Transverse momentum dependent self energy of a quasi-one-dimensional system: a 2-loop renormalization group calculation

Rina Tazai, Youichi Yamakawa, Hiroshi Kondani

Masahisa Tsuchiizu

We derive two-loop renormalization group equations for the half filled one-dimensional Hubbard chains coupled by the interchain hopping. Our renormalization group scheme for the quasi-one-dimensional electron system is a natural extension of that for the purely one-dimensional systems in the sense that transverse-momentum dependences are introduced in the $g$-ological coupling constants and we regard the transverse momentum as a patch index. We develop symmetry arguments for the particle-hole symmetric half filled Hubbard model and obtain constraints on the $g$-ological coupling constants by which resultant renormalization equations are given in a compact form. By solving the renormalization group equations numerically, we estimate the magnitude of excitation gaps and clarify that the charge gap is suppressed due to the interchain hopping but is always finite even for the relevant interchain hopping. To show the validity of the present analysis, we also apply this to the two-leg ladder system. By utilizing the field-theoretical Bosonization and Fermionization method, we derive low-energy effective theory and analyze the magnitude of all the excitation gaps in detail. It is shown that the low-energy excitations in the two-leg Hubbard ladder have $\mathrm{SO}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{SO}(3)\ifmmode\times\else\texttimes\fi{}\mathrm{U}(1)$ symmetry when the interchain hopping exceeds the magnitude of the charge gap.

Jean-Paul Blaizot, Ramón Méndez-Galain, Nicolás Wschebor

Agnese Tagliavini, Cornelia Hille, Fabian B. Kugler +3 more

We present a functional renormalization group (fRG) study of the two dimensional Hubbard model, performed with an algorithmic implementation which lifts some of the common approximations made in fRG calculations. In particular, in our fRG flow; (i) we take explicitly into account the momentum and the frequency dependence of the vertex functions; (ii) we include the feedback effect of the self-energy; (iii) we implement the recently introduced multiloop extension which allows us to sum up all the diagrams of the parquet approximation with their exact weight. Due to its iterative structure based on successive one-loop computations, the loop convergence of the fRG results can be obtained with an affordable numerical effort. In particular, focusing on the analysis of the physical response functions, we show that the results become independent from the chosen cutoff scheme and from the way the fRG susceptibilities are computed, i.e., either through flowing couplings to external fields, or through a “post-processing” contraction of the interaction vertex at the end of the flow. The presented substantial refinement of fRG-based computation schemes paves a promising route towards future quantitative fRG analyses of more challenging systems and/or parameter regimes.

Eberth de Almeida Correa, Hermann Freire, Álvaro Antônio Bandeira Ferraz

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.

Stéphane Daul, Michael Dzierzawa