Scale-free localization in pseudo-Hermitian quantum walk

summary

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.

In short

A single non-Hermitian defect in a one-dimensional quantum walk induces scale-free localization, which is linked to pseudo-Hermitian (PH) phase transitions. Increasing non-Hermiticity causes a second transition marked by bound states. The phenomenon is universal in two dimensions, confirming the existence and characteristics of these novel localized states.

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.

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This episode discusses

Scale-free localization in pseudo-Hermitian quantum walk · Paper Radio

The paper

Scale-free localization in pseudo-Hermitian quantum walk · Read on arXiv

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

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.

Transcript

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.

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