Ground-State Phase Diagram, Higher-Winding Topology, and Lifshitz Criticality in an Anisotropic Four-Spin XX Chain

Abstract: We investigate the quantum critical and topological properties of a spin-$1/2$ XX chain with anisotropic four-spin cluster interactions in a transverse magnetic field. The interplay of exchange anisotropy, cluster interactions, and the external field produces a rich phase diagram with multiple gapped topological phases and distinct quantum critical boundaries. We identify several Lifshitz-type transitions arising from reconstructions of the low-energy Fermi-point structure, including an unconventional multicritical point where distinct critical branches intersect and the momentum-space topology undergoes a singular reorganization. In the isotropic limit, a conventional Lifshitz transition emerges through the merging and annihilation of Fermi points, revealing distinct mechanisms of Fermi-point reconstruction within the same model. The gapped phases are characterized by quantized winding numbers $ν=0,\pm1,-2,$ and $\pm3$, with the higher-winding phases induced by the extended cluster interactions. The $ν=\pm3$ phases exhibit a richer topological structure than conventional short-range Kitaev-type chains and are separated from other sectors by field- and interaction-driven gap closings. These topological distinctions are further reflected in characteristic degeneracy patterns of the lowest levels of the bulk entanglement spectrum. Our results demonstrate that anisotropic multispin interactions provide a versatile route to realizing higher-winding topological phases and unconventional multicriticality in one-dimensional quantum systems.