Anomalous behavior of the spin gap of a spin-1/2 two-leg antiferromagnetic ladder with Ising-like rung interactions

Using mainly numerical methods, we investigate the width of the spin gap of a spin-1/2 two-leg ladder described by $\cH= J_\rl \sum_{j=1}^{N/2} [ \vS_{j,a} \cdot \vS_{j+1,a} + \vS_{j,b} \cdot \vS_{j+1,b} ] + J_\rr \sum_{j=1}^{N/2} [\lambda (S^x_{j,a} S^x_{j,b} + S^y_{j,a} S^y_{j,b}) + S^z_{j,a} S^z_{j,b}] $, where $S^\alpha_{j,a(b)}$ denotes the $\alpha$-component of the spin-1/2 operator at the $j$-th site of the $a (b)$ chain. We mainly focus on the $J_\rr \gg J_\rl > 0$ and $|\lambda| \ll 1$ case. The width of the spin gap as a function of $\lambda$ anomalously increases near $\lambda = 0$; for instance, for $-0.1 < \lambda < 0.1$ when $J_{\rm l}/J_{\rm r} = 0.1$. The gap formation mechanism is thought to be different for the $\lambda < 0$ and $\lambda > 0$ cases. Since, in usual cases, the width of the gap becomes zero or small at the point where the gap formation mechanism changes, the above gap-increasing phenomenon in the present case is anomalous. We explain the origin of this anomalous phenomenon by use of the degenerate perturbation theory. We also draw the ground-state phase diagram.