Two-loop renormalization group calculation of response functions for a two-dimensional flat Fermi surface

Abstract

We present the formalism for a two-loop renormalization group ͑RG͒ calculation of some order-parameter susceptibilities associated with a two-dimensional ͑2D͒ 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 ͑AF͒ spindensity-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.

Superconductivity in the 2d Attractive Hubbard Model Within a Functional Field-Theoretical Rg

“…This is due to the fact that the two-loop self-energy diagrams contribute with an opposite sign to the FFTRG equations for the interactions in which the divergences are retarded if compared with one-loop calculations. 25 In addition, the complete divergence of some of the susceptibilities seems to occur in a rate slower than those for the couplings. To get an improvement for the temperatures at any density we should go to the finite temperature formalism for the FFTRG equations in which the correlation functions are those at a given temperature in the Λ → 0 limit, but this is out of the scope of this work.…”

“…Here we follow the same numerical procedure detailed in our previous RG works. 17,25 All the integrals are calculated considering N = 32 FS patches. Despite this limited number of points, the distance between two different points at the FS is such that the integral step necessary for the calculations is at most 0.5 at half-filling.…”

“…As a result, the pair formation in the present case is now favored, for the PH fluctuations are less divergent. 5,25 Therefore, the main effect of the correlations into the RG equations is manifest by means of the critical energy scale in which the quasi-long-range order takes place. By contrast, another important point we wish to emphasize here is that the aforementioned scenario is very different from the corresponding model in the repulsive regime.…”

“…The next step in the FFTRG treatment is to calculate the quasiparticle weight which plays an important role in the repulsive HM. 16,17,24,25 It is worthwhile emphasizing that in one-loop approximation the self-energy produces no contribution to Z Λ . The first nonzero contribution shows up in two-loop order and is produced by the so-called sunset diagram displayed in Fig.…”

“…[19][20][21][22][23] However, in higher dimensions, the corresponding FFTRG equations become much more complex to be solved analytically and, for this reason, one has to resort necessarily to numerical methods. In earlier works, 16,17,24,25 we pointed out the importance of self-energy corrections for the repulsive HM in 2D. These effects are typically of higher order and, therefore, one has to go beyond one-loop in the truncation scheme, before finally arriving at the FFTRG equations for the renormalized couplings.…”