Competing Orders in Coupled Luttinger Liquids

TL;DR

This paper investigates the phase diagram and competing orders in coupled Luttinger liquids at various fillings using bosonization and renormalization group techniques, revealing complex ordered phases and quantum disordered states.

Contribution

It provides a detailed analysis of phase boundaries and order types in coupled Luttinger liquids, including effects of symmetry breaking at half-filling and low doping.

Findings

Charge density wave and related orders form doublets with quasi-long range order.

Half-filling induces Ising-type ordered phases due to umklapp symmetry breaking.

Quantum disordered Haldane phases with finite pairing amplitudes are identified.

Abstract

We consider the problem of two coupled Luttinger liquids both at half filling and at low doping levels, to investigate the problem of competing orders in quasi-one-dimensional strongly correlated systems. We use bosonization and renormalization group equations to investigate the phase diagrams, to determine the allowed phases and to establish approximate boundaries among them. Because of the chiral translation and reflection symmetry in the charge mode away from half filling, orders of charge density wave (CDW) and spin-Peierls (SP) diagonal current (DC) and $d$-density wave (DDW) form two doublets and thus can be at most quasi-long range ordered. At half-filling, umklapp terms break this symmetry down to a discrete group and thus Ising-type ordered phases appear as a result of spontaneous breaking of the residual symmetries. Quantum disordered Haldane phases are also found, with finite amplitudes of pairing orders and triplet counterparts of CDW, SP, DC and DDW. Relations with recent numerical results and implications to similar problems in two dimensions are discussed.