertex functions to the single-particle Green functions as well as the generalised Ward identities between the four- and three-point vertex functions are derived analytically for the Tomonaga-Luttinger model by using explicitly the boson algebra of the particle-hole operators.
DOI
10.1088/0022-3719/16/4/013

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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.
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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.
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