The Quantum Geometric Tensor in a Parameter Dependent Curved Space
arXiv:2209.07728 · doi:10.3390/e24091236
Abstract
We introduce a quantum geometric tensor in a curved space with a parameter-dependent metric, which contains the quantum metric tensor as the symmetric part and the Berry curvature corresponding to the antisymmetric part. This parameter-dependent metric modifies the usual inner product, which induces modifications in the quantum metric tensor and Berry curvature by adding terms proportional to the derivatives with respect to the parameters of the determinant of the metric. The quantum metric tensor is obtained in two ways: By using the definition of the infinitesimal distance between two states in the parameter-dependent curved space and via the fidelity susceptibility approach. The usual Berry connection acquires an additional term with which the curved inner product converts the Berry connection into an object that transforms as a connection and density of weight one. Finally, we provide three examples in one dimension with a nontrivial metric: an anharmonic oscillator, a Morse-like potential, and a generalized anharmonic oscillator; and one in two dimensions: the coupled anharmonic oscillator in a curved space.
arXiv comment22 pages, 6 figures
References in corpus (15)
- The electronic properties of graphene
- Continuous Quantum Phase Transitions
- Fidelity approach to quantum phase transitions
- Quantum critical scaling of the geometric tensors
- Circuit complexity in quantum field theory
- Scaling of geometric phases close to quantum phase transition in the XY chain
- Geometric phases and criticality in spin chain systems
- Geometry of quantum phase transitions
- Quantum phase transitions in transverse field spin models: from statistical physics to quantum information
- Coupled harmonic oscillators and their quantum entanglement
- Three Lectures on Complexity and Black Holes
- Quantum Information Metric and Berry Curvature from a Lagrangian Approach
- From the Jordan product to Riemannian geometries on classical and quantum states
- Geometry of the parameter space of a quantum system: Classical point of view
- Conformal bootstrap in Liouville Theory