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

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 $|\delta| \ll J$