From local weight selection to Zeno slowdown in an open Su-Schrieffer-Heeger chain with a single local loss

Technical Summary: From Local Weight Selection to Zeno Slowdown in an Open Su-Schrieffer-Heeger Chain with a Single Local Loss

Problem Statement
The paper investigates the relaxation dynamics of a quadratic open quantum system, specifically a Su-Schrieffer-Heeger (SSH) chain with a single-site local loss. The central challenge addressed is determining the Liouvillian gap ($\Delta_L$), defined as the smallest nonzero decay rate of the full Lindblad generator, which sets the longest intrinsic relaxation time. While direct time evolution can extract this scale, it does not reveal the spectral mechanisms by which a local defect selects slow modes or how the slow sector reorganizes as dissipation strength ($\gamma$) varies. The authors aim to provide an exact formulation where long-time dynamics are traced to an explicit spectral problem, bridging the gap between local dissipation, topology, and geometry.

Methodology
The authors employ third quantization to solve the many-body Lindblad master equation exactly. Because the Hamiltonian is quadratic and the jump operator (representing the loss) is linear in fermion operators, the many-body problem reduces exactly to a finite non-Hermitian single-particle matrix problem.

• Reduction: The full Liouvillian spectrum is generated by fermionic subset sums of the eigenvalues (rapidities) of two $N_s \times N_s$ non-Hermitian blocks ($P_\pm$). Due to chiral symmetry, these blocks are isospectral, allowing the problem to be mapped to a single matrix $P = P_0 + i(\gamma/2)\Pi_s$, where $P_0$ is the clean SSH hopping matrix and $\Pi_s$ is a rank-one projector onto the lossy site $s$.
• Spectral Analysis: The eigenvalues $E$ of $P$ are determined by a scalar secular equation derived via the matrix determinant lemma: $1 - i(\gamma/2)G_0(s, s; E) = 0$, where $G_0$ is the local resolvent of the clean chain. This equation is solved analytically using Chebyshev polynomials to handle the tridiagonal structure of the SSH chain.
• Regime Analysis: The study systematically analyzes three distinct regimes of dissipation strength: weak loss ($\gamma \to 0$), intermediate loss, and strong loss ($\gamma \to \infty$).

Key Results and Mechanisms

1. Weak-Loss Regime ($\gamma \ll |t_{1,2}|$): Local Spectral Weight Selection

   • In this limit, the decay rate of a mode is proportional to the local spectral weight of the corresponding clean eigenstate at the lossy site: $\Delta_L \sim \frac{\gamma}{2} |\phi_\alpha(s)|^2$.
   • Generic Bulk: For extended standing waves in a trivial phase, the slowest mode has a weight scaling as $N^{-3}$, yielding the generic law $\Delta_L \sim \gamma N^{-3}$.
   • Topological Phase: If the loss overlaps with an exponentially localized edge mode, the gap becomes exponentially small, scaling as $\Delta_L \sim \gamma r^{2N}$ (where $r = |t_1/t_2| < 1$). If the loss is on a bulk site in the topological phase, the gap is determined by the minimum of the bulk $N^{-3}$ law and the exponentially small edge overlap.

2. Intermediate-Loss Regime: Exceptional Points and Rapidity Reorganization

   • As $\gamma$ increases, rapidity branches reorganize. The authors identify that an exact Exceptional Point (EP) on the real-$\gamma$ axis requires balanced local spectral weights for symmetry-related pairs.
   • In a centered-loss geometry (loss at the middle of the chain), chiral symmetry enforces this balance, leading to an exact EP at $\gamma_{EP} \approx 2.62$.
   • Dynamical Signature: At this lower-edge EP, the system exhibits a defective Jordan block structure. While the leading decay rate remains exponential, the real-space dynamics (specifically one-body correlation functions) display a polynomially enhanced late-time prefactor (e.g., $t^2 e^{-\Delta t}$). This is contrasted with a matched high-energy control point (same decay edge but no lower-edge coalescence), which remains nearly exponential, confirming that the polynomial enhancement is a specific signature of lower-edge defectiveness.

3. Strong-Loss Regime ($\gamma \gg |t_{1,2}|$): Zeno Slowdown

   • Contrary to the intuition that strong loss freezes the system, the authors demonstrate a dissipative Zeno effect where the lossy site is adiabatically eliminated from the slow dynamics.
   • One ultrafast defect mode ($E \approx i\gamma/2$) separates from the spectrum. The remaining slow sector is governed by two disconnected subchains (left and right of the lossy site) coupled only virtually through the lossy site.
   • The effective dynamics are described by a reduced matrix $P_Z = P_R + \frac{2i}{\gamma} vv^T$, where $v$ represents hopping to neighbors.
   • Scaling Law: The induced slow decay scales as $\Delta_L \sim \gamma^{-1}$. The prefactor depends on the geometry (lengths of the subchains) and the termination (trivial vs. topological edge modes). In the topological regime with "dark" terminations, the gap is exponentially suppressed even in the strong-loss limit.

Significance and Claims
The paper establishes that the relaxation of an open dimerized lattice is jointly organized by topology, defect geometry, and local dissipation through three distinct mechanisms:

1. Local-weight selection at weak loss.
2. Lower-edge rapidity reorganization (including exact EPs) at intermediate loss.
3. Cut-chain Zeno reduction at strong loss.

The authors emphasize that while the full Liouvillian spectrum (including operator-parity sectors and subset-sum structures) is specific to fermionic statistics, the elementary one-body decay spectrum and the three associated mechanisms are governed by a finite-dimensional linear drift matrix. Consequently, the spectral signatures (local-weight scaling, EP coalescence, and Zeno $\gamma^{-1}$ law) are universal and can be accessed in bosonic and classical-wave platforms engineered to realize the same effective matrix. The work provides an exactly solvable framework to trace how a single tunable defect reorganizes the full relaxation problem, distinguishing robust asymptotic mechanisms from disorder-sensitive spectral details.