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International Journal of Geometric Methods in Modern Physics

Quantum state manifold and geometric, dynamic and topological phases for an interacting two-spin system

Brahim Amghar, Mohammed Daoud A

Published Dec 16, 2019• 8 citations
Publisher

Abstract

We consider a two-spin system of [Formula: see text] Heisenberg type submitted to an external magnetic field. Using the associated [Formula: see text] geometry, we investigate the dynamics of the system. We explicitly give the corresponding Fubini–Study metric. We show that for arbitrary pure initial states, the dynamics occurs on a torus. We compute the geometric phase, the dynamic phase and the topological phase. We investigate the interplay between the torus geometry and the entanglement of the two spins. In this respect, we provide a detailed analysis of the geometric phase, the dynamics velocity and the geodesic distance measured by the Fubini–Study metric in terms of the degree of entanglement between the two spins.

DOI
10.1142/s0219887820500309

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We consider a two-spin system of [Formula: see text] Heisenberg type submitted to an external magnetic field. Using the associated [Formula: see text] geometry, we investigate the dynamics of the system. We explicitly give the corresponding Fubini–Study metric. We show that for arbitrary pure initial states, the dynamics occurs on a torus. We compute the geometric phase, the dynamic phase and the topological phase. We investigate the interplay between the torus geometry and the entanglement of the two spins. In this respect, we provide a detailed analysis of the geometric phase, the dynamics velocity and the geodesic distance measured by the Fubini–Study metric in terms of the degree of entanglement between the two spins.
We consider a physical system consisting of two interacting spins governed by the [Formula: see text]-type Heisenberg Hamiltonian in an external magnetic field. We investigate the quantum evolution and the Riemannian geometry of the two-spin state space by means of the relevant Fubini–Study metric. The geometrical phase accumulated by the two-spin state is also examined under arbitrary and cyclic evolutions. By computing the evolution speed and the corresponding geodesic distance, we solve the quantum brachistochrone problem. The entanglement between the two spins is also studied via two approaches: the first one deals with the entanglement effect on the Fubini–Study metric and the geometrical phase, while the second one treats the entanglement effect on the evolution speed and the corresponding geodesic distance. Finally, we solve the quantum brachistochrone problem using the entanglement degree.
Comprehending the connections between the geometric, topological, and dynamical structures of integrable quantum systems with quantum phenomena exploitable in quantum information tasks, such as quantum entanglement, is a major problem in geometric information science. In this work we investigate these issues in a physical system of N\mathcal{N}N interacting spin-s\mathtt{s}s under the long-range Ising model. We discover the relevant dynamics, identify the corresponding quantum phase space, and we derive the associated Fubini-Study metric. Through the application of the Gauss-Bonnet theorem and the derivation of the Gaussian curvature, we proved that the dynamics occurs on a spherical topology manifold. Afterwards, we analyze the gained geometrical phase under the arbitrary and cyclic evolution processes and solve the quantum brachistochrone problem by establishing the time-optimal evolution. Moreover, by narrowing the system to a two spin-s\mathtt{s}s system, we explore the relevant entanglement from two different perspectives. The first is geometrical in nature and involves the investigation of the interplay between the entanglement degree and the geometrical structures, such as the Fubini-Study metric, the Gaussian curvature, and the geometrical phase. The second is dynamical in nature and tackles the entanglement effect on the evolution speed and geodesic distance. Additionally, we resolve the quantum brachistochrone problem based on the entanglement degree.
This thesis, explores the quantum entanglement and evolution through both a geometric and dynamical perspective. The first part focuses on classical phase space and its central role in Hamiltonian mechanics, emphasizing the importance of symplectic structures in describing mechanical states. The study highlights the formal analogy between classical phase space and the Hilbert space used in quantum mechanics. The second part is devoted to the geometric description of quantum states through the projective structure of Hilbert space. Emphasis is placed on the geometric interpretation of quantum evolution, particularly via the Fubini-Study metric, associated symplectic structures, and the geometric phase acquired during unitary evolutions. The final two parts are dedicated to the study of spin systems (both two-body and many-body) under different interaction models (XXZ Heisenberg and all-range Ising). Both the dynamical aspects (evolution speed, entanglement, and the quantum brachistochrone problem) and the geometric and topological structures of the corresponding quantum states are analyzed.
We present a unified geometric and dynamical framework for a physical system consisting of nnn spin-1/21/21/2 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 provide explicit geometric description of state manifolds obtained from evolution governed by a four-parameter family of time-independent Hamiltonians. We cover most cases related to the real interacting two-qubit systems and discuss possible types of evolutions in terms of the defining parameters. The relevant description of the pure state spaces and their Riemannian geometry with the Fubini–Study metric is given. In particular, we analyze the modification of known geometry of quantum state manifold by the linear noncommuting perturbation of the Hamiltonian. Finally, we investigate the behavior of the entanglement for obtained families of states resulting from the unitary evolution.