Geometry of the parameter space of a quantum system: Classical point of view

Abstract

The local geometry of the parameter space of a quantum system is described by the quantum metric tensor and the Berry curvature, which are two fundamental objects that play a crucial role in understanding geometrical aspects of condensed matter physics. We consider classical integrable systems and report a new approach to obtain the classical analogs of the quantum metric tensor and the Berry curvature. An advantage of this approach is that it can be applied to a wide variety of classical systems corresponding to quantum systems with bosonic and fermionic degrees of freedom. Our approach arises from the semiclassical approximation of the Berry curvature and the quantum metric tensor in the Lagrangian formalism. We also exploit this semiclassical approximation to establish, for the first time, the relation between the quantum metric tensor and its classical counterpart. We illustrate and validate our approach by applying it to five systems: the generalized harmonic oscillator, the symmetric and linearly coupled harmonic oscillators, the singular Euclidean oscillator, and a spin-half particle in a magnetic field. Finally, we mention some potential applications of this approach and possible generalizations that can be of interest in the field of condensed matter physics

arXiv comment16 pages, accepted for publication in Annalen der Physik

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Cited by in corpus (8)

• Nontrivial quantum geometry of degenerate flat bands
• Phase space formulation of the Abelian and non-Abelian quantum geometric tensor
• Classical description of the parameter space geometry in the Dicke and Lipkin-Meshkov-Glick models
• Classical analogs of generalized purities, entropies, and logarithmic negativity
• The Quantum Geometric Tensor in a Parameter Dependent Curved Space
• Generalized quantum geometric tensor for excited states using the path integral approach
• Quantum geometric tensors from sub-bundle geometry
• Classical analogs of the covariance matrix, purity, linear entropy, and von Neumann entropy