Quantum phase transition and critical behavior between the gapless topological phases
Hao-Long Zhang, Han-Ze Li, Sheng Yang, Xue-Jia Yu
Abstract
The phase transition between gapped topological phases represents a class of unconventional criticality beyond the Landau paradigm. However, recent research has shifted attention to topological phases without a bulk gap, where the phase transitions between them are still elusive. In this work, based on large-scale density-matrix renormalization-group techniques, we investigate the critical behaviors of the extended quantum $XXZ$ model obtained by the Kennedy-Tasaki transformation. Using fidelity susceptibility as a diagnostic, we obtain a complete phase diagram, which includes both topological nontrivial and trivial gapless phases. Furthermore, as the $XXZ$-type anisotropy parameter $\mathrm{\ensuremath{\Delta}}$ varies, both the critical points ${h}_{c}$ and correlation length exponent $\ensuremath{\nu}$ remain the same as in the $\mathrm{\ensuremath{\Delta}}=0$ case, characterized by $c=3/2$ (Ising plus free boson) conformal field theory. Our results indicate that fidelity susceptibility can effectively detect and reveal a stable unconventional critical line between the topologically distinct gapless phases for general $\mathrm{\ensuremath{\Delta}}$. This work serves as a valuable reference for further research on phase transitions within the gapless topological phase of matter.