Introducing one non-Hermitian defect into quantum walks, systems describing how probability spreads through space, induces scale-free localization; this means localized states appear whose spatial extent grows linearly with system size. This research reveals two phase transitions driven by increasing non-Hermiticity: first, a change in pseudo-Hermitian symmetry coinciding with these new localized states, then another marked by bound state formation at critical points of 1.32 and 2.09.

Introducing even slight imperfections into quantum systems can lead to unexpected results; introducing just one “defect” within these systems induces scale-free localization, where energy states are contained in areas growing proportionally to overall size. Two distinct changes occur with increasing defect intensity: first, a shift in symmetry coinciding with localized states, then bound state formation at specific points.

Research into quantum walks, systems describing how probability spreads through space, akin to flipping a coin and moving left or right but utilising quantum mechanics allowing simultaneous states, reveals that even minor imperfections can sharply alter behaviour. Specifically, introducing just one “defect” induces scale-free localization; imagine dropping pebbles into still water where ripples normally fade quickly, this phenomenon creates persistent ripples maintaining strength over larger areas due to specific conditions.

This research identifies two phase transitions driven by increasing defect intensity: first, a shift in pseudo-Hermitian (PH) symmetry, think of balancing a seesaw slightly off-centre changing its fundamental properties while retaining structure, and then bound state formation at critical points of 1.32 and 2.09.

Single Non-Hermitian Defects Drive Scale-Free Localization and Bound State Emergence

Introducing even a single non-Hermitian defect into one-dimensional quantum walks induces scale-free localisation, a shift from initial eigenstates spreading evenly to exhibiting localisation lengths scaling linearly with system size. Previously impossible without substantial complexity, this transition occurs at critical points of 1.32 and 2.09; the first marks pseudo-Hermitian symmetry breaking accompanied by localized states while the second signals tightly confined energy levels emerging as bound states within the system. Further analysis using inverse participation ratios confirmed these behaviours, revealing higher values for bound states compared to those experiencing scale-free localisation allowing precise identification of both phase transitions through their distinct characteristics.

Spatial profiles and inverse participation ratios provided further evidence supporting this localisation effect; the latter measurements quantify how spread out an eigenstate is within the system. At a non-Hermiticity level around 1.32, initial localisation was observed alongside pseudo-Hermitian symmetry breaking. Increasing this value to approximately 2.09 triggered emergence of tightly confined bound states, distinct energy levels appearing with elevated IPR values enabling clear differentiation between phases.

Single defect localization clarifies foundational principles in quantum walk behaviour

This work convincingly demonstrates scale-free localization arising from a single defect, a remarkable feat given previous studies requiring substantial complexity for similar effects; it deliberately confines itself to that specific scenario while existing investigations have explored localized states within more disordered systems or those containing multiple imperfections. The authors acknowledge an open question: will these findings hold true when faced with varied arrangements of defects, or even different types of disruption beyond simple gain and loss. This research focuses on a simplified scenario as real materials are invariably more complex than laboratory models.

Demonstrating scale-free localisation from such basic disruption establishes a fundamental principle for understanding how imperfections impact quantum systems. A team at Beijing Computational Science Research Centre, Southeast University, China University of Petroleum-Beijing, and College of Science has demonstrated that introducing a single imperfection into ‘quantum walks’ reliably creates scale-free localization; meaning energy states are contained within areas growing proportionally to overall size. This finding is significant because previous studies required far more complex arrangements to achieve similar localized behaviour in these systems.

Future work will explore if these findings extend to more realistic scenarios with multiple defects, potentially unlocking new quantum technologies here.

This research demonstrates that scale-free localization can occur from even a single non-Hermitian defect within one-dimensional and two-dimensional quantum walk systems. Scale-free localization means the area containing an energy state grows linearly alongside the system’s total size. This work provides insight into how imperfections affect quantum systems and offers a basis for further investigation into defects and novel localized states.

👉 More information
🗞 Scale-free localization in pseudo-Hermitian quantum walk
✍️ Haiting Zhang (Affiliation: Beijing Computational Science Research Center); Lei Xiao, Quan Lin and Peng Xue (Southeast University); Cui-Xian Guo (China University of Petroleum-Beijing)
🧠 ArXiv: https://arxiv.org/abs/2610.31067

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