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arXiv:cond-mat/0512098 · Journal of Physics A Mathematical and General

Field theory renormalization group: the Tomonaga–Luttinger model revisited

A Ferraz

Published Jun 7, 2006• 5 citations
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Abstract

We apply field theoretical renormalization group (RG) methods to describe the Tomonaga–Luttinger model as an important test ground to deal with spin–charge separation effects in higher spatial dimensions. We calculate the RG equations for the renormalized forward couplings g 2 R and g 4 R up to two-loop order and demonstrate that they do not flow in the vicinities of the Fermi points (FPs). We solve the Callan–Symanzik equation for G ( a ) R in the vicinities of the FPs. We calculate the related spectral function and the momentum distribution function at p = k F , p 0 = ω. We compute the renormalized one-particle irreducible function Γ (2) (+) R ( p , p 0 = 0; Λ) and show that it carries important spin–charge separation effects in agreement with well-known results. Finally, we discuss the implementation of the RG scheme taking into account the important simplifications produced by the Ward identities.

DOI
10.1088/0305-4470/39/25/s11

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We apply field theoretical renormalization group (RG) methods to describe the Tomonaga–Luttinger model as an important test ground to deal with spin–charge separation effects in higher spatial dimensions. We calculate the RG equations for the renormalized forward couplings g 2 R and g 4 R up to two-loop order and demonstrate that they do not flow in the vicinities of the Fermi points (FPs). We solve the Callan–Symanzik equation for G ( a ) R in the vicinities of the FPs. We calculate the related spectral function and the momentum distribution function at p = k F , p 0 = ω. We compute the renormalized one-particle irreducible function Γ (2) (+) R ( p , p 0 = 0; Λ) and show that it carries important spin–charge separation effects in agreement with well-known results. Finally, we discuss the implementation of the RG scheme taking into account the important simplifications produced by the Ward identities.
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.
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.
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The absence of fermionic, asymptotical one-particle states in the Luttinger model raises the suspicion that the interactions are actually strong at the vicinity of the Fermi points. The functional internal space renormalization group method, a systematical scheme for the computation of effective coupling strengths off the Fermi points, is applied to shed some light on this issue. A simple truncation of the evolution equation shows that the theory is indeed strongly coupled at the Fermi points for arbitrarily small value of the bare coupling constant and the removal of the cutoff is blocked by Landau poles. A peculiar feature of the normal ordering scheme is pointed out to explain the absence of these effects in the bosonized solution.
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
We study the renormalization group flow of the Luttinger-Ward functional and of its two-particle-irreducible vertex functions, given a cutoff in the two-particle interaction. We derive a conserving approximation to the flow and relate it to the fluctuation exchange approximation as well as to nonconserving approximations introduced in an earlier publication [J. F. Rentrop, S. G. Jakobs, and V. Meden, J. Phys. A: Math. Theor. 48, 145002 (2015)]. We apply the different approximate flow equations to the single-impurity Anderson model in thermal equilibrium at vanishing temperature. Numerical results for the effective mass, the spin susceptibility, the charge susceptibility, and the linear conductance reflect the similarity of the methods to the fluctuation exchange approximation. We find the majority of the approximations to deviate stronger from the exact results than one-particle-irreducible functional renormalization group schemes. However, we identify a simple static two-particle-irreducible flow scheme which performs remarkably well and produces an exponential Kondo-like scale in the renormalized level position.