Gapless Phases in an s=1/2 Quantum Spin Chain with Bond Alternation

TL;DR

This paper investigates gapless phases in an s=1/2 quantum spin chain with bond alternation, revealing conditions for XY and spin nematic phases, and analyzing their properties and transitions.

Contribution

It introduces an exactly solvable model showing gapless phases with Majorana fermions and characterizes the transition to a spin nematic state due to bond alternation.

Findings

Gapless XY-like and spin nematic phases identified.

Spectrum described by two massless Majorana fermions with different velocities.

Transition to spin nematic state occurs at |δ| > J.

Abstract

The $S=1/2$ XXZ spin chain with the staggered XY anisotropy $$ H = J \sum_{n}^{N} (S^{x}_{n} S^{x}_{n+1} + S^{y}_{n} S^{y}_{n+1} + \Delta S^{z}_{n} S^{z}_{n+1}) - \delta \sum_{n}^{N} (-1)^{n} (S^{x}_{n} S^{x}_{n+1} - S^{y}_{n} S^{y}_{n+1}) $$ is shown to possess gapless, Luttinger-liquid-like phases in a wide range of its parameters: the XY-like phase and spin nematic phases, the latter characterized by a two-spin order parameter breaking translational and spin rotation symmetries. In the simplest, exactly solvable case $\Delta = 0$, the spectrum remains gapless at arbitrary $J$ and $\delta$ and is described by two massless Majorana (real) fermions with different velocities $v_{\pm} = |J \pm \delta|$. At $|\delta| < J$ the staggered XY anisotropy does not influence the ground state of the system (XY phase). At $|\delta| > J$, due to level crossing, a spin nematic state is realized, with $\uparrow \uparrow \downarrow \downarrow$ and $\uparrow \downarrow \downarrow \uparrow$ local symmetry of the $xx$ and $yy$ spin correlations. The spin correlation functions are calculated and the effect of thermally induced spin nematic ordering in the XY phase ("order from disorder") is discussed. The role of a finite $\Delta$ is studied in the limiting cases $|\delta| \ll J$