Symmetry and quantization

H. D. Doebner, J. Tolar

A. Hr. Anastassov

Geonwoo Hwang, Jun-Won Rhim

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.

A. Hr. Anastassov

Ling-Yan Hung, Xiao-Gang Wen

We study the quantized topological terms in a weak-coupling gauge theory with gauge group ${G}_{g}$ and a global symmetry ${G}_{s}$ in $d$ space-time dimensions. We show that the quantized topological terms are classified by a pair $(G,{\ensuremath{\nu}}_{d})$, where $G$ is an extension of ${G}_{s}$ by ${G}_{g}$ and ${\ensuremath{\nu}}_{d}$ an element in group cohomology ${\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})$. When $d=3$ and/or when ${G}_{g}$ is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (i.e., gapped long-range-entangled phases with symmetry). Thus, those SET phases are classified by ${\mathcal{H}}^{d}(G,\mathbb{R}/\mathbb{Z})$, where $G/{G}_{g}={G}_{s}$. We also apply our theory to a simple case ${G}_{s}={G}_{g}={Z}_{2}$, which leads to 12 different SET phases in $2+1$ dimensions [(2+1)D], where quasiparticles have different patterns of fractional ${G}_{s}={Z}_{2}$ quantum numbers and fractional statistics. If the weak-coupling gauge theories are gapless, then the different quantized topological terms may describe different gapless phases of the gauge theories with a symmetry ${G}_{s}$, which may lead to different fractionalizations of ${G}_{s}$ quantum numbers and different fractional statistics [if in ($2+1$)D].

Maria Clara Nucci

A procedure which obviates the constraint imposed by the conflict between consistent quantization and the invariance of the Hamiltonian description under nonlinear canonical transformation is proposed. This new quantization scheme preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrödinger's equation. As an example, the quantization of the `goldfish' many-body problem extensively studied by Calogero et al. is presented.