$\mathbb{Z}_{2}$ Skin Channels and Scale-Dependent Dynamical Quantum Phase Transitions

1. Problem Statement

Non-Hermitian systems exhibit two distinct types of skin effects: the ordinary skin effect (driven by nonreciprocity and point gaps) and the $Z_2$ skin effect (driven by Anomalous Time-Reversal Symmetry, ATRS). While the ordinary skin effect has been extensively studied dynamically, the $Z_2$ skin effect remains poorly understood analytically, particularly regarding:

• The rigorous connection between ATRS and characteristic dynamical evolutions.
• The nature of wavepacket propagation under Periodic Boundary Conditions (PBC).
• The emergence of Dynamical Quantum Phase Transitions (DQPTs) in this specific symmetry class.

Existing literature lacks a unified analytical framework to describe how wavepackets evolve in $Z_2$ skin systems, specifically how they separate into distinct channels and how these channels relate to topological invariants like winding numbers.

2. Methodology

The author employs a combination of analytical derivation and semiclassical worldline perspectives to characterize the dynamics of a 1D non-Hermitian chain with ATRS.

• Model System: The study focuses on a symplectic Hatano-Nelson Hamiltonian ($H_s$) that preserves ATRS ($T H T^{-1} = H(-k)$ with $T^2 = -1$) but breaks pseudo-Hermiticity ($|g| > |\Delta|$). This regime ensures complex eigenenergies and the existence of a $Z_2$ skin effect.
• Wavepacket Construction: Initial states are constructed as Gaussian wavepackets in momentum space, composed of right eigenstates from two bands. Crucially, the initial momenta ($k_0^\pm$) are set as Kramers pairs to respect the underlying symmetry.
• Semiclassical Worldline Approach: The author adapts the semiclassical worldline picture (previously used for the ordinary skin effect via Quantum Monte Carlo Stochastic Series Expansion, QMC-SSE) to the $Z_2$ case. This involves analyzing the imaginary velocity derived from the imaginary part of the energy spectrum ($E_I(k)$).
• Analytical Derivation:
  • Derivation of the time-evolution of amplitude maxima in momentum space ($k_{max}^\pm$).
  • Derivation of the Center-of-Mass (COM) position in real space ($X_c^\pm(t)$).
  • Calculation of the Loschmidt echo ($L(t)$) and rate function ($\lambda(t)$) to identify DQPTs.

3. Key Contributions

1. Analytical Description of Dynamically Separated Channels: The paper rigorously proves that under ATRS, an initial wavepacket splits into two distinct dynamical $Z_2$ skin channels. Unlike the ordinary skin effect where a single channel dominates, the $Z_2$ effect creates two channels driven by counter-circulating spectral loops.
2. Winding-Control Mechanism Extension: The work extends the "winding-control mechanism" to the $Z_2$ case. It demonstrates that even though the total spectral winding number is zero (due to symmetry), the individual band winding numbers are non-zero and govern the unidirectional motion of quasiparticles.
3. Discovery of Scale-Dependent DQPTs: The paper identifies a novel class of DQPTs driven by the circulation of these skin channels. A key finding is that the time intervals between DQPT critical points are scale-dependent (scaling with system size $N$), distinguishing them from conventional DQPTs where intervals are typically system-size independent.
4. Unification of Skin Effects: The work provides a unified physical picture showing that the core physics of the $Z_2$ skin effect (worldline windings, circulation) is shared with the ordinary skin effect, despite their different topological origins (point gap vs. $Z_2$ invariant).

4. Key Results

• Momentum Space Evolution:
  The amplitude maxima of the two channels evolve toward specific target momenta where the imaginary part of the energy, $E_I(k)$, is maximized. The evolution follows:
  $$k_{max}^\pm(t) \approx k_0^\pm + \sigma^2 \frac{dE_I^\pm(k)}{dk} t$$
  The channels move along the spectral loop in opposite directions.

• Real Space Circulation (Worldlines):
  In real space, the two channels circulate around the 1D chain, tracing semiclassical worldlines.

  • The COM position evolves as $X_c^\pm(t) = n_0 + V_g^\pm(t)t$.
  • Due to PBC, when a channel reaches a boundary, it wraps around to the opposite end, creating a cyclic motion.
  • This circulation leads to quantum revivals, where the wavepacket periodically returns to its initial state.

• Scale-Dependent DQPTs:

  • Mechanism: DQPTs occur when the wavepacket amplitude overlap with the initial state is minimized (Loschmidt echo $L(t) \to 0$), which happens when the circulating channels reach the system boundaries.
  • Scaling: The time interval $\Delta t_c$ between successive DQPTs scales linearly with the system size $N$:
    $$\Delta t_c \sim \frac{N}{\bar{v}_g}$$
    where $\bar{v}_g$ is the average group velocity. This contrasts with conventional DQPTs, which typically exhibit time intervals independent of $N$.
  • Phase Independence: These DQPTs occur regardless of wavepacket phase interference because the exponential growth/decay factors (driven by the skin effect) dominate the dynamics.

• Initial State Dependence:
  The separation of channels is robust. Whether the initial excitation is at a Kramers degenerate point ($k=0, \pi$) or a generic Kramers pair, the system evolves into two separated channels. However, if the initial wavepacket widths ($\sigma$) are mismatched between the two bands, the symmetry breaks, and the distinct $Z_2$ channels may not form.

5. Significance

• Theoretical Advancement: This work bridges the gap between static topological characterization (winding numbers, $Z_2$ invariants) and dynamic phenomena in non-Hermitian systems. It provides the first rigorous analytical framework for the $Z_2$ skin effect dynamics.
• New Class of Phase Transitions: The identification of scale-dependent DQPTs challenges the conventional understanding of dynamical phase transitions, suggesting that in non-Hermitian skin systems, the "thermodynamic limit" behavior is fundamentally different due to the interplay between system size and skin-driven circulation.
• Experimental Relevance: The predicted phenomena (circulating worldlines, quantum revivals, and scale-dependent DQPTs) are proposed to be detectable in current experimental platforms, including acoustic crystals, photonic quantum walks, and electrical circuits.
• Future Directions: The paper highlights the challenge of extending these findings to Open Boundary Conditions (OBC) for $Z_2$ systems, where the lack of a unified circular Generalized Brillouin Zone (GBZ) complicates the application of global imaginary gauge transformations.

In summary, this paper establishes that the $Z_2$ skin effect is not merely a static spectral feature but a dynamic engine that drives wavepackets into circulating worldlines, leading to unique, size-dependent dynamical phase transitions.