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arXiv:2606.21499

Localization pattern of a mobile impurity in the disordered Kitaev chain

А. V. Sadovnikov, A. N. Rubtsov

Published Jun 19, 2026• 0 citations
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Abstract

We study a mobile impurity coupled to a Kitaev chain with chemical-potential disorder and ask whether the impurity behavior distinguishes different regimes of the host system. Exact diagonalization calculations for small periodic chains shows that in the deep topological regime the impurity localizes only partially, with a smooth increase of $\mathrm{IPR}_d$, whereas in the deep trivial regime it undergoes a much sharp transition to nearly single-site localization. For open chains at strong interaction, DMRG shows edge-localized impurity density near the Kitaev sweet spot. With increasing chemical potential, the impurity weight spreads into the bulk and eventually becomes almost uniform. We explain the edge preference analytically from the Majorana-dimer structure: a bulk impurity rearranges two neighboring dimers, while an edge impurity affects only one. Disorder competes with this clean edge bias and can pin the impurity in the bulk. Thus, the impurity is sensitive to the regime of the host system, although we do not find a strict one-to-one correspondence between the impurity localization pattern and the host topology. Instead, the disorder-averaged behavior suggests only an indirect correlation between impurity localization and the underlying phase of the chain.

DOI
10.48550/arxiv.2606.21499

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We study a mobile impurity coupled to a Kitaev chain with chemical-potential disorder and ask whether the impurity behavior distinguishes different regimes of the host system. Exact diagonalization calculations for small periodic chains shows that in the deep topological regime the impurity localizes only partially, with a smooth increase of IPRd\mathrm{IPR}_dIPRd​, whereas in the deep trivial regime it undergoes a much sharp transition to nearly single-site localization. For open chains at strong interaction, DMRG shows edge-localized impurity density near the Kitaev sweet spot. With increasing chemical potential, the impurity weight spreads into the bulk and eventually becomes almost uniform. We explain the edge preference analytically from the Majorana-dimer structure: a bulk impurity rearranges two neighboring dimers, while an edge impurity affects only one. Disorder competes with this clean edge bias and can pin the impurity in the bulk. Thus, the impurity is sensitive to the regime of the host system, although we do not find a strict one-to-one correspondence between the impurity localization pattern and the host topology. Instead, the disorder-averaged behavior suggests only an indirect correlation between impurity localization and the underlying phase of the chain.
We study the physics of a mobile impurity immersed in a 1d1d1d topological superconductor. We discuss the system's phase diagram obtained with exact diagonalization. We argue that the character of the transition from a weak to strong coupling regime depends on the phase of the host superconductor. A smooth crossover between a weakly coupled polaron and a molecular state is observed in the topological phase. In contrast, the impurity undergoes a sharp phase transition in a topologically trivial background.
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To clarify the relationship between edge electronic states in open-boundary crystalline systems and their corresponding bulk electronic structure, A. Alase et al., [Phys. Rev. Lett. 117, 076804 (2016)] have recently generalized Bloch's theorem to lattice models with broken translational symmetry. Their formalism provides a bulk-boundary correspondence indicator, DDD, sensitive directly to the localization of edge states. We extend this formalism in two significant ways. First, we explicitly classify the edge-state basis provided by the theory according to their underlying protecting symmetries. Second, acknowledging that the true topological invariant is the Zak phase, inherently defined only for periodic boundary conditions, we introduce an analogous quantity ${\ensuremath{\gamma}}_{Z}$, suitable for open-boundary systems. We illustrate these developments on the example of the Kitaev wire model. We demonstrate that both indicators, DDD and our open-boundary analog, ${\ensuremath{\gamma}}_{Z}$, effectively capture topological localization: while DDD diverges within the topological phase, ${\ensuremath{\gamma}}_{Z}=\ensuremath{\pi}$. We further examine eigenvalues (zzz) of the lattice shift operator, showing that variations in these eigenvalues as a function of chemical potential provide additional signatures of topological phase transitions. Additionally, we study a periodic Kitaev wire containing a bond impurity, revealing that while the topological phase remains characterized by ${\ensuremath{\gamma}}_{Z}=\ensuremath{\pi}$, significant state localization near the impurity emerges only at critical values of the chemical potential.