Ground state properties and exact thermodynamics of a 2-leg anisotropic spin ladder system

We study a frustrated two-leg spin ladder with alternate isotropic Heisenberg and Ising rung exchange interactions, whereas, interactions along legs and diagonals are Ising-type. All the interactions in the ladder are anti-ferromagnetic in nature and induce frustration in the system. This model shows four interesting quantum phases: (i) stripe rung ferromagnetic (SRFM), (ii) stripe rung ferromagnetic with edge singlet (SRFM-E), (iii) anisotropic antiferromagnetic (AAFM), and (iv) stripe leg ferromagnetic (SLFM) phase. We construct a quantum phase diagram for this model and show that in stripe rung ferromagnet (SRFM), the same type of sublattice spins (either $S$ or $\sigma$-type spins) are aligned in the same direction. Whereas, in anisotropic antiferromagnetic phase, both $S$ and $\sigma$-type of spins are anti-ferromagnetically aligned with each other, two nearest $S$ spins along the rung form an anisotropic singlet bond whereas two nearest $\sigma$ spins form an Ising bond. In large Heisenberg rung exchange interaction limit, spins on each leg are ferromagnetically aligned, but spins on different legs are anti-ferromagnetically aligned. The thermodynamic quantities like $Cv(T)$, $\chi(T)$ and $S(T)$ are also calculated using the transfer matrix method for different phase. The magnetic gap in the SRFM and the SLFM can be notice from $\chi(T)$ and $Cv(T)$ curves.