Scale-free localization in pseudo-Hermitian quantum walk

arXiv:2610.31067 · quant-ph, cond-mat.mes-hall · Submitted 2026-10-22 · Read on arXiv

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Introduction to the show: ident: Quantum Radio. Generated commentary on the latest quantum physics and condensed matter papers.

Kai: We're looking at the paper: "Scale-free localization in pseudo-Hermitian quantum walk".

Mira: Here's the gist of the paper. The gist: A single non-Hermitian defect in one-dimensional quantum walks induces scale-free localization, which is linked to pseudo-Hermitian (PH) phase transitions and is universal in two dimensions.

Kai: Model Setup The study considers a split-step QW on a 1D lattice under periodic boundary conditions with a single non-Hermitian defect introduced via on-site gain and loss terms defined by U = D−1SR(θ)DSR(θ) The coin operator is given by R(θ) = Xxx⟩⟨x ⊗ cos θ − sin θsin θ cos θ The conditional position shift operator is given by S = Xxx + 1⟩⟨x ⊗ 0⟩⟨0 + x − 1⟩⟨x ⊗ 1⟩⟨1> Pseudo-Hermitian Symmetry and Phase Transitions The system satisfies the PH condition, i.e., there exists an invertible matrix η such that ηU −1η−1 = U† When the system is in the PH-symmetry unbroken phase, all eigenvalues are real Conversely, in the PH-symmetry broken phase, paired complex-conjugate eigenvalues emerge.

Mira: The spectral transitions are characterized by two quantities: ρcom = Pcom/2L and (ImE)max = max i=1,…,2L ImEi.

Kai: In the PH-symmetry unbroken phase, ρcom = 0 and (ImE)max = 0, indicating the system is PH-symmetry unbroken.

Paper summary: Kai: The paper, "Scale-free localization in pseudo-Hermitian quantum walk," investigates how specific eigenstates, known as scale-free localized states or LSs, have localization lengths that scale linearly with the system size in non-Hermitian systems.

Mira: The core thesis they present is that a single non-Hermitian defect placed within a one-dimensional quantum walk induces this scale-free localization.

Lev: It’s driven by the fact that this QW system undergoes a pseudo-Hermitian phase transition, and breaking of that PH symmetry is what accompanies the emergence of these scale-free localized states.

Kai: They also note that as the non-Hermiticity is increased further, the system goes through a second phase transition which results in two pairs of bound states.

Mira: What matters here is how they identify the critical points for both transitions using localization properties, rather than just spectral analysis alone.

Conclusion: Kai: So looking at the full picture, the paper "Scale-free localization in pseudo-Hermitian quantum walk" shows that scale-free localization isn't just a feature of some exotic non-Hermitian systems, but it appears here when you have a simple defect in a one-dimensional quantum walk.

Mira: It really matters because it connects the breaking of the pseudo-Hermitian symmetry directly to these spatial scaling properties, giving us a concrete way to link symmetry breaking to physical localization behavior.

Lev: For someone building real hardware, understanding where those critical points are—like gamma c equals one point three two and gamma b equals two point zero nine for the first transition—that tells you exactly where you need to tune your system parameters to observe these specific states in action <ref:2610.31067#pg2>.

Kai: And extending this to two dimensions shows that this behavior isn't just a fluke of one dimension; it’s universal, meaning you see this same phenomenon in higher dimensional quantum walks too.

Mira: It gives us a new tool for exploring defect physics and identifying novel types of localized states in non-Hermitian quantum systems that we haven't seen before.

Haiting Zhang, *, Lei Xiao, *, Cui-Xian Guo, Quan Lin, Peng Xue

Beijing Computational Science Research Center of China and Southeast University, Nanjing University, Key Laboratory of Quantum Materials and Devices of Ministry of Education, China University of Petroleum-Beijing

quant-ph, cond-mat.mes-hall

Submitted: 2026-10-22

Updated: 2026-10-22

Comments: 7 pages, 4 figures

License: http://creativecommons.org/licenses/by/4.0/

The gist: The gist: A single non-Hermitian defect in one-dimensional quantum walks induces scale-free localization, which is linked to pseudo-Hermitian (PH) phase transitions and is universal in two dimensions.

Key concepts

Scale-free Localization (LS): This describes specific quantum states in non-Hermitian systems where their localization length scales linearly with the system size. This means as you make the system bigger, these special states don't localize more or less; they maintain a fixed scaling relationship, which is a distinctive feature of non-Hermitian physics.
Pseudo-Hermitian (PH) Symmetry: This condition checks if a non-Hermitian Hamiltonian behaves like a Hermitian one under a specific transformation. If the system has unbroken PH symmetry, all its energy eigenvalues are real. Breaking this symmetry leads to the appearance of complex-conjugate eigenvalue pairs, which is crucial for understanding spectral transitions.
Localization Measures (IPR and NPR): These metrics quantify how localized or extended an eigenstate is within the system. The Inverse Participation Ratio (IPR) scales linearly with 1/L for scale-free states, while the Normalized Participation Ratio (NPR) remains constant. These measures are used to distinguish between extended states and the newly found scale-free localized states.
Universality: Universality means that a specific physical phenomenon, like scale-free localization induced by a defect, occurs in the same way across different systems, even if they have different dimensions. The paper shows this effect holds true when moving from one dimension (1D) to two dimensions (2D), confirming its fundamental nature.

Terminology

Localization — 1× in this paper · explained in 2 episodes

Summary

The gist: A single non-Hermitian defect in one-dimensional quantum walks induces scale-free localization, which is linked to pseudo-Hermitian (PH) phase transitions and is universal in two dimensions.

Model Setup

The study considers a split-step QW on a 1D lattice under periodic boundary conditions with a single non-Hermitian defect introduced via on-site gain and loss terms defined by U = D−1SR(θ)DSR(θ) The coin operator is given by R(θ) = Xxx⟩⟨x ⊗ cos θ − sin θsin θ cos θ The conditional position shift operator is given by S = Xxx + 1⟩⟨x ⊗ 0⟩⟨0 + x − 1⟩⟨x ⊗ 1⟩⟨1>

Pseudo-Hermitian Symmetry and Phase Transitions

The system satisfies the PH condition, i.e., there exists an invertible matrix η such that ηU −1η−1 = U† When the system is in the PH-symmetry unbroken phase, all eigenvalues are real Conversely, in the PH-symmetry broken phase, paired complex-conjugate eigenvalues emerge. The spectral transitions are characterized by two quantities: ρcom = Pcom/2L and (ImE)max = max i=1,…,2L ImEi. In the PH-symmetry unbroken phase, ρcom = 0 and (ImE)max = 0, indicating the system is PH-symmetry unbroken. The critical point of the PH phase transition is analytically expressed as γc = ln tan θ/2 for θ!= π/2 + kπ, 0 for θ = π/2 + kπ. Increasing the non-Hermitian strength to γb induces another phase transition, namely the emergence of bound states.

Characterization of Localization

To clarify the types of LSs, the normalized spatial profile is introduced as Ψn(x)2 = ψn(x)2 max [ψn(x)2] Additionally, the IPR for a given eigenstate is defined as IPRn = Xxψn(x)4 (P xψn(x)2) 2. The paper presents eigenstates of H for γ = 1.8 and 2.2 in the PH-symmetry broken phase, revealing that scale-free LSs exist at both strengths, whereas bound states appear only at γ = 2.2. For scale-free LSs, the IPR scales linearly with 1/L. In contrast, for bound states, the spatial profiles vary with system size while their IPR remains constant as a function of 1/L.

Critical Points Identification

The critical points of these two phase transitions are identified by introducing the averaged IPR ⟨IPR⟩ = 1/L X L n=1 IPRn and the averaged NPR ⟨NPR⟩ = 1/L X L n=1 NPRn. In the PH-symmetry unbroken phase, all eigenstates are extended, so ⟨IPR⟩ assumes a small value while ⟨NPR⟩ remains comparatively large. When the system enters the PH-symmetry broken phase, some eigenstates become scalefree LSs, resulting in an increase in ⟨IPR⟩ and a corresponding decrease in ⟨NPR⟩. A further increase in the non-Hermitian strength leads to the emergence of bound states, which enhance ⟨IPR⟩ and suppress ⟨NPR⟩, marking another phase transition. Both ln⟨IPR⟩ and ln⟨NPR⟩ exhibit clear singularities at γc ≈ 1.32 and γb ≈ 2.09, in agreement with the critical points determined from the spectral analysis.

Universality in Two Dimensions

The study extends to a 2D QW with a single non-Hermitian defect, defining U2 = D−1/2 S2R2(θ)D2S2R2(θ). The 5th largest imaginary part of the eigenvalues for a scale-free LS scales linearly with 1/L. Furthermore, the normalized spatial profiles of these states almost overlap regardless of system size, demonstrating that these states are scale-free LSs. This confirms the universality of this phenomenon in two dimensions. The results highlight the characteristics of scale-free LSs and offer a new perspective for studying non-Hermitian defect physics.

Conclusion

The emergence of scale-free localization is deeply connected to the breaking of PH-symmetry in the 1D QW. A further increase in non-Hermitian strength induces another phase transition, giving rise to two additional pairs of bound states. By analyzing spectral and localization properties, the critical points of these two transitions are identified. The investigation extends to 2D QWs, where a single non-Hermitian defect also induces scale-free localization, thereby demonstrating the universality of this phenomenon. Our results highlight the characteristics of scale-free LSs and offer a new perspective for studying non-Hermitian defect physics. The findings pave the way for future explorations in higher-dimensional QWs.

Acknowledgments

This work is supported by the National Key R&D Program of China (Grant No. 2023YFA1406701) and National Natural Science Foundation of China (Grants No. 92265209, 12474352, and 92476106) L. Xiao acknowledges support from Beijing National Laboratory for Condensed Matter Physics (No. 2024BNLCMPKF010) C. X. Guo acknowledges support from the National Natural Science Foundation of China (Grant No. 12405030) and Beijing National Laboratory for Condensed Matter Physics (No. 2025BNLCMPKF023) Data availability is available from the corresponding authors upon reasonable request.

References

[1] H. J. Carmichael, Quantum trajectory theory for cascaded open systems, Phys. Rev. Lett. 70, 2273 (1993)

[2] C. M. Bender and S. Boettcher, Real spectra in nonHermitian Hamiltonians having P T symmetry, Phys. Rev. Lett. 80, 5243 (1998)

[3] K. Kawabata, K. Shiozaki, M. Ueda, and M. Sato, Symmetry and topology in non-Hermitian physics, Phys. Rev. X 9, 041015 (2019)

[4] H. Zhou and J. Y. Lee, Periodic table for topological bands with non-Hermitian symmetries, Phys. Rev. B 99, 235112 (2019)

[5] S. Weimann, M. Kremer, Y. Plotnik, Y. Lumer, S. Nolte, K. G. Makris, M. Segev, M. C. Rechtsman, and A Szameit, Topologically protected bound states in photonic parity-time-symmetric crystals, Nat. Mater. 16, 433 (2017)

[6] Z. Gu, H. Gao, H Xue, J Li, Z Su, and J Zhu, Transient non-Hermitian skin effect, Nat. Commun. 13, 7668 (2022)

[7] F.

Improvements for AI systems

Non-Hermitian defect physics modeling can be improved by incorporating scale-free localized states (LSs) as emergent features in quantum walk systems, allowing AI to predict localization behavior based on non-Hermiticity strength and dimensionality.
AI systems can develop the ability to identify critical points of phase transitions by analyzing the averaged IPR and NPR observables, as these quantities exhibit clear singularities at gamma c ≈ 1.32 and gamma b ≈ 2.09.
The system can be enhanced to simulate two distinct regimes within a single non-Hermitian defect: one where some eigenstates become LSs and another where two pairs of bound states appear, based on the non-Hermiticity parameter gamma.
In two-dimensional quantum walks, the AI can predict universality by recognizing that a single non-Hermitian defect also induces scale-free localization, confirming this phenomenon across dimensions.
The AI can characterize LSs and bound states by analyzing their spatial profiles, specifically noting that for LSs, the spatial profiles of all eigenstates overlap, while for bound states, they show characteristics where their IPR remains constant as a function of 1/L.

Abstract

Scale-free localization is a distinctive phenomenon in non-Hermitian systems, where specific eigenstates, known as scale-free localized states (LSs), possess localization lengths that scale linearly with the system size. Here, we show that a single non-Hermitian defect in one-dimensional quantum walks (QWs) induces scale-free localization. Driven by non-Hermiticity, the QW system undergoes a pseudo-Hermitian (PH) phase transition and the PH symmetry breaking is accompanied by the emergence of scale-free localization. As non-Hermiticity is further increased, the system undergoes a second phase transition, marked by the emergence of two pairs of bound states. Crucially, we identify the critical points of both transitions using localization properties. Furthermore, we extend our investigation to two-dimensional QWs, where a single non-Hermitian defect also induces scale-free localization, thereby demonstrating the universality of this phenomenon. Our work provides a valuable perspective for further exploring defect physics and a type of novel LS in non-Hermitian quantum systems.