Entanglement-entropy analysis of critical and topological quantum phases in a frustrated spin-1/2 Heisenberg ladder

We investigate the ground-state phase diagram of a frustrated spin-1/2 Heisenberg ladder in the transverse magnetic field with an anisotropic inter-rung exchange coupling $\alpha$. This bond-anisotropic parameter continuously interpolates among the Ising-like limit $\alpha=0$, the isotropic point $\alpha=1$, and the XY-dominated regime $\alpha \gg 1$. Using density-matrix renormalization group calculations within a matrix-product-state framework, we analyze bipartite entanglement, magnetization, and spin correlations to characterize the emergent quantum phases. We identify six distinct ground-state phases, including rung-singlet, Haldane-like, Tomonaga-Luttinger liquid, canted Ising-ordered, XY-polarized, and ordered ferromagnetic states. While magnetization and local correlations provide a first insight into phase classifications, the finite-size scaling of the entanglement measures offers a more sensitive and unified diagnostics of both gapped and gapless regimes. In the gapless regime, we extract the central charge $c \simeq 1$, confirming Tomonaga-Luttinger liquid behavior, while at the transition between the canted Ising and ferromagnetic phases, we observe $c \simeq 1/2$, consistent with the Ising universality. Finally, we find that both the Haldane-like behavior and the extended critical Tomonaga-Luttinger liquid regime are strongly confined to the vicinity of the isotropic point $\alpha = 1$.