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arxiv: 2411.16843 · v2 · submitted 2024-11-25 · 🪐 quant-ph · cond-mat.mes-hall

Mobility edges in pseudo-unitary quasiperiodic quantum walks

Christopher Cedzich , Jake Fillman This is my paper

Pith reviewed 2026-05-23 08:23 UTC · model grok-4.3

classification 🪐 quant-ph cond-mat.mes-hall
keywords mobility edgequasiperiodic quantum walkspseudo-unitaryPT-symmetryFloquet quasicrystalAubry dualityspectral winding numbernon-reciprocal hopping
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The pith

Non-reciprocal hopping in a discrete-time quasicrystal produces a mobility edge separating metallic and insulating phases plus a PT-symmetry breaking transition.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The authors build a Floquet quasicrystal that mimics Bloch electrons moving in a magnetic field but stepped in discrete time. Allowing non-reciprocal hopping and shifting a phase parameter off the real axis via generalized Aubry duality turns the evolution operator pseudo-unitary. In this setting the spectrum splits into metallic and insulating regions divided by a sharp mobility edge. When hopping remains reciprocal in one direction the model is PT-symmetric and the spectrum stays on the unit circle until a critical point; beyond that point PT symmetry breaks spontaneously and a topological winding number jumps. Both transitions are tied directly to spectral features, with the second one appearing only in the discrete-time case.

Core claim

The introduction of non-reciprocal hopping in a Floquet quasicrystal leads to a pseudo-unitary model that supports a novel mobility edge between metallic and insulating phases, sharply dividing the parameter space. Additionally, when hopping is reciprocal in one direction, PT-symmetry confines the spectrum to the unit circle until a critical point where symmetry breaks spontaneously, accompanied by a topological transition measured by the spectral winding number. This second transition appears unique to the discrete-time setting.

What carries the argument

The pseudo-unitary Floquet operator obtained by applying generalized Aubry duality to a non-reciprocal quasiperiodic quantum walk.

If this is right

  • The mobility edge sharply separates regions of extended and localized eigenstates in the spectrum.
  • PT symmetry holds and keeps the spectrum on the unit circle only up to a critical point determined by the non-reciprocity strength.
  • At the critical point the spectral winding number changes, marking a topological transition.
  • The second transition has no counterpart in continuous-time models and is therefore tied to the discrete-time stepping.
  • Both transitions can be read off from properties of the spectrum without separate localization calculations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same construction could be used to explore how synthetic dimensions interact with non-Hermitian hopping in other Floquet systems.
  • Experimental platforms that realize discrete-time quantum walks with tunable non-reciprocity could test the predicted mobility edge directly.
  • The uniqueness of the discrete-time PT transition suggests that continuous-time limits may miss entire classes of spectral reorganizations.
  • Extending the duality argument to other quasiperiodic potentials might produce additional mobility edges at different parameter values.

Load-bearing premise

The generalized Aubry duality continues to hold after the phase parametrizing the synthetic dimension is moved off the real axis, allowing the resulting non-unitary model to be classified as pseudo-unitary and to support the claimed spectral and localization properties.

What would settle it

Direct numerical diagonalization of the Floquet operator for non-reciprocal parameters beyond the predicted critical value, checking whether eigenvalues leave the unit circle or whether inverse participation ratios jump at the mobility edge location.

Figures

Figures reproduced from arXiv: 2411.16843 by Christopher Cedzich, Jake Fillman.

Figure 2
Figure 2. Figure 2: FIG. 2. The spectral phase diagram of the unitary almost [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Phase diagrams of the PUAMO in ( [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: , and the states transition from extended to local￾ized. Similarly, increasing η through the transition point Lλ1,λ2,0,ε/(2π) produces a transition from localization to delocalization in which the non-reciprocal hopping over￾powers the localization effects of the coin. These phase transitions may be absent: for example, in the subcritical case λ1 > λ2 for 0 ≤ |ε| ≤ L ♯ λ1,λ2,0,ε/(2π), delocalization holds … view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. The spectrum of the walk [PITH_FULL_IMAGE:figures/full_fig_p005_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: This transition is topological and measured by a spectral winding number. To define this invariant, fix η = 0 and let z ∈ ∂D lie in a gap of the UAMO which is a Cantor set [25, 30]. Consider the continued fraction expansion of Φ with convergents Φk = nk/mk. The corresponding PUAMO is N := mk-periodic, and we denote the resulting matrix with periodic boundary conditions by WN = WN (ϑ). We show in Appendix D… view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Illustrations of graphs of the Lyapunov exponents [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
read the original abstract

We introduce a Floquet quasicrystal that simulates the motion of Bloch electrons in a homogeneous magnetic field in discrete time steps. We admit the hopping to be non-reciprocal which, via a generalized Aubry duality, leads us to push the phase that parametrizes the synthetic dimension off of the real axis. This breaks unitarity, but we show that the model is still ``pseudo-unitary''. We unveil a novel mobility edge between a metallic and an insulating phase that sharply divides the parameter space. Moreover, for the first time, we observe a second transition that appears to be unique to the discrete-time setting. We quantify both phase transitions and relate them to properties of the spectrum. If the hopping is reciprocal either in the lattice direction or the synthetic dimension, the model is $\mathcal{PT}$-symmetric, and the spectrum is confined to the unit circle up to a critical point. At this critical point, $\mathcal{PT}$-symmetry is spontaneously broken and the spectrum leaves the unit circle. This transition is topological and measured by a spectral winding number.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a Floquet quasicrystal that simulates Bloch electrons in a homogeneous magnetic field via discrete-time quantum walks. Non-reciprocal hopping is incorporated, and a generalized Aubry duality is used to shift the synthetic-dimension phase off the real axis, yielding a pseudo-unitary (non-unitary) model. The central results are a novel mobility edge sharply separating metallic and insulating phases, plus a second transition unique to the discrete-time setting; both are quantified and related to spectral properties. When hopping is reciprocal in either direction the model is PT-symmetric, with the spectrum confined to the unit circle up to a critical point at which PT symmetry breaks spontaneously; this transition is topological and diagnosed by a spectral winding number.

Significance. If the duality extension and pseudo-unitarity are rigorously established, the work would be significant for extending mobility-edge and localization studies to non-unitary Floquet systems and for identifying discrete-time-specific transitions. The topological characterization via winding number and the explicit construction of a pseudo-unitary quasiperiodic walk constitute concrete advances that could guide both theory and quantum-simulation experiments.

major comments (2)
  1. [Model construction] Model-construction paragraph (abstract and §2): the mobility-edge claim and the phase diagram rest on the generalized Aubry duality continuing to map spectrum and localization properties after the synthetic phase is taken complex. No explicit verification or derivation is supplied showing that the duality operator remains well-defined and that the resulting operator is pseudo-unitary without extra assumptions; this is load-bearing for both the metallic/insulating division and the second transition.
  2. [PT-symmetry section] PT-symmetry and spectral section (abstract and §4): the statement that the PT-breaking point is topological and measured by a spectral winding number requires an explicit relation between the winding-number jump and the mobility edge. Without a theorem or numerical protocol linking the two quantities at the same parameter value, the topological diagnosis of the second transition remains unsupported.
minor comments (2)
  1. [Introduction] Define 'pseudo-unitary' with a precise operator equation at first use rather than relying on the duality construction alone.
  2. [Numerical results] Phase diagrams should include error bars or convergence checks on the mobility-edge location obtained from finite-size scaling.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and constructive comments. We address the major comments point by point below.

read point-by-point responses

  1. Referee: [Model construction] Model-construction paragraph (abstract and §2): the mobility-edge claim and the phase diagram rest on the generalized Aubry duality continuing to map spectrum and localization properties after the synthetic phase is taken complex. No explicit verification or derivation is supplied showing that the duality operator remains well-defined and that the resulting operator is pseudo-unitary without extra assumptions; this is load-bearing for both the metallic/insulating division and the second transition.

    Authors: We agree that the derivation of the generalized Aubry duality for complex phases and the resulting pseudo-unitarity could be presented more explicitly. Although §2 introduces the duality and states that the model is pseudo-unitary, a step-by-step verification that the operator remains well-defined and satisfies the pseudo-unitary condition without extra assumptions is not detailed. In the revised manuscript we will add this explicit derivation in §2 to rigorously support the mobility edge, phase diagram, and second transition. revision: yes

  2. Referee: [PT-symmetry section] PT-symmetry and spectral section (abstract and §4): the statement that the PT-breaking point is topological and measured by a spectral winding number requires an explicit relation between the winding-number jump and the mobility edge. Without a theorem or numerical protocol linking the two quantities at the same parameter value, the topological diagnosis of the second transition remains unsupported.

    Authors: We appreciate the referee highlighting the need for an explicit link. Section §4 contains numerical results indicating that the winding-number jump and mobility edge occur at the same parameter values, but no dedicated protocol or theorem connecting them is stated. In the revision we will add a subsection in §4 that outlines the numerical protocol for the winding number and explicitly shows its coincidence with the mobility edge, thereby supporting the topological diagnosis. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation self-contained via model construction and duality extension

full rationale

The paper introduces a non-reciprocal hopping model, invokes generalized Aubry duality to move the synthetic phase off the real axis, explicitly shows the resulting model remains pseudo-unitary, and derives the mobility edge and PT-symmetry breaking transition from the resulting spectrum and localization properties. None of these steps reduce by construction to their inputs (no self-definitional reparameterization, no fitted quantity renamed as prediction, no load-bearing self-citation chain). The abstract and model description frame the duality extension and pseudo-unitarity as shown results rather than assumed definitions, making the phase diagram and transitions independent consequences of the construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claims rest on the applicability of generalized Aubry duality to the non-reciprocal case and on the definition of pseudo-unitarity preserving enough structure for spectral analysis; both are invoked without independent verification in the abstract.

axioms (1)
  • domain assumption Generalized Aubry duality extends to non-reciprocal hopping and permits shifting the synthetic-dimension phase off the real axis while preserving pseudo-unitarity
    Invoked to construct the model and justify the pseudo-unitary property after breaking unitarity.
invented entities (1)
  • pseudo-unitary quantum walk no independent evidence
    purpose: To retain analyzable spectral properties after non-reciprocal hopping moves the phase off the real axis
    Introduced as the key property enabling the mobility-edge and transition analysis

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