The Zak phase provides a fundamental geometric characterization of one-dimensional band structures and, when quantized by symmetry, serves as a bulk indicator of polarization and boundary phenomena. Conventionally, its exact quantization is tied to a global symmetry of the Bloch Hamiltonian, such as inversion or reflection symmetry. Here, we demonstrate a distinct mechanism by which the Zak phase remains exactly quantized even when the corresponding protecting symmetry is explicitly broken at the level of the full Hamiltonian. The mechanism is based on local support symmetry and destructive interference: symmetry-breaking degrees of freedom are coupled to a topological parent system in such a way that the relevant Bloch states cannot leak into them. The preserved states therefore retain their parent wave functions and inherit their quantized Zak phase exactly, despite the absence of the protecting global symmetry. We establish this mechanism for general one-dimensional insulators and demonstrate it in an inversion-broken minimal model and a reflection-broken armchair nanoribbon, where a quantized $π$ Zak phase and associated boundary-localized states persist. We further show that perturbations violating the exact destructive-interference condition continuously dequantize the Zak phase, while a finite regime with a near- $π$ Zak phase and boundary response can survive. Our results establish wave-function support, rather than global symmetry alone, as a mechanism for protecting quantized quantum geometry in one-dimensional systems.The Zak phase provides a fundamental geometric characterization of one-dimensional band structures and, when quantized by symmetry, serves as a bulk indicator of polarization and boundary phenomena. Conventionally, its exact quantization is tied to a global symmetry of the Bloch Hamiltonian, such as inversion or reflection symmetry. Here, we demonstrate a distinct mechanism by which the Zak phase remains exactly quantized even when the corresponding protecting symmetry is explicitly broken at the level of the full Hamiltonian. The mechanism is based on local support symmetry and destructive interference: symmetry-breaking degrees of freedom are coupled to a topological parent system in such a way that the relevant Bloch states cannot leak into them. The preserved states therefore retain their parent wave functions and inherit their quantized Zak phase exactly, despite the absence of the protecting global symmetry. We establish this mechanism for general one-dimensional insulators and demonstrate it in an inversion-broken minimal model and a reflection-broken armchair nanoribbon, where a quantized $π$ Zak phase and associated boundary-localized states persist. We further show that perturbations violating the exact destructive-interference condition continuously dequantize the Zak phase, while a finite regime with a near-$π$ Zak phase and boundary response can survive. Our results establish wave-function support, rather than global symmetry alone, as a mechanism for protecting quantized quantum geometry in one-dimensional systems.

quant-pharXiv:2610.13250Oct 12, 2026

Bloch surface wave-atom coupling in one-dimensional photonic crystal structure

Mahnaz Asadolah Salmanpour, Mohammad Mosleh, Seyedeh Mehri Hamidi

Considering efforts for hot atomic vapor-nanophotonics integration as a new paradigm in quantum optics, in this paper, we introduce 1D photonic crystal-Rb vapor cell as structure with miniaturized interaction volume. The Bloch surface wave (BSW) excited on surface of a photonic crystal as electromagnetic hosting photonic mode, and altered the optical response of Rb atoms in the vicinity of surface. Coupling of atomic states with BSW confined modes would lead to quantum interference effects and results in nonlinearities in resonant coupling of atoms with BSW. We show Bloch surface wave induced transparency is highly stable under a change of incidence angle. Our results show slight changes in transitions detuning's due to nonlinear interactions like the Casimire-Polder effect under change of localized density of optical states.Considering efforts for hot atomic vapor-nanophotonics integration as a new paradigm in quantum optics, in this paper, we introduce 1D photonic crystal-Rb vapor cell as structure with miniaturized interaction volume. The Bloch surface wave (BSW) excited on surface of a photonic crystal as electromagnetic hosting photonic mode, and altered the optical response of Rb atoms in the vicinity of surface. Coupling of atomic states with BSW confined modes would lead to quantum interference effects and results in nonlinearities in resonant coupling of atoms with BSW. We show Bloch surface wave induced transparency is highly stable under a change of incidence angle. Our results show slight changes in transitions detuning's due to nonlinear interactions like the Casimire-Polder effect under change of localized density of optical states.

arXiv:2210.12356Optics ExpressJan 12, 20238 citations

On the solution of the Bloch equations

E. Arimondo, Davide Bertolini

Lettere al nuovo cimento della societa italiana di fisica/Lettere al nuovo cimentoAug 1, 19720 citations

π

Zak phase and associated boundary-localized states persist. We further show that perturbations violating the exact destructive-interference condition continuously dequantize the Zak phase, while a finite regime with a near-

π

Zak phase and boundary response can survive. Our results establish wave-function support, rather than global symmetry alone, as a mechanism for protecting quantized quantum geometry in one-dimensional systems.