Equilibrium and nonequlibrium scaling behaviors of localization transition in a non-Hermitian Aubry-André model with onsite gain and loss

Technical Summary: Equilibrium and Nonequilibrium Scaling Behaviors of Localization Transition in a Non-Hermitian Aubry-André Model with Onsite Gain and Loss

Problem Statement
While the interplay between non-Hermiticity and localization has attracted significant attention, the driven dynamics of localization transitions in non-Hermitian systems featuring onsite gain and loss remain largely unexplored. Existing studies on non-Hermitian Aubry-André (AA) models have primarily focused on nonreciprocal hopping mechanisms. However, the gain-loss mechanism exhibits fundamentally different spectral and topological characteristics compared to nonreciprocal hopping (e.g., the correspondence between extended/localized phases and real/complex spectra is reversed). Furthermore, while the Kibble-Zurek mechanism (KZM) and Finite-Time Scaling (FTS) have been applied to non-Hermitian systems, their applicability to models with gapless initial states (extended phases) under the gain-loss mechanism requires systematic verification.

Methodology
The authors investigate a one-dimensional single-particle non-Hermitian AA model with onsite gain and loss, described by the Hamiltonian:
$$H = -J \sum_{j=1}^{L} (c^\dagger_j c_{j+1} + \text{H.c.}) + \sum_{j=1}^{L} V_n(j) c^\dagger_j c_j$$
where the onsite potential $V_n(j) = V \cos(2\pi\gamma j + \phi)$ is complexified via the phase $\phi = \theta + ih$, introducing gain and loss. The system size $L$ is approximated using Fibonacci numbers to satisfy periodic boundary conditions (PBC) in the thermodynamic limit.

The study employs two main analytical approaches:

1. Static Critical Behavior: The authors perform finite-size scaling (FSS) analyses on three observables: the localization length ($\xi$), the inverse participation ratio (IPR), and the energy gap ($E_g$). They extract critical exponents ($\nu$, $s$, $z$) by collapsing numerical data for various system sizes onto universal curves near the critical point.
2. Driven Dynamics: The system is linearly driven across the critical point by varying the potential strength $\varepsilon(t) = \varepsilon_0 + Rt$. The initial state is prepared as a gapless extended state. The authors test the applicability of the FTS framework, which generalizes KZM to gapless initial states, provided the criterion $z' < r$ is met (where $z'$ characterizes gap closure in the initial phase and $r = z + 1/\nu$).

Key Results

• Critical Exponents and Universality Class:
  Through FSS analysis, the authors extract the following critical exponents for the gain-loss AA model:

  • Correlation length exponent: $\nu = 1.00(2)$
  • IPR exponent: $s = 0.7965(2)$
  • Dynamical exponent: $z = 1.999(2)$

  These results establish that the gain-loss AA model belongs to a distinct universality class from both the Hermitian AA model and the nonreciprocal hopping AA model. While $\nu$ and $z$ are nearly identical to the nonreciprocal model, the IPR exponent $s$ differs significantly ($s \approx 0.80$ vs. $0.12$ for nonreciprocal and $0.33$ for Hermitian). This indicates that the multifractal properties of wavefunctions at the critical point are highly sensitive to the specific non-Hermitian mechanism.

• Spectral and Spatial Characteristics:
  The phase diagram confirms a transition line $h = \log(2J/V)$. Unlike the nonreciprocal model, the extended phase in the gain-loss model possesses a purely real spectrum and trivial topology, while the localized phase exhibits a complex spectrum and nontrivial topology. Spatial analysis of wavefunctions reveals symmetric exponential decay in the localized phase under both PBC and open boundary conditions (OBC), contrasting with the asymmetric skin-effect localization observed in nonreciprocal models.

• Driven Dynamics and FTS Validity:
  The authors verify that the FTS framework remains applicable for the gain-loss model starting from a gapless extended state.

  • The gap closure exponent in the extended phase is found to be $z' = 1.999(2)$.
  • The criterion $z' < r$ is satisfied, as $r = z + 1/\nu \approx 2.999$.
  • Numerical simulations of the time-dependent Schrödinger equation confirm the predicted FTS scaling forms for the IPR across a wide range of system sizes and driving rates ($R$). The data collapses onto universal curves consistent with the theoretical ansatz:
    $$\text{IPR} \sim L^{-s/\nu} g_1(RL^r, \varepsilon R^{-1/r\nu}) \quad (\text{small } R)$$
    $$\text{IPR} \sim L^{-1} R^{(s-\nu)/r\nu} g_2(RL^r, \varepsilon R^{-1/r\nu}) \quad (\text{large } R)$$

Significance and Claims
The paper claims two primary contributions:

1. New Universality Class: It establishes the gain-loss type non-Hermitian AA model as a new universality class for localization transitions, distinguished primarily by the critical exponent $s$ governing the multifractal nature of the wavefunctions.
2. Extension of FTS Framework: It demonstrates that the unified FTS scaling description, previously established for Hermitian systems and nonreciprocal non-Hermitian models, can be successfully generalized to non-Hermitian systems with gapless initial states (specifically the gain-loss AA model).

The authors conclude that their work provides a comprehensive perspective on the interplay between non-Hermiticity, localization criticality, and nonequilibrium driving. They note that given the experimental accessibility of non-Hermitian quasiperiodic systems in photonic and cold-atom platforms, these predictions are expected to stimulate further experimental investigations.