Phase diagram of the anisotropic spin- 1 2 X Z chain through the lens of bifurcation in ground-state fidelity, fidelity susceptibility, and local quantum uncertainty

Abstract

In this study we investigate the ground-state phase diagram of the quantum spin-1/2 anisotropic XZ chain using the infinite time-evolving block decimation algorithm. The model exhibits four distinct phases: FM_{x}, FM_{z}, AFM_{z}, and PM. In terms of the transverse magnetic field h and anisotropy parameter J, our analysis focuses on the lines h=0 and 0.5 with J≥0, where different types of quantum phase transitions (QPTs) occur. We compute the bifurcation behavior of the ground-state fidelity per lattice site. On the line h=0, continuous behavior at the bifurcation point identifies a continuous QPT between the ordered FM_{x} and FM_{z} phases at the critical point J_{c1}, with bifurcations on both sides reflecting Z_{2} symmetry breaking in both phases. On the line h=0.5, a continuous bifurcation indicates a QPT between the ordered FM_{x} and disordered PM phases at J_{c2}, with bifurcation occurring only on one side, corresponding to Z_{2} symmetry breaking in a single phase. We further compute the ground-state fidelity susceptibility per lattice site. When the ground state is obtained from random initial states, the fidelity susceptibility exhibits oscillatory behavior in the symmetry-broken phase, providing a clear signature of ground-state degeneracy. These fidelity-based measures serve as effective tools for identifying critical points and characterizing the nature of symmetry breaking. To complement these results, we evaluate the two-spin Wigner-Yanase skew information and local quantum uncertainty (LQU) as indicators of quantum correlations. While both detect QPTs, we find that not all nonanalytic features in the LQU correspond to genuine phase transitions. A critical behavior analysis is carried out at J_{c1} and J_{c2} to extract the critical exponents η^{I}, η^{C}, η^{x}, and η^{z}, associated with mutual information I(r), classical correlation C(r), and spin-spin correlations 〈σ_{i}^{x}σ_{j}^{x}〉, 〈σ_{i}^{z}σ_{j}^{z}〉. Our results demonstrate that η^{I}=η^{C} and η^{x/z}=η^{α}/2 for α=I,C, consistent with known universal scaling relations. We also extract the order parameter exponent β, correlation length exponent ν, dynamic exponent z, and the central charge c. A transition at J_{c1} between the FM_{x} and FM_{z} phases is identified as a Gaussian transition with central charge c=1, involving the breaking of two Z_{2} symmetries and lying beyond the conventional Landau-Ginzburg-Wilson symmetry-breaking paradigm. The transition at J_{c2} between the FM_{x} and PM phases belongs to the quantum Ising universality class, characterized by c=1/2 and breaking of a single Z_{2} symmetry.