Instabilities in Chains Coupled by Two-Body Interactions

TL;DR

This paper investigates the stability of Luttinger states in coupled one-dimensional fermionic chains with two-body interactions, revealing how boundary conditions influence the emergence of massless or massive states and their relevance to 2D fermionic systems.

Contribution

It derives scaling equations and analyzes the stability of Luttinger states in coupled chains, highlighting the impact of boundary conditions on system instabilities.

Findings

Periodic boundary conditions lead to massive states in most of the phase diagram.

Open boundary conditions allow massless states but with proximity to instabilities.

The analysis is relevant for understanding instabilities in two-dimensional fermionic systems.

Abstract

We derive a general set of Poor Man's scaling equations and analyze the stability of the Luttinger state in a system composed of a finite number N of one dimensional spinless fermionic chains, coupled through a general two body interaction. The effect of processes with momentum transfer parallel to the Fermi surface in destroying massless states is investigated. It will be shown that there are two processes competing: one in which two electrons exchange chains and the other in which they jump into a same chain. When periodic boundary conditions in the transverse direction are taken into account this competition leads always to massive states (except in hyperplanes of the phase diagram), a well known example being the generalized sine-Gordon model. If instead open boundary conditions are taken, massless states are possible but due to this competition the system is placed near instabilities. We argue that this kind of analysis has relevance for understanding the instabilities of 2D fermionic systems.