Ground-State Phase Diagram of an Anisotropic S=1/2 Ladder with Different Leg Interactions

We explore the ground-state phase diagram of the $S=1\mathrm{/}2$ two-leg ladder with different leg interactions. The $xy$ and $z$ components of the leg interactions between nearest-neighbor spins in the $a$ ($b$) leg are respectively denoted by $J_{\mathrm{l},a}$ and ${\mathrm{\Delta }}_{\mathrm{l}}{J}_{\mathrm{l},a}$ ($J_{\mathrm{l},b}$ and ${\mathrm{\Delta }}_{\mathrm{l}}{J}_{\mathrm{l},b}$). On the other hand, the $xy$ and $z$ components of the uniform rung interactions are respectively denoted by $\Gamma_{\rm r} J_{{\rm r}}$and$J_{{\rm r}}$\mathrm{.}Intheabove,$\Delta_{\rm l}$and$\Gamma_{\rm r}$arethe$XXZ$-type anisotropy parameters for the leg and rung interactions, respectively. This system has a frustration when $J_{{\rm l},a} J_{{\rm l},b}<0$irrespectiveofthesignof$J_{\rm r}$\mathrm{.}Thephasediagramonthe$\Delta_{\rm l}$($|\Delta_{\rm l}| \leq 1.0$)versus$J_{{\rm l},b}$($-2.0\leq J_{{\rm l},b}\leq 3.0$)planeinthecasewhere$J_{{\rm l},a}=0.2$,$J_{{\rm r}}=-1.0$,and$\Gamma_{\rm r} = 0.5$ is determined numerically. We employ the physical consideration, and the level spectroscopy and phenomenological renormalization-group analyses of the numerical date obtained by the exact diagonalization method. The resultant phase diagram contains the ferromagnetic, Haldane, N{'e}el, nematic Tomonaga-Luttinger liquid (TLL), partial ferrimagnetic, and $XY1$ phases. Interestingly enough, the nematic TLL phase appears in the strong-rung unfrustrated region as well as in the strong-rung frustrated one. We perform the first-order perturbational calculations from the strong rung coupling limit to elucidate the characteristic features of the phase diagram. Furthermore, we make the density-matrix renormalization-group calculations for some physical quantities such as the energy gaps, the local magnetization, and the spin correlation functions to supplement the reliability of the phase diagram.