One-Dimensional Spin Liquids

Abstract

This chapter is intended as a brief overview of some of the zerotemperature quantum spin liquid phases with unbroken SU(2) spin symmetry that have been found in one dimension. The main characteristics of these phases are discussed by means of the bosonization approach. A special emphasis is laid on the interplay between frustration and quantum fluctuations in one dimension. IntroductionA central issue in the study of strongly correlated systems is the classification of all possible Mott insulating phases at zero temperature. The general strategy for describing the possible phases associated with the spin degrees of freedom (the so-called "quantum magnetism") of Mott insulators consists of analysing localized spin models such as the antiferromagnetic (AF) Heisenberg model:where Si are quantum spin-5 operators on sites i of a lattice, and the sum < i,j > is over the nearest neighbor sites with an antiferromagnetic exchange (J > 0); the ellipsis represents additional terms like second-neighbor competing exchange interaction or ring exchange that can be eventually added to the Heisenberg Hamiltonian (1). Generically, the ground state of the model (1) without these additional terms displays long-range Neel ordering (< Si >^ 0) which breaks spontaneously the SU(2) symmetry of the 307 Frustrated Spin Systems Downloaded from www.worldscientific.com by UNIVERSITY OF BIRMINGHAM on 08/24/15. For personal use only.

Numerical study of incommensurate and decoupled phases of spin-1/2 chains with isotropic exchangeJ1,J2between first and second neighbors

“…The eigenstates of H mf are products of HAF eigenstates in sectors with S A = S B ⩽ n. We define J 2 E(S,2n) as the lowest energy for S ⩽ n. The S = 0 energy per site is E(0,2n)/2n = ε 0 = 1/4 − ln2 in the thermodynamic limit. The GS is the combination of S = S A + S B and S A = S B that minimizes the energy in equation (18). The QLRO(π/2) phase with S = S A = S B = 0 is the GS for J 1 < 0 until the FM state with S = S A + S B = 2n is reached at J 1 /J 2 = −4ln2 in the thermodynamic limit.…”

“…Sandvik [17] has reviewed numerical studies of the HAF and related spin chains, including H(J 1 ,J 2 ) at g < 1. An earlier review by Lecheminant [18] addresses frustrated 1D spin systems mainly in terms of field theory, also for g < 1.…”

“…The dimer phase has doubly degenerate GS, broken inversion symmetry at sites and finite energy gap E m to the lowest triplet state. The dimer phase and P4 were the initial focus of theoretical and numerical studies [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Attention has recently shifted to the J 1 -J 2 model with ferromagnetic J 1 < 0 that is the starting point for the magnetic properties of Cu(II) chains in some cupric oxides [19][20][21][22].…”

Level crossing, spin structure factor and quantum phases of the frustrated spin-1/2 chain with first and second neighbor exchange

“…The critical point g ON = 0.2411 is fully consistent with field theory. As perhaps another indication of low priority, the critical point 1/g** = 0.45 does not appear in field theories [12][13][14]25] since level crossing as a function of 1/g had not been reported.…”

“…Sandvik [24] has reviewed numerical approaches to the HAF and related spin chains. An earlier review by Lecheminant [25] addresses frustrated 1D spin systems mainly in terms of field theory.…”