🔬 optics

Non-Bloch Dynamical Quantum Phase Transitions with Memory

This paper identifies a new class of non-Bloch dynamical quantum phase transitions in Hermitian chains with memory that emerge in open systems due to complex momenta and a non-zero point-gap winding number, establishing a dynamical bulk-boundary correspondence where non-Hermitian topology arises from Hermitian dynamics.

Original authors: Zhesen Yang, Wei-Wei Yang, Henning Schomerus, Jiangping Hu

Published 2026-10-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Zhesen Yang, Wei-Wei Yang, Henning Schomerus, Jiangping Hu

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 quiet world of quantum physics, scientists often study how materials behave when they are perfectly still, or when they are gently nudged. For decades, a fundamental rule has guided this work: in a large enough system, the edges do not matter. Whether a chain of atoms is connected in a loop or left open at the ends, the bulk of the material behaves the same way. This idea holds true for materials in equilibrium, where properties like temperature and pressure are stable. However, when a system is pushed out of balance—when it is suddenly jolted or driven by a rapid change—new rules can emerge. These are called dynamical phase transitions. They are moments in time, rather than points in space, where the system's behavior shifts abruptly, much like water freezing into ice, but happening in the flow of time itself. For a long time, physicists believed that even in these frantic, time-dependent scenarios, the edges of a system would remain irrelevant, and the behavior would be dictated entirely by the interior.

A team of researchers has now challenged this long-held assumption, revealing that in certain quantum systems, the edges can indeed dictate the fate of the whole, even when the system is infinitely large. They studied a specific setup: a one-dimensional chain of atoms, which acts as the main stage, connected to a vast, silent reservoir of other atoms that acts as a memory bank. When the chain is disturbed, it does not just react and move on; it interacts with this reservoir, which remembers its past states and feeds that information back. This interaction creates a complex, non-ordinary evolution where the system's history matters. By simulating this process, the researchers discovered that the way the chain behaves depends entirely on whether its ends are open or closed in a loop. In a loop, the system follows one set of rules; with open ends, it follows a completely different script, producing new critical moments in time that simply do not exist in the looped version.

The key to this discovery lies in a hidden topological property, a kind of winding number that counts how the system's internal states twist and turn as time passes. The researchers found that when this winding number is zero, the edges do not matter, and the open chain behaves just like the looped one. But when the winding number is not zero, the open chain undergoes a dramatic transformation. The system develops a "skin effect," where particles or energy accumulate heavily at the ends of the chain, a phenomenon usually associated with systems that are not perfectly balanced. However, in this case, the system is perfectly balanced in its fundamental laws; the imbalance arises purely from the memory of the reservoir. This memory forces the system to behave as if it were governed by complex, non-ordinary mathematics, even though the underlying physics remains standard.

What makes this finding particularly striking is that the researchers showed how to detect this invisible topological twist using a simple counting method. They proposed a way to watch the system evolve and count how many particles move across a specific point in the chain. By filtering out the chaotic noise and looking only at the moments when the system returns to a specific, ordered state, they found that the average number of particles that had crossed the line directly revealed the winding number. This turns a highly abstract mathematical concept into a tangible, countable physical quantity. It means that the strange, edge-dependent behavior is not just a theoretical curiosity but a real, measurable process where the system effectively pumps particles through its own interior, driven by the memory of its past interactions.

The study also clarifies why this happens. In the open chain, the critical moments in time—when the system's behavior shifts—occur at different times than in the looped chain, or sometimes not at all. The researchers traced this difference to the way the system's internal states are arranged in a complex mathematical space. In the looped chain, these states are arranged in a way that cancels out the edge effects. In the open chain, the memory of the reservoir forces the system to explore a different part of this space, one where the edges become dominant. This creates a new kind of phase transition, one that is determined not by the real, physical momentum of the particles, but by a complex, mathematical version of momentum that only exists when the boundaries are open. This "non-Bloch" behavior, as the authors call it, shows that the boundary conditions of a system can fundamentally alter its dynamical history.

The implications of this work reach beyond the specific model studied. It suggests that non-ordinary, topological behaviors can emerge naturally in ordinary, closed quantum systems simply because they have memory. This challenges the idea that such exotic effects require engineered imperfections or external noise. Instead, the memory of a simple environment is enough to generate complex, edge-sensitive dynamics. The researchers propose that this phenomenon could be observed in various experimental platforms, from waves traveling through arrays of optical fibers to quantum simulators that can track individual atoms. By measuring the flow of particles and the timing of these transitions, scientists could directly observe how the past shapes the future in a quantum system, revealing a deep connection between memory, topology, and the very nature of time in the quantum world.

Technical Summary: Non-Bloch Dynamical Quantum Phase Transitions with Memory

Problem Statement
Dynamical Quantum Phase Transitions (DQPTs) describe non-analyticities in the dynamical free energy during real-time evolution, typically studied under periodic boundary conditions (PBC). In equilibrium systems with short-range interactions, boundaries are expected to affect bulk properties only by vanishing surface terms in the thermodynamic limit. While exceptions have been noted, the general expectation is that DQPTs, like equilibrium phase transitions, are bulk phenomena independent of boundary conditions. However, recent work on "subspace DQPTs"—where a closed system is partitioned into a subspace $S$ and an environment $E$ —suggests that boundaries might play a more significant role. In such setups, the effective Hamiltonian governing the subspace is non-Hermitian and time-dependent due to non-Markovian memory effects induced by the environment. Since non-Hermitian systems are known to be sensitive to boundaries (e.g., the Non-Hermitian Skin Effect, NHSE), it remains an open question whether DQPTs in these subspace systems depend on boundary conditions and, if so, what the underlying mechanism is.

Methodology
The authors investigate subspace DQPTs in a Hermitian Su-Schrieffer-Heeger (SSH) chain coupled to a semi-infinite two-dimensional substrate. The system is defined by a block Hamiltonian where the chain ( $S$ ) and substrate ( $E$ ) are coupled. By integrating out the substrate degrees of freedom, the dynamics of the chain are described by a projected propagator $U_{SS}(t)$ , which is non-unitary and non-Markovian.

The study employs the following theoretical tools:

Subspace DQPT Definition: The dynamical free energy is defined as $r(t) = -\frac{1}{N} \ln |\det U_{SS}(t)|$ . DQPTs occur when Fisher zeros (zeros of the determinant in the complex time plane) approach the real axis.
Non-Bloch Band Theory: Guided by non-Bloch band theory for static non-Hermitian systems, the authors analytically continue the momentum $k$ to complex values $k - i\mu$ . This allows them to relate the open boundary condition (OBC) dynamics to the PBC dynamics via a specific path in the complex momentum plane.
Topological Invariants: The study focuses on the point-gap winding number $W(\tau)$ of the projected propagator in the Brillouin zone (BZ), defined as $W(\tau) = \oint_{BZ} \frac{dk}{2\pi} \partial_k \arg \det U_{SS}(k, \tau)$ .
Many-Body Extension: The single-particle results are extended to non-interacting spinless fermions, where the Loschmidt echo corresponds to the subspace dynamical amplitude.
Detection Protocol: A scheme based on full counting statistics and site-resolved fermion detection is proposed to measure the winding number $W$ directly through particle counting.

Key Results

Boundary Dependence in the Thermodynamic Limit: The authors demonstrate that the dynamical free energy $r(t)$ for an open chain differs from that of a periodic chain in specific time intervals, even in the thermodynamic limit. This difference is strictly determined by the winding number $W$ . When $W=0$ , $r_{OBC}(t) = r_{PBC}(t)$ ; when $W \neq 0$ , the two diverge.
Non-Bloch DQPTs: A new class of DQPTs, termed "non-Bloch DQPTs," is identified. These occur exclusively in open chains at critical times $t_p$ where the zero-winding plateau in the complex momentum plane closes. At these times, the OBC dynamical free energy exhibits a kink (non-analyticity) that is absent in the PBC case. Conversely, PBC DQPTs (occurring when $W$ changes) are suppressed in the OBC case.
Mechanism of Boundary Dependence: The difference between OBC and PBC free energies is derived from the analytic continuation of the momentum. Specifically, $r_{OBC}(t) = r_{PBC}(\mu^*(t), t)$ , where $\mu^*(t)$ is a path with zero winding number. The discrepancy arises from the distance between the real BZ ( $\mu=0$ ) and the winding boundary in the complex plane.
Non-Markovian Skin Effect: The system exhibits a time-dependent skin effect where eigenstates accumulate at the boundaries. Unlike static non-Hermitian systems, this accumulation pattern changes dynamically due to the frequency-dependent self-energy (memory). However, the skin effect alone does not cause boundary-dependent DQPTs; the non-zero winding number $W$ is the necessary condition.
Measurement of Topology: The authors propose that the winding number $W(t)$ corresponds to a physical particle pump. By preparing a filled chain and an empty substrate, switching on the coupling, and post-selecting snapshots where a central window of the chain remains fully occupied, the mean number of particles transferred across a cut ( $\langle m \rangle$ ) converges to $W(t)$ as the window size increases.

Significance and Claims
The paper claims several significant theoretical advances:

Dynamical Bulk-Boundary Correspondence: It reveals a dynamical analog of the bulk-boundary correspondence where a bulk topological invariant (the winding number $W$ ) dictates whether boundary conditions affect the thermodynamic limit of the dynamical free energy.
Emergent Non-Hermitian Topology: The study demonstrates that non-Hermitian topology (point-gap winding and skin effects) can emerge naturally from strictly Hermitian dynamics with memory, without engineered gain and loss. This places non-Hermitian topology on a microscopic footing within ordinary closed quantum systems.
Role of Memory: The work highlights that non-Markovian memory effects are not merely a complication but an essential resource for generating topological phenomena in dynamics. The topology exists in the time domain, as causality forbids such windings at real frequencies.
Extension of Non-Bloch Theory: The results extend non-Bloch band theory from static non-Hermitian Hamiltonians to non-equilibrium dynamics with memory, providing a practical method (analytic continuation of momentum) to compute open-boundary dynamics in the thermodynamic limit.
Experimental Feasibility: The authors assert that their predictions are testable on existing platforms, including classical waveguide arrays (for PBC DQPTs), quantum simulators (for broken time-reversal and memory), and fermionic simulators with site-resolved imaging (for direct measurement of $W$ via particle counting).

The authors distinguish their findings from previous reports of boundary-dependent DQPTs by attributing the effect to a bulk topological invariant rather than edge modes or dynamical zero modes. They conclude that this mechanism offers a new route to engineering topological phenomena via non-Markovian environments.

Technical Summary: Non-Bloch Dynamical Quantum Phase Transitions with Memory

More like this