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arXiv:cond-mat/0005277 · Physical review. B, Condensed matter

Third-order renormalization group applied to the attractive one-dimensional Fermi gas

P. Carta

Published Oct 15, 2000• 1 citation
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Abstract

We consider a Callan-Symanzik and Wilson renormalization-group (RG) approach to the infrared problem for interacting fermions in one dimension with backscattering. We compute the third-order (two-loop) approximation of the $\ensuremath{\beta}$ function using both methods, and compare it with the well-known multiplicative Gell--Mann Low approach. We point out a previously unnoticed strong instability of the third-order fixed point with respect to an arbitrary dimensionless parameter, which suggests a RG flow toward a strong-coupling phase.

DOI
10.1103/physrevb.62.10687

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We consider a Callan-Symanzik and Wilson renormalization-group (RG) approach to the infrared problem for interacting fermions in one dimension with backscattering. We compute the third-order (two-loop) approximation of the \ensuremathβ\ensuremath{\beta}\ensuremathβ function using both methods, and compare it with the well-known multiplicative Gell--Mann Low approach. We point out a previously unnoticed strong instability of the third-order fixed point with respect to an arbitrary dimensionless parameter, which suggests a RG flow toward a strong-coupling phase.
A weakly interacting Bose–Fermi mixture model was investigated using Wisonian renormalization group (RG). This model includes one boson–boson interaction term and one boson–fermion interaction term. The scaling dimensions of the two interaction coupling constants were calculated as 2-D at tree level and the flow equations were derived at one-loop level. We find that in the flow equations the contributions from the fermion loops go to zero as the length scale approaches infinity. In three-dimensional case two fixed points are calculated. One is the Gaussian fixed point and the other one is Wilson–Fisher fixed point. We observe that the boson–fermion interaction decouples at the Wilson–Fisher fixed point. We also find that under RG transformation the boson–fermion interaction coupling constant runs to negative infinity with a small negative initial value, which indicates a boson–fermion pairing instability. Furthermore, the possibility of emergent supersymmetry in this model was discussed.
We implement an explicit two-loop calculation of the coupling functions and the self-energy of interacting fermions with a two-dimensional flat Fermi surface in the framework of the field theoretical renormalization group (RG) approach. Throughout the calculation both the Fermi surface and the Fermi velocity are assumed to be fixed and unaffected by interactions. We show that in two dimensions, in a weak coupling regime, there is no significant change in the RG flow compared to the well-known one-loop results available in the literature. However, if we extrapolate the flow to a moderate coupling regime there are interesting new features associated with an anisotropic suppression of the quasiparticle weight ZZZ along the Fermi surface, and the vanishing of the renormalized coupling functions for several choices of the external momenta.
We present a general method to study weak-coupling instabilities of a large class of interacting electron models in a controlled and unbiased way. Quite generally, the electron gas is unstable towards a superconducting state even in the absence of phonons, since high-energy spin fluctuations create an effective attraction between the quasi-particles. As an example, we show the occurrence of d-wave pairing in the repulsive Hubbard model in two dimensions. In one dimension or if the Fermi surface is nested, there are several competing instabilities. The required renormalization group formalism for this case is presented to lowest (one-loop) order on a most elementary level, connecting the idea of the “parquet summation” to the more modern concept of Wilson's effective action. The validity and restrictions of the one-loop approximation are discussed in detail. As a result, three different renormalization group approaches known in the literature are shown to be equivalent within the regime of applicability. We also briefly discuss the open problem of a two-dimensional Fermi system at Van Hove filling without nesting.
We study critical Fermi surfaces in generic dimensions arising from coupling finite-density fermions with transverse gauge fields by applying a dimensional regularization scheme developed previously [Phys. Rev. B 92, 035141 (2015)]. We consider the cases of U (1) and U (1) × U (1) transverse gauge couplings and extract the nature of the renormalization group (RG) flow fixed points as well as the critical scalings.Our analysis allows us to treat a critical Fermi surface of a generic dimension m perturbatively in an expansion parameter = (2m)/(m + 1).One of our key results is that although the two-loop corrections do not alter the existence of an RG flow fixed line for certain U (1) × U (1) theories, which was identified earlier for m = 1 at one-loop order, the third-order diagrams do.However, this fixed line feature is also obtained for m > 1, where the answer is one-loop exact due to UV/IR mixing.
We present the formalism for a two-loop renormalization group (RG) calculation of some order-parameter susceptibilities associated with a two-dimensional (2D) flat Fermi-surface model. In this order of perturbation theory, one must take into account the self-energy effects directly in all RG flow equations. In one-loop order, our calculation reproduces the well-known results obtained previously by other RG schemes. That is, for repulsive interactions all susceptibilities diverge in the low-energy limit and the antiferromagnetic (AF) spin-density-wave correlations produce indeed the leading instability in the system. In contrast, in two-loop order, only the AF susceptibility diverges for this model. However, even this divergence takes place at a much slower rate than in the one-loop RG approach. The purpose of this paper is to show in a very simple setting how to assess the importance of two-loop quantum fluctuations in 2D interacting fermionic models. With some modifications, the present formalism can also be extended to discuss more realistic models such as the paradigmatic 2D Hubbard model.
The one-dimensional Fermi system with Tomonaga-type (g2{g}_{2}g2​) and back-scattering (g1{g}_{1}g1​) interactions is studied by a weak-coupling renormalization-group method at the three-loop level. When the transfer cutoff procedure is used, the Gell-Mann-Low function for g1{g}_{1}g1​ depends on g1{g}_{1}g1​ alone, $\frac{d{g}_{1}}{d\ensuremath{\xi}}={{g}_{1}}^{2}+\frac{1}{2}{{g}_{1}}^{3}+\frac{1}{8}{{g}_{1}}^{4}$. Further, in the absence of umklapp processes the combination ${g}_{1}\ensuremath{-}2{g}_{2}$ is a scale invariant. These conclusions agree with the results of Luther and Emery derived by the boson representation method. Some novel calculational techniques for the derivation of the scaling equations are described and compared to the conventional scheme.