Quantum criticality and universality in the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>p</mml:mi></mml:math>-wave-paired Aubry-André-Harper model
Ting Lv, Tian-Cheng Yi, Liangsheng Li, Gaoyong Sun, Wen‐Long You
Abstract
We investigate the quantum criticality and universality in Aubry-Andr\'e-Harper (AAH) model with $p$-wave superconducting pairing $\mathrm{\ensuremath{\Delta}}$ in terms of the generalized fidelity susceptibility (GFS). We show that the higher order GFS is more efficient in spotlighting the critical points than lower order ones, and thus the enhanced sensitivity is propitious for extracting the associated universal information from the finite-size scaling in quasiperiodic systems. The GFS obeys power-law scaling for localization transitions and thus scaling properties of the GFS provide compelling values of critical exponents. Specifically, we demonstrate that the fixed modulation phase $\ensuremath{\phi}=\ensuremath{\pi}$ alleviates the odd-even effect of scaling functions across the Aubry-Andr\'e transition with $\mathrm{\ensuremath{\Delta}}=0$, while the scaling functions for odd and even numbers of system sizes with a finite $\mathrm{\ensuremath{\Delta}}$ cannot coincide irrespective of the value of $\ensuremath{\phi}$. A thorough numerical analysis with odd number of system sizes reveals the correlation-length exponent $\ensuremath{\nu}\ensuremath{\simeq}$ 1.000 and the dynamical exponent $z\phantom{\rule{4pt}{0ex}}\ensuremath{\simeq}$ 1.388 for transitions from the critical phase to the localized phase, suggesting the unusual universality class of localization transitions in the AAH model with a finite $p$-wave superconducting pairing lies in a different universality class from the Aubry-Andr\'e transition. The results may be testified in near term state-of-the-art experimental settings.