Twisting symmetries provides an efficient method to diagnose symmetry-protected topological (SPT) phases. In this paper, edge theories of (2+1)-dimensional topological phases protected by reflection as well as other symmetries are studied by twisting reflection symmetry, which effectively puts the edge theories on an unoriented space-time, such as the Klein bottle. A key technical step taken in this paper is the use of the so-called cross-cap states, which encode entirely the unoriented nature of space-time, and can be obtained by rearranging the space-time geometry and exchanging the role of space and time coordinates. When the system is in a nontrivial SPT phase, we find that the corresponding cross-cap state is noninvariant under the action of the symmetries of the SPT phase, but acquires an anomalous phase. This anomalous phase, with a proper definition of a reference state, on which symmetry acts trivially, reproduces the known classification of (2+1)-dimensional bosonic and fermionic SPT phases protected by reflection symmetry, including in particular the ${\mathbb{Z}}_{8}$ classification of topological crystalline superconductors protected by reflection and time-reversal symmetries.Twisting symmetries provides an efficient method to diagnose symmetry-protected topological (SPT) phases. In this paper, edge theories of (2+1)-dimensional topological phases protected by reflection as well as other symmetries are studied by twisting reflection symmetry, which effectively puts the edge theories on an unoriented space-time, such as the Klein bottle. A key technical step taken in this paper is the use of the so-called cross-cap states, which encode entirely the unoriented nature of space-time, and can be obtained by rearranging the space-time geometry and exchanging the role of space and time coordinates. When the system is in a nontrivial SPT phase, we find that the corresponding cross-cap state is noninvariant under the action of the symmetries of the SPT phase, but acquires an anomalous phase. This anomalous phase, with a proper definition of a reference state, on which symmetry acts trivially, reproduces the known classification of (2+1)-dimensional bosonic and fermionic SPT phases protected by reflection symmetry, including in particular the ${\mathbb{Z}}_{8}$ classification of topological crystalline superconductors protected by reflection and time-reversal symmetries.

arXiv:1501.07285Physical Review BMay 26, 201546 citations

Topological phases in one dimension

Antônio Sérgio Teixeira Pires

Institute of Physics eBooksNov 1, 20210 citations

Operator geometry and an index formula for Floquet topological phases

Chen Zeqian, Sun Mu, Zhi Yin

For giving a geometric interpretation of the observable-geometric phase with application to quantum computation, operator geometry on a finite-dimensional complex Hilbert space was developed by Chen (2020), i.e., geometry over the operator-principal G-bundle of the unitary operator group based on the observable space. In this paper, we further develop operator geometry on an infinite-dimensional separable complex Hilbert space. To this end, we use the theory of unbounded operators for describing the differential structure of infinite-dimensional observable spaces and defining quantum connections and quantum parallel transportation. Then, we introduce the notion of the observable-geometric phase for infinite-dimensional quantum systems and give its geometric interpretation in terms of operator geometry. Finally, we present a topological index for Floquet topological phases and prove a local formula for it in terms of the observable-geometric phases, which are measurable in the physical sense.For giving a geometric interpretation of the observable-geometric phase with application to quantum computation, operator geometry on a finite-dimensional complex Hilbert space was developed by Chen (2020), i.e., geometry over the operator-principal G-bundle of the unitary operator group based on the observable space. In this paper, we further develop operator geometry on an infinite-dimensional separable complex Hilbert space. To this end, we use the theory of unbounded operators for describing the differential structure of infinite-dimensional observable spaces and defining quantum connections and quantum parallel transportation. Then, we introduce the notion of the observable-geometric phase for infinite-dimensional quantum systems and give its geometric interpretation in terms of operator geometry. Finally, we present a topological index for Floquet topological phases and prove a local formula for it in terms of the observable-geometric phases, which are measurable in the physical sense.

Scientia Sinica MathematicaDec 21, 20220 citations

Geometry and quantum brachistochrone analysis of multiple entangled spin-1/2 particles under all-range Ising interaction

B. Amghar, M. Yachi, M. Amghar +3 more

We present a unified geometric and dynamical framework for a physical system consisting of $n$ spin- $1/2$ particles with all-range Ising interaction. Using the Fubini-Study formalism, we derive the metric tensor of the associated quantum state manifold and compute the corresponding Riemann curvature. Our analysis reveals that the system evolves over a smooth, compact, two-dimensional manifold with spherical topology and a dumbbell-like structure shaped by collective spin interactions. We further investigate the influence of the geometry and topology of the resulting state space on the behavior of geometric and topological phases acquired by the system. We explore how this curvature constrains the system's dynamical behavior, including its evolution speed and Fubini-Study distance between the quantum states. Within this geometric framework, we address the quantum brachistochrone problem and derive the minimal time required for optimal evolution, a result useful for time-efficient quantum circuit design. Subsequently, we explore the role of entanglement in shaping the state space geometry, modulating geometric phase, and controlling evolution speed and brachistochrone time. Our results reveal that entanglement enhances dynamics up to a critical threshold, beyond which geometric constraints begin to hinder evolution. Moreover, entanglement induces critical shifts in the geometric phase, making it a sensitive indicator of entanglement levels and a practical tool for steering quantum evolution. This approach offers valuable guidance for developing quantum technologies that require time-efficient control strategies rooted in the geometry of quantum state space.We present a unified geometric and dynamical framework for a physical system consisting of $n$ spin-$1/2$ particles with all-range Ising interaction. Using the Fubini-Study formalism, we derive the metric tensor of the associated quantum state manifold and compute the corresponding Riemann curvature. Our analysis reveals that the system evolves over a smooth, compact, two-dimensional manifold with spherical topology and a dumbbell-like structure shaped by collective spin interactions. We further investigate the influence of the geometry and topology of the resulting state space on the behavior of geometric and topological phases acquired by the system. We explore how this curvature constrains the system's dynamical behavior, including its evolution speed and Fubini-Study distance between the quantum states. Within this geometric framework, we address the quantum brachistochrone problem and derive the minimal time required for optimal evolution, a result useful for time-efficient quantum circuit design. Subsequently, we explore the role of entanglement in shaping the state space geometry, modulating geometric phase, and controlling evolution speed and brachistochrone time. Our results reveal that entanglement enhances dynamics up to a critical threshold, beyond which geometric constraints begin to hinder evolution. Moreover, entanglement induces critical shifts in the geometric phase, making it a sensitive indicator of entanglement levels and a practical tool for steering quantum evolution. This approach offers valuable guidance for developing quantum technologies that require time-efficient control strategies rooted in the geometry of quantum state space.

quant-pharXiv:2512.21400Scientific ReportsDec 26, 20250 citations

Topological phases in two-dimensional systems

Emmanuel V. C. Lopes

Jul 18, 20250 citations

{\mathbb{Z}}_{8}

classification of topological crystalline superconductors protected by reflection and time-reversal symmetries.

We present a unified geometric and dynamical framework for a physical system consisting of

n

spin-

1/2

particles with all-range Ising interaction. Using the Fubini-Study formalism, we derive the metric tensor of the associated quantum state manifold and compute the corresponding Riemann curvature. Our analysis reveals that the system evolves over a smooth, compact, two-dimensional manifold with spherical topology and a dumbbell-like structure shaped by collective spin interactions. We further investigate the influence of the geometry and topology of the resulting state space on the behavior of geometric and topological phases acquired by the system. We explore how this curvature constrains the system's dynamical behavior, including its evolution speed and Fubini-Study distance between the quantum states. Within this geometric framework, we address the quantum brachistochrone problem and derive the minimal time required for optimal evolution, a result useful for time-efficient quantum circuit design. Subsequently, we explore the role of entanglement in shaping the state space geometry, modulating geometric phase, and controlling evolution speed and brachistochrone time. Our results reveal that entanglement enhances dynamics up to a critical threshold, beyond which geometric constraints begin to hinder evolution. Moreover, entanglement induces critical shifts in the geometric phase, making it a sensitive indicator of entanglement levels and a practical tool for steering quantum evolution. This approach offers valuable guidance for developing quantum technologies that require time-efficient control strategies rooted in the geometry of quantum state space.