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arXiv:2601.20081

Spectral Transitions and Singular Continuous Spectrum in A New Family of Quasi-periodic Quantum Walks

Xinyu Yang, Long Li, Qi Zhou

quant-phmath.SPSubmitted Jan 27, 2026• Announced Jan 29, 2026
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Abstract

This paper introduces and rigorously analyzes a new class of one-dimensional discrete-time quantum walks whose dynamics are governed by a parametrized family of extended CMV matrices. The model generalizes the unitary almost Mathieu operator (UAMO) and exhibits a richer spectral phase diagram, closely resembling the extended Harper's model. It provides the first example of a solvable quasi-periodic quantum walk that exhibits a stable region of purely singular continuous spectrum.

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29 pages, 1 figure

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We define coined Quantum Walks on the infinite rooted binary tree given by unitary operators U(C)U(C)U(C) on an associated infinite dimensional Hilbert space, depending on a unitary coin matrix C∈U(3)C\in U(3)C∈U(3), and study their spectral properties. For circulant unitary coin matrices CCC, we derive an equation for the Carathéodory function associated to the spectral measure of a cyclic vector for U(C)U(C)U(C). This allows us to show that for all circulant unitary coin matrices, the spectrum of the Quantum Walk has no singular continuous component. Furthermore, for coin matrices CCC which are orthogonal circulant matrices, we show that the spectrum of the Quantum Walk is absolutely continuous, except for four coin matrices for which the spectrum of U(C)U(C)U(C) is pure point.
We study the unitary almost Mathieu operator (UAMO), a one-dimensional quasi-periodic unitary operator arising from a two-dimensional discrete-time quantum walk on Z2\mathbb Z^2Z2 in a homogeneous magnetic field. In the positive Lyapunov exponent regime 0≤λ1<λ2≤10\le λ_1<λ_2\le 10≤λ1​<λ2​≤1, we establish an arithmetic localization statement governed by the frequency exponent β(ω)β(ω)β(ω). More precisely, for every irrational ωωω with β(ω)<Lβ(ω)<Lβ(ω)<L, where L>0L>0L>0 denotes the Lyapunov exponent, and every non-resonant phase θθθ, we prove Anderson localization, i.e. pure point spectrum with exponentially decaying eigenfunctions. This extends our previous arithmetic localization result for Diophantine frequencies (for which β(ω)=0β(ω)=0β(ω)=0) to a sharp threshold in frequency.
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Non-Hermitian extensions of the Aubry-André-Harper (AAH) model reveal a rich variety of phase transitions arising from the interplay of quasiperiodicity and nonhermiticity. Despite their theoretical significance, experimental explorations remain challenging due to intricacies in realizing controlled nonhermiticity. Here, we present the first experimental realization of the unitary almost-Mathieu operator that simulates the AAH model by employing single-photon quantum walks. We systematically observe a phase transition between localized and delocalized regimes in the Hermitian limit. Subsequently, by introducing a nonreciprocal hopping, we experimentally probe the parity-time symmetry-breaking transition that is characterized by the emergence of complex quasienergies. Moreover, we identify a novel spectral transition exclusive to discrete-time settings, beyond which the real parts of all quasienergies vanish. These results clarify the interplay between localization, symmetry breaking, and topology in non-Hermitian quasicrystals, thereby paving the way for future exploration of synthetic quantum matter.