Quantum criticality and universality in the $p$-wave paired Aubry-André-Harper model

Abstract

We investigate the quantum criticality and universality in Aubry-André-Harper (AAH) model with p-wave superconducting pairing ∆ in terms of the generalized fidelity susceptibility (GFS). We show that the higher-order GFS is more efficient in spotlighting the critical points than lowerorder 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 φ = π alleviates the odd-even effect of scaling functions across the Aubry-André transition with ∆ = 0, while the scaling functions for odd and even numbers of system sizes with a finite ∆ cannot coincide irrespective of the value of φ. A thorough numerical analysis with odd number of system sizes reveals the correlationlength exponent ν 1.000 and the dynamical exponent z 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-André transition. The results may be testified in near term state-of-the-art experimental settings.

Topological properties of the long-range Kitaev chain with Aubry-André-Harper modulation

“…We presented an in-depth study of the localization properties of the AAH model with long-range hopping and superconducting pairing. We first reviewed a tool-box for the characterization of localization properties on two known limits of the Hamiltonian, namely the shortrange Kitaev with AAH modulation [20,21] and the AAH model with long-range hopping [25,28,30]. In particular, we focused on global characterization methods for systems with no mobility edges and local characterization methods to investigate energy dependent transitions and hybridization of bands with different type of localization behavior.…”

“…In particular, this means that the system described by the Hamiltonian in eq. ( 1) can host coexisting localized, multifractal and ergodic states, which was not possible in the limits of the model that have been previously studied [20,21,25,28,30]. Such regimes can be understood as the system being in a mix of regimes in energy space, where different bands show different localization properties.…”

“…For a nonzero AAH chemical potential and both ξ, α 1, we recover the short-range Kitaev model with an AAH modulation studied in Refs. [24,25,28,30]. For ∆ = 0 and ξ 1, we retrieve the AAH model with long-range hopping amplitudes characterized in Refs.…”

“…For small values of µ t, the totality of the states of the system are ergodic, while for large values of µ t, all the eigenvectors are localized. In between, in contrast to the conventional Aubry-André-Harper model without pairing, which features a direct transition between ergodic and localized states, this system also has a multifractal region for µ t ∈ [|∆ − t|, |∆ + t|] [24,25,28,30]. In this Section, we characterize the phase diagram of the Hamiltonian H ξ,α 1 with the help of three global observables.…”

“…In addition, on can also study the effect of interactions. In particular, density-density interactions [22] or Hamiltonians with superconducting pairing term [23][24][25][26][27][28][29][30] have been considered, and lead to an intermediate extended multifractal regime, which exists for a finite range of chemical potential strengths. Other works have considered coupling of more than one Aubry-André-Harper chains [31,32] or two-dimensional Aubry-André-Harper models [33], which leads to hybridization of ergodic and localized bands.…”