Renormalization group calculation of the uniform susceptibilities in low-dimensional systems

Abstract

We analyze the one-dimensional (1D) and the two-dimensional (2D) repulsive Hubbard models (HM) for densities slightly away from half-filling through the behavior of two central quantities of a system: the uniform charge and spin susceptibilities. We point out that a consistent renormalization group treatment of them can only be achieved within a two-loop approach or beyond. In the 1D HM, we show that this scheme reproduces correctly the metallic behavior given by the well-known Luttinger liquid fixed-point result. Then, we use the same approach to deal with the more complicated 2D HM. In this case, we are able to show that both uniform susceptibilities become suppressed for moderate interaction parameters as one take the system towards the Fermi surface. Therefore, this result adds further support to the interpretation that those systems are in fact insulating spin liquids. Later, we perform the same calculations in 2D using the conventional random phase approximation, and establish clearly a comparison between the two schemes.

Breakdown of Fermi liquid behavior near the hot spots in a two-dimensional model: A two-loop renormalization group analysis

“…Thus, we must regularize these divergences by defining the renormalized uniform response functions and the cor- responding counterterms as follows 47,48 T αα…”

The absence of logarithmic divergences in the spin and charge density correlations of the 1d Hubbard model

“…In some recent papers, see [5], [6], [7], the spin and charge density correlations in the 1d repulsive Hubbard model have been considered from a Renormalization Group (RG) point of view. In particular it has been found in [7] that the interaction apparently produces logarithmic divergences at zero momentum (see (15),(16) of [7]), which would be in contrast with the expected Luttinger liquid behaviour of the 1d Hubbard model.…”

“…In some recent papers, see [5], [6], [7], the spin and charge density correlations in the 1d repulsive Hubbard model have been considered from a Renormalization Group (RG) point of view. In particular it has been found in [7] that the interaction apparently produces logarithmic divergences at zero momentum (see (15),(16) of [7]), which would be in contrast with the expected Luttinger liquid behaviour of the 1d Hubbard model. Aim of this paper is to rigorously compute the spin and charge density correlations, and to prove that there are indeed no logarithmic divergences at zero momentum in the spin or charge correlations; their apparence in [7] is just an artifact due to the approximations involved in the analyis.…”

“…If there is no interaction U = 0 the two point Schwinger function g(x − y) is given by The density correlations for U = 0 are given by N ε 0 (x) = g(x)g(−x) (7) and the static density correlations for x = 0 can be written as…”

Confinement and Quantum Anomaly in Quasi-1d Spinless Fermion Chains

“…From (17), (18), (19) and (20), we can rewrite (24) in the diagonalized form By defining the following dispersion relations…”