🔬 condensed matter

Amplified Memory and Finite-Time Regularity in Driven Non-Hermitian Systems

This paper investigates finite-time driven non-Hermitian fermionic dynamics in an imbalanced-pairing Kitaev chain, revealing that transient passage through broken-spectrum regions amplifies preparation memory via accumulated imaginary-energy action while exceptional points induce distinct finite-time regularity in correlation matrices despite characteristic gap closures.

Original authors: H. Yavartanoo, R. Jafari, Alireza Akbari

Published 2026-09-25

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: H. Yavartanoo, R. Jafari, Alireza Akbari

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the microscopic world of quantum physics, matter is often described by mathematical rules that dictate how particles behave and interact. For decades, scientists have relied on a specific set of these rules, known as Hermitian physics, which guarantees that energy levels are real numbers and that information is never lost as a system evolves. However, a newer and more complex branch of physics explores systems where these rules are broken. These non-Hermitian systems, which can model materials with gain and loss or open environments, allow for energy values that are complex numbers and introduce strange points where the usual distinction between different states of matter blurs together. Understanding how these systems behave when pushed and pulled by external forces is crucial, because it reveals how memory and information survive in environments that are inherently unstable or dissipative. The question remains: if you briefly push a quantum system into a chaotic, unstable region and then pull it back to safety, does the system remember the journey, or does it settle down as if nothing happened?

A team of researchers has now mapped out exactly what happens when a specific quantum chain is driven through such a turbulent landscape. Using a theoretical model of a chain of particles that can pair up in unbalanced ways, they simulated a process where the system is slowly swept across a region where its energy spectrum becomes complex and unstable, before returning it to a stable, real-energy state. They found that the system does indeed retain a vivid memory of this transient journey. Even after the external force is removed and the system settles into a stable configuration, a measurable "excess" remains. This excess is not a random fluctuation but a direct imprint of the time the system spent in the unstable zone, amplified by the imaginary energy components it accumulated during the passage. The researchers demonstrated that this memory persists indefinitely in the post-ramp evolution, surviving long after the system has returned to a calm state.

To uncover this phenomenon, the scientists focused on a specific type of quantum chain where the pairing between particles is imbalanced. In the stable parts of this chain, the energy levels are real numbers, much like the familiar energy states of ordinary matter. However, when the researchers tuned a control parameter to push the system into a specific range, the energy levels turned into complex numbers, creating a region of broken symmetry. In this zone, the system's behavior is dominated by exponential growth or decay rather than simple oscillation. The team simulated a scenario where the system starts in a stable state, is driven through this complex region at a controlled speed, and is then stopped in a new stable state. By carefully comparing the final state of the driven system to a static reference state that had never been driven, they isolated a specific signal: a residual amount of "spectral nonpositivity." This term describes a deviation in the mathematical weights of the system's states that would be impossible in a standard, stable system. The key finding is that the size of this deviation is not random; it is strictly controlled by the total "action" accumulated while the system was in the unstable region. The slower the drive, the more time the system spends in the amplifying zone, and the larger the memory imprint becomes.

This memory is not just a mathematical curiosity hidden in abstract equations; it manifests in the physical correlations between particles across the chain. The researchers tracked how the connection between particles at different distances changed after the drive. They found that the pattern of these connections carried the exact same signature of the drive's history as the abstract memory measure. The strength of these correlations grew in direct proportion to the time spent in the unstable region, confirming that the memory is a genuine physical feature of the system's state, not an artifact of the measurement method. Furthermore, they discovered that if the drive is halted while the system is still inside the unstable region, the amplification does not stop; instead, the system continues to grow and amplify indefinitely. This contrasts sharply with the behavior when the drive stops in a stable region, where the memory remains fixed and finite, a clear distinction between a system that is still in a runaway state and one that has settled with a scar from its past.

The study also investigated what happens when the system is stopped exactly at the boundary between the stable and unstable regions, a point known as an exceptional point. At this precise location, the mathematical rules governing the system become singular, and the energy gap between states vanishes in a specific way. Intuitively, one might expect such a singularity to cause a violent, unpredictable reaction in the system's dynamics. However, the researchers found that for any finite amount of time, the system's evolution remains smooth and regular. The strange, singular behavior of the energy levels does not immediately translate into a singularity in the observable properties of the system. The system responds linearly to how close it is to this boundary, and the dramatic effects only emerge after a very long time, as the system slowly resolves the tiny energy differences. This "finite-time regularity" means that even at the most extreme points of instability, the system behaves predictably in the short term, with the full impact of the singularity only revealing itself over long durations.

In addition to the memory of the drive, the team examined how information spreads through the chain when the system is stopped exactly at this exceptional boundary. They tracked the movement of correlations across the chain and found that they spread out in a sharp, ballistic front, moving at a constant speed. This speed is determined by the overall structure of the energy bands rather than the local instability at the boundary. Interestingly, the researchers also measured how far the "memory" of the system extends across the chain as it evolves. They found that the region of the chain affected by the drive grows linearly over time, establishing a distinct spatial scale. While this growing region moves at a speed similar to the correlation front, it is not identical, suggesting that the way the system remembers its past and the way it transmits information are related but separate processes. The study concludes that these non-Hermitian systems offer a rich playground where transient excursions into instability leave permanent, measurable marks, and where the singularities that define their boundaries behave with a surprising degree of order in the short term.

Technical Summary: Amplified Memory and Finite-Time Regularity in Driven Non-Hermitian Systems

Problem Statement
The paper investigates the dynamical response of non-Hermitian fermionic systems to finite-rate driving, specifically focusing on two distinct phenomena: the memory effects arising from transient passages through broken-spectrum regions and the behavior of observables near exceptional points (EPs). While static biorthogonal entanglement in non-interacting systems is well-understood, the dynamics of finite-rate ramps across complex-energy windows and the specific signatures of EPs in time-evolved states remain largely unexplored. The authors address two central questions: (1) What residual memory of a temporary broken-spectrum passage survives after a finite-rate ramp returns the system to a real-spectrum phase? and (2) When does the singular scale of an exceptional endpoint become visible in the dynamics of reduced states?

Methodology
The authors utilize the Nambu biorthogonal correlation matrix as the primary diagnostic tool for a driven, imbalanced-pairing Kitaev chain. The system is modeled by a quadratic fermionic Hamiltonian with a time-dependent chemical potential $\mu(t)$ sweeping linearly at rate $v$ .

Framework: The study employs a biorthogonal framework where right ( $|R\rangle$ ) and left ( $\langle L|$ ) eigenstates span different bases. The many-body state is characterized by a correlation projector $P_k(t)$ constructed from these frames. The reduced density matrix for a subsystem $A$ is defined via overlap normalization, $\rho_{LR}^A = \text{Tr}_{\bar{A}} (|\Psi_R\rangle\langle\Psi_L| / \langle\Psi_L|\Psi_R\rangle)$ , ensuring unit trace and idempotence, distinct from Dirac-normalized schemes.
Metrics: Key observables include:
- Spectral Nonpositivity ( $N_A$ ): A measure of eigenvalues $\lambda_\nu$ of the subsystem correlation matrix $C_A$ that fall outside the interval $[0, 1]$ , indicating negative many-body weights.
- Entanglement Entropy ( $S_A$ ): Calculated from the biorthogonal spectrum, with careful handling of branch cuts for complex eigenvalues.
- Connected Longitudinal Correlations ( $C_{zz}$ ): Real-space spin correlations derived from fermionic contractions to verify memory effects independent of entanglement diagnostics.
Model: The specific realization is an imbalanced Kitaev chain with pairing imbalance parameter $\gamma$ $γ$ .
- For $\gamma < 0$ , the system exhibits a broken-spectrum region (complex eigenenergies) bounded by exceptional points.
- For $\gamma > 0$ , the system is mapped to a Hermitian Kitaev chain via a local similarity transformation, serving as a benchmark.

Key Contributions and Results

1. Amplified Finite-Rate Memory in Real-Spectrum Endpoints
When the drive sweeps through a broken-spectrum region (where eigenenergies are imaginary) and halts in a real-spectrum regime ( $\mu_f > \mu_{EP}$ ), the system retains a "preparation memory."

Excess Nonpositivity: The authors define an excess nonpositivity $\Delta N_A$ by subtracting the static baseline of the final Hamiltonian. This excess is generated by the transient passage through the complex-energy window.
Scaling Law: The excess scales exponentially with the inverse ramp velocity $v$ , governed by the accumulated imaginary action $A(k) = \int \kappa_k dt$ . Specifically, $\Delta N_A^{prep} \sim \frac{1}{\sqrt{v}} \exp\left(\frac{\pi |\gamma| \Delta^2}{v}\right)$ .
Persistence: This memory effect persists indefinitely during the post-ramp hold time under the real-spectrum Hamiltonian, manifesting as a finite, non-relaxing offset.
Correlation Confirmation: The connected longitudinal correlation $C_{zz}$ exhibits an identical drive-history dependence. The integrated memory norm $M_C$ scales as $v^{3/4} \exp(2A_{max}/v)$ , confirming the memory is a genuine dynamical feature and not an artifact of the entanglement definition.
Contrast with Broken Endpoints: If the drive halts inside the broken-spectrum region, the system exhibits unstable, indefinite amplification ( $\sim \tau^{-1/2} e^{\Gamma_{max}\tau}$ ) rather than saturated memory.

2. Finite-Time Regularity at Exceptional Endpoints
The paper challenges the assumption that exceptional points (where the Hamiltonian becomes defective) necessarily induce singularities in finite-time dynamics.

Analyticity: Despite the quasiparticle gap closing as $\sqrt{\delta\mu}$ (where $\delta\mu$ is the distance to the EP), the finite-time propagator and the subsystem correlation matrix remain analytic in the endpoint parameter $\delta\mu$ for any fixed finite time $\tau$ .
Linear Scaling: The deviation of the correlation matrix from the exact EP limit scales linearly with $\delta\mu$ ( $D_C \propto \delta\mu$ ), and the entropy follows the same linear behavior on a chosen branch. The square-root singularity of the spectrum does not translate to a singularity in the finite-time evolution operator.
Long-Time Crossover: The singular scale of the EP only emerges in the asymptotic long-time limit. A crossover occurs when the time $\tau$ satisfies $\tau \sim \delta\mu^{-1/2}$ , at which point the dynamics resolves the vanishing gap.

3. Spatial Dynamics at the Exact Exceptional Halt
When the drive halts exactly at the EP ( $\delta\mu = 0$ ):

Ballistic Fronts: Both the correlation projector and the connected longitudinal correlation spread ballistically with a single dominant wavefront velocity $v_{ridge} \approx 2.39$ . This velocity is determined by the global band dispersion ( $v_{max} = \sqrt{6}$ ) rather than local EP kinematics, contrasting with previous findings of supersonic modes in other non-Hermitian quenches.
Saturation Length: The subsystem saturation length $\ell_{sat}$ (where entanglement or nonpositivity saturates with subsystem size) also expands linearly in time but with a distinct, slower velocity ( $u \approx 1.9$ ) compared to the correlation front. This establishes that the correlation wavefront and the subsystem saturation scale are distinct spatial scales.

Significance and Claims
The paper claims to separate finite-rate preparation memory from exceptional-point regularity.

Memory: The transient amplification of imaginary energy during a ramp leaves a permanent, measurable imprint (excess nonpositivity and correlation memory) even after the system returns to a stable, real-spectrum phase. This memory is strictly controlled by the integrated imaginary action and is robust across different final endpoints and imbalance parameters.
Regularity: The study demonstrates that the spectral singularity of an exceptional point does not induce immediate non-analyticity in finite-time observables. The "defectiveness" of the Hamiltonian is smoothed out over finite durations, with singular behavior only appearing as a long-time crossover.
Benchmarking: The work provides a rigorous benchmark for non-Hermitian dynamics using the $\gamma > 0$ Hermitian-equivalent case, validating the biorthogonal correlation matrix approach.

The authors conclude that while broken-spectrum passages act as amplifiers for preparation memory, exceptional endpoints exhibit a unique "finite-time regularity" where the singular nature of the spectrum is deferred to the asymptotic long-time regime. The study remains within the realm of quadratic (Gaussian) systems, noting that the interplay of these effects with many-body interactions remains an open problem.

Technical Summary: Amplified Memory and Finite-Time Regularity in Driven Non-Hermitian Systems

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