Analysis of complex Bloch vectors, descriptions of a system’s quantum state before and after a sudden change, reveals anomalous structures within Fisher zero lines during non-Hermitian quenches, instances where energy levels shift unexpectedly. The work identifies critical times when rate functions become unpredictable, improving upon existing methods used to pinpoint such moments in dynamic quantum phase transitions.

Examination of how quantum systems respond to abrupt changes, particularly those lacking typical symmetry, resulted in new analytical techniques focused on “density of Fisher zeros”. This metric precisely identifies critical moments during these shifts; conventional methods struggle when ‘exceptional points’ cause unusual system behaviour.

These advances enable detailed analysis of complex dynamics in non-Hermitian systems beyond previous capabilities. Researchers at Dalian University of Technology have refined techniques for identifying critical moments in dynamic quantum phase transitions within systems experiencing abrupt changes; these shifts are particularly complex when dealing with non-Hermitian systems, those lacking typical symmetry. The team focused on “Fisher zeros”, which can be understood as locations on a graph signalling important points of change during a system’s evolution, much like key frames in a movie reveal its narrative arc.

By employing what they call a ‘biorthogonal framework’, essentially comparing two related perspectives simultaneously to analyse how states evolve, they pinpointed conditions where traditional methods fail due to the unusual behaviour around ‘exceptional points‘. This advancement allows detailed analysis beyond previous capabilities and introduces a new metric, the density of Fisher zeros, to identify critical times within regions exhibiting unpredictable rate functions; but will this refined approach fully resolve the challenges posed by these anomalous structures.

Non-Hermitian Momentum Reveals Critical Times via Anomalous Fisher Zero Structures

The dot product between pre- and post-quench Bloch vectors (m_k) now exceeds unity, a value previously impossible within Hermitian systems. This reveals critical momentum conditions during dynamical quantum phase transitions induced by abrupt changes to open quantum systems. Detailed examination alongside energy spectrum analysis after quenching shows how anomalous structures within Fisher zero lines (divergences or separations on a complex time plane) originate from this m_k value.

When these zeros densely populate the real time axis, ‘density of Fisher zeros’ emerges as a new metric for pinpointing precise critical times when rate functions become unpredictable, surpassing methods reliant solely on identifying zero existence. This work identifies conditions for unpredictable rate functions but does not yet explain practical realization of such precise control over complex quantum systems.

A generalised non-Hermitian Su-Schrieffer-Heeger model verified these findings by abruptly changing both the Peierls phase and gain/loss strength; even dynamics at exceptional points were analysed, a regime where standard methods fail. The study focuses on understanding implications within this specific SSH framework while leaving open questions about broader applicability to other models or parameter regimes.

Fisher zero behaviour reveals dynamics near dynamical quantum phase transitions

During rapid shifts (quenches) in non-Hermitian systems, Fisher zeros, points in complex time when calculations relating to quantum changes become undefined, exhibit unusual patterns like divergence, separation, or restriction to specific axes. These anomalies are directly linked to the dot product between pre- and post-quench Bloch vectors, offering insight into identifying conditions for dynamical quantum phase transitions. Beyond simply detecting when such transitions occur, researchers now examine ‘density of Fisher zeros’ which allows them to pinpoint precise moments of change within those critical regions.

A generalised non-Hermitian SSH model, a mathematical framework simulating materials with unique electronic properties, verified these findings by altering both energy levels and gain/loss strengths during quenches. Jordan form analysis was also employed at exceptional points where standard methods fail; they represent singularities in system behaviour requiring alternative analytical approaches. Current results apply specifically to parameters around exceptional points, acknowledging limitations and leaving open questions about broader applicability across different types of non-Hermitian systems or challenges calculating Bloch vectors for other models.

Biorthogonal Framework Analysis of Dynamical Quantum Phase Transitions

Research into dynamical quantum phase transitions (DQPTs), critical phenomena occurring during changes in quantum systems, has expanded beyond traditional closed systems to include scenarios like slow quenches and open systems, experiments have been conducted using trapped ions, cold atomic gases, and quantum simulation platforms. Initial DQPT studies in non-Hermitian systems relied on a self-normal framework utilising conventional probability calculation techniques. A subsequent biorthogonal framework was proposed, employing different mathematical bases suited to non-Hermitian systems with complex eigenvalues; this defined the Loschmidt echo as a real probability between zero and one.

Single photon experiments recently demonstrated differing critical times when comparing rate functions within these two frameworks. This research adopts that approach to analyse Fisher zeros, points where the Loschmidt amplitude equals zero, providing an avenue for determining conditions for DQPTs and understanding how ‘rate function’ becomes nonanalytic, particularly useful since standard formulas fail at exceptional points in postquench dynamics. The findings were verified using a generalised Su-Schrieffer-Heeger (SSH) model by altering both the Peierls phase and gain/loss strength of the system; this clarifies how unusual structures appear within Fisher zero lines, signalling change during quantum shifts, revealing they originate from the relationship between initial and final states after abrupt disturbance, quantified through similarity measure analysis. This extends beyond standard methods because it accounts for systems lacking typical symmetry where conventional calculations break down at so-called ‘exceptional points’. By introducing a new metric to track these systems, researchers can move past limitations in current models and explore broader applicability across all types or parameter regimes of non-Hermitian Hamiltonians, the energy functions governing these systems.

This research clarified the origins of anomalous structures found in Fisher zeros, points where Loschmidt amplitude equals zero, during rapid changes in quantum systems. The study demonstrated that these structures arise from the relationship between the system’s state before and after a disturbance, quantified by analyzing dot products between complex Bloch vectors.

Using a generalized Su-Schrieffer-Heeger model with altered Peierls phase and gain/loss strength, researchers showed this approach identifies critical times when rate function behavior becomes unpredictable. This work extends analysis beyond standard methods applicable to systems lacking symmetry or encountering exceptional points during dynamic shifts.

👉 More information
🗞 Anomalous Structures of Fisher Zero Lines in Biorthogonal Dynamical Quantum Phase Transitions
✍️ Wen-Hui Nie, Chong Li and Lin-Cheng Wang (Dalian University of Technology)
🧠 ArXiv: https://arxiv.org/abs/2610.36521

More like this