arXiv subjects

Gabriele Peluso

Publications and source records attributed to Gabriele Peluso.

5 recordsLinked to original sources

arXiv2026

The Zak phase in topologically insulating chains: invariants and limitations

In this work we investigate the topological content of the Zak phase in one-dimensional translation-invariant topological insulators endowed with time-reversal, particle-hole and/or chiral symmetries, extending results from \cite{Monaco_2023}. We analyze the extent to which the Zak phase captures the topology of all Altland--Zirnbauer--Cartan (AZC) symmetry classes in $1$D. Building on the framework of fibered Hamiltonians and spectral projections, we construct symmetric Bloch bases adapted to the discrete symmetries of the system and define a $\mathbb{Z}_2$-valued topological invariant $\mathrm{I}^{(\mathrm{AZC-class})}(H)$ obtained from the abelian Zak phase. Moreover, we demonstrate that in symmetry classes admitting a quaternionic structure, i.e. anti-unitary symmetries squaring to minus the identity, the Zak phase is further constrained, leading to the vanishing of the $\mathbb{Z}_2$ invariant mentioned above. This highlights the sensitivity of the Zak phase to additional geometric structures of the manifold of occupied energy states, as well as its limitations in being an effective marker for topological phases of insulating chains. As an example, we discuss the case of generalized Kitaev chains with arbitrary finite-range hopping and single or multiple chiral channels, and show how the Zak phase only retains partial information about their different topological phases.

math-ph↗

arXiv2026

Zak phase and bulk-boundary correspondence in a generalized Dirac-Kronig-Penney model

We investigate the topological properties of a generalized Dirac--Kronig--Penney model, a continuum one-dimensional model for a relativistic quantum chain. By tuning the coupling parameters this model can accommodate five Altland--Zirnbauer--Cartan symmetry classes, three of which (AIII, BDI and D) support non-trivial topological phases in dimension one. We characterize analytically the spectral properties of the Hamiltonian in terms of a spectral function, and numerically compute the Zak phase to probe the bulk topological content of the insulating phases. Our findings reveal that, while the Zak phase is quantized in classes AIII and BDI, it exhibits non-quantized values in class D, challenging its traditional role as a topological marker in continuum settings. We also discuss the bulk-boundary correspondence for a truncated version of the chain, analyzing how the emergence of edge states depends on both the truncation position and the boundary conditions. In classes AIII and BDI, we find that the Zak phase effectively detects edge states as a relative boundary topological index, although the correspondence is highly sensitive to the parameters characterizing the truncation.

quant-ph↗

arXiv2025

Splitting obstructions and $\mathbb{Z}_2$ invariants in time-reversal symmetric topological insulators

The Fu-Kane-Mele $\mathbb{Z}_2$ index characterizes two-dimensional time-reversal symmetric topological phases of matter. We shed some light on some features of this index by investigating projection-valued maps endowed with a fermionic time-reversal symmetry. Our main contributions are threefold. First, we establish a decomposition theorem, proving that any such projection-valued map admits a splitting into two projection-valued maps that are related to each other via time-reversal symmetry. Second, we provide a complete homotopy classification theorem for these maps, thereby clarifying their topological structure. Third, by means of the previous analysis, we connect the Fu-Kane-Mele index to the Chern number of one of the factors in the previously-mentioned decomposition, which in turn allows to exhibit how the $\mathbb{Z}_2$-valued topological obstruction to constructing a periodic and smooth Bloch frame for the projection-valued map, measured by the Fu-Kane-Mele index, can be concentrated in a single pseudo-periodic Kramers pair.

math-ph↗

arXiv2025

Moduli space of spin connections on three-dimensional homogeneous spaces

In this manuscript, we aim to classify and characterize the moduli space of homogeneous spin connections and homogeneous SU(2) connections on three-dimensional Riemannian homogeneous spaces. An analysis of the topology of the associated moduli spaces reveals that they are finite-dimensional topological manifolds (possibly with boundary) possessing trivial homotopy groups. Owing to their deep connection with cosmological models in the Ashtekar-Barbero-Immirzi formulation of General Relativity, this study offers a mathematically rigorous interpretation of the Ashtekar-Barbero-Immirzi-Sen connection within a cosmological context. In particular, we show that a correct formulation of the theory relies crucially on identifying the moduli space of homogeneous spin connections, thereby emphasizing the essential role of the spin structure in ensuring consistency with the physical content of the theory. The favorable topological properties of these moduli spaces circumvent many of the usual difficulties associated with singularities and the definition of regular measures in Quantum Field Theory and Quantum Gravity. As a result, they provide a solid foundation for the rigorous implementation of quantum theory in the cosmological setting.

math-ph↗

arXiv2023

A $\mathbb{Z}_2$ invariant for chiral and particle-hole symmetric topological chains

We define a $\mathbb{Z}_2$-valued topological and gauge invariant associated to any 1-dimensional, translation-invariant topological insulator which satisfies either particle-hole symmetry or chiral symmetry. The invariant can be computed from the Berry phase associated to a suitable basis of Bloch functions which is compatible with the symmetries. We compute the invariant in the Su-Schrieffer-Heeger model for chiral symmetric insulators, and in the Kitaev model for particle-hole symmetric insulators. We show that in both cases the $\mathbb{Z}_2$ invariant predicts the existence of zero-energy boundary states for the corresponding truncated models.

math-ph↗