Topological Properties and Classification of Quantum Systems with Discrete Symmetries

Discrete quantum walks are periodically driven systems with discrete time evolution. In contrast to ordinary Floquet systems, no microscopic Hamiltonian exists, and the one-period time evolution is given directly by a series of unitary operators. Regarding each constituent unitary operator as a discrete time step, we formulate discrete space-time symmetry in quantum walks and evaluate the corresponding symmetry protected topological phases. In particular, we study chiral and/or time-glide symmetric topological quantum walks in this formalism. Due to the discrete nature of time evolution, the topological classification is found to be different from that in conventional Floquet systems. As a concrete example, we study a two-dimensional quantum walk having both chiral and time-glide symmetries and identify the anomalous edge states protected by these symmetries.Discrete quantum walks are periodically driven systems with discrete time evolution. In contrast to ordinary Floquet systems, no microscopic Hamiltonian exists, and the one-period time evolution is given directly by a series of unitary operators. Regarding each constituent unitary operator as a discrete time step, we formulate discrete space-time symmetry in quantum walks and evaluate the corresponding symmetry protected topological phases. In particular, we study chiral and/or time-glide symmetric topological quantum walks in this formalism. Due to the discrete nature of time evolution, the topological classification is found to be different from that in conventional Floquet systems. As a concrete example, we study a two-dimensional quantum walk having both chiral and time-glide symmetries and identify the anomalous edge states protected by these symmetries.

arXiv:2004.09332Physical review. B./Physical review. BJul 13, 202016 citations

Classification of topological states with symmetries

Antônio Sérgio Teixeira Pires

Institute of Physics eBooksNov 1, 20210 citations

Classification of two-dimensional quantum systems with Z3-graded topological symmetries

D. Pouladsaz

Dec 1, 20120 citations

Classification of Interacting Topological Crystalline Superconductors in Three Dimensions and Beyond

Shang-Qiang Ning, Xing-Yu Ren, Qing-Rui Wang +2 more

Although classification for free-fermion topological superconductors (TSC) is established, systematically understanding the classification of 3D interacting TSCs remains difficult, especially those protected by crystalline symmetries like the 230 space groups. We build up a general framework for systematically classifying 3D interacting TSCs protected by crystalline symmetries together with discrete internal symmetries. We first establish a complete classification for fermionic symmetry protected topological phases (FSPT) with purely discrete internal symmetries, which determines the crystalline case via the crystalline equivalence principle. Using domain wall decoration, we obtain classification data and formulas for generic FSPTs, what are suitable for systematic computation. The four layers of decoration data $(n_1, n_2, n_3, ν_4)$ characterize a 3D FSPT with symmetry $G_b\times_{ω_2}Z_2^f$ , corresponding to $p+ip$ , Kitaev chain, complex fermion, and bosonic SPT layers. Inspired by previous works, a crucial aspect is the $p+ip$ layer, where classification involves two possibilities: anti-unitary and infinite-order symmetries (e.g., translation). We show the former maps to some mirror FSPT classification with the mirror plane decorated by a $p+ip$ superconductor, while the latter is determined by the free part of $H^1(G_b, Z_T)$ , corresponding to weak TSCs. Another key point is the Kitaev chain decoration for the anti-unitary symmetries, which differs essentially from unitary ones. We explicitly obtain formulas for all three layers of decoration $(n_2, n_3, ν_4)$ , which are amenable to automatic computation. As an application, we classify the 230 space-group topological crystalline superconductors in interacting electronic systems.Although classification for free-fermion topological superconductors (TSC) is established, systematically understanding the classification of 3D interacting TSCs remains difficult, especially those protected by crystalline symmetries like the 230 space groups. We build up a general framework for systematically classifying 3D interacting TSCs protected by crystalline symmetries together with discrete internal symmetries. We first establish a complete classification for fermionic symmetry protected topological phases (FSPT) with purely discrete internal symmetries, which determines the crystalline case via the crystalline equivalence principle. Using domain wall decoration, we obtain classification data and formulas for generic FSPTs, what are suitable for systematic computation. The four layers of decoration data $(n_1, n_2, n_3, ν_4)$ characterize a 3D FSPT with symmetry $G_b\times_{ω_2}Z_2^f$, corresponding to $p+ip$, Kitaev chain, complex fermion, and bosonic SPT layers. Inspired by previous works, a crucial aspect is the $p+ip$ layer, where classification involves two possibilities: anti-unitary and infinite-order symmetries (e.g., translation). We show the former maps to some mirror FSPT classification with the mirror plane decorated by a $p+ip$ superconductor, while the latter is determined by the free part of $H^1(G_b, Z_T)$, corresponding to weak TSCs. Another key point is the Kitaev chain decoration for the anti-unitary symmetries, which differs essentially from unitary ones. We explicitly obtain formulas for all three layers of decoration $(n_2, n_3, ν_4)$, which are amenable to automatic computation. As an application, we classify the 230 space-group topological crystalline superconductors in interacting electronic systems.

arXiv:2512.25069Dec 31, 20250 citations

Topology of Discrete Quantum Feedback Control

Masaya Nakagawa, Masahito Ueda

A general framework for analyzing the topology of quantum channels of single-particle systems is developed to find a class of genuinely dynamical topological phases that can be realized by means of discrete quantum feedback control. We provide a symmetry classification of quantum channels by identifying ten symmetry classes of discrete quantum feedback control with projective measurements. We construct various types of topological feedback control by using topological Maxwell demons that achieve robust feedback-controlled chiral or helical transport against noise and decoherence. Topological feedback control thus offers a versatile tool for creating and controlling nonequilibrium topological phases in open quantum systems that are distinct from non-Hermitian and Lindbladian systems and should provide a guiding principle for topology-based design of quantum feedback control.A general framework for analyzing the topology of quantum channels of single-particle systems is developed to find a class of genuinely dynamical topological phases that can be realized by means of discrete quantum feedback control. We provide a symmetry classification of quantum channels by identifying ten symmetry classes of discrete quantum feedback control with projective measurements. We construct various types of topological feedback control by using topological Maxwell demons that achieve robust feedback-controlled chiral or helical transport against noise and decoherence. Topological feedback control thus offers a versatile tool for creating and controlling nonequilibrium topological phases in open quantum systems that are distinct from non-Hermitian and Lindbladian systems and should provide a guiding principle for topology-based design of quantum feedback control.

arXiv:2403.08406Physical Review XApr 15, 20255 citations

Discrete symmetries and supersymmetries of quantum-mechanical systems

Jiri Niederle

AIP conference proceedingsJan 1, 19980 citations

(n_1, n_2, n_3, ν_4)

characterize a 3D FSPT with symmetry

G_b\times_{ω_2}Z_2^f

, corresponding to

p+ip

, Kitaev chain, complex fermion, and bosonic SPT layers. Inspired by previous works, a crucial aspect is the

p+ip

layer, where classification involves two possibilities: anti-unitary and infinite-order symmetries (e.g., translation). We show the former maps to some mirror FSPT classification with the mirror plane decorated by a

p+ip

superconductor, while the latter is determined by the free part of

H^1(G_b, Z_T)

, corresponding to weak TSCs. Another key point is the Kitaev chain decoration for the anti-unitary symmetries, which differs essentially from unitary ones. We explicitly obtain formulas for all three layers of decoration

(n_2, n_3, ν_4)

, which are amenable to automatic computation. As an application, we classify the 230 space-group topological crystalline superconductors in interacting electronic systems.