We take as a starting point an expression for the quantum geometric tensor recently derived in the context of the gauge/gravity duality. We proceed to generalize this formalism in such way it is possible to compute the geometrical phases of quantum systems. Our scheme provides a conceptually complete description and introduces a different point of view of earlier works. Using our formalism, we show how this expression can be applied to well-known quantum mechanical systems.

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. Classical integrable systems are considered and a new approach is reported 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. The approach used arises from the semiclassical approximation of the Berry curvature and the quantum metric tensor in the Lagrangian formalism. This semiclassical approximation is exploited to establish, for the first time, the relation between the quantum metric tensor and its classical counterpart. The approach described is illustrated and validated 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, some potential applications of this approach and possible generalizations that can be of interest in the field of condensed matter physics are mentioned.

We introduce a new method to compute the Quantum Geometric Tensor, this procedure allow us to compute the Quantum Information Metric and the Berry curvature perturbatively for a theory with an arbitrary interaction Hamiltonian. The calculation uses the generating function method, and it is illustrated with the harmonic oscillator with a linear and a quartic perturbation.

Starting from N = 4 SYM and using an appropriate Higgs mechanism we reconsider the construction of a scalar field theory non-minimally coupled to a Coulomb potential with a relativistic SO(4) symmetry and check for scalar field consistency conditions. This scalar field theory can also be obtained from a relativistic particle Lagrangian with a proper implementation of the non-minimal coupling. We provide the generalization of the non-relativistic construction of the Runge-Lenz vector to the relativistic case and show explicitly that this new vector generates the SO(4) algebra. Using the power of the SO(4) symmetry, we calculate the relativistic hydrogen atom spectrum. We provide a generalization of the Kustaanheimo-Stiefel transformation to the relativistic case and relate our results with the corresponding relativistic oscillator. Finally, in the light of these results, we reconsider the calculation of the hydrogen atom spectrum from the cusp anomalous dimension given in [2].

The quantum metric tensor was introduced for defining the distance in the parameter space of a system. However, it is also useful for other purposes, like predicting quantum phase transitions. Due to the physical information this tensor provides, its gauge independence sounds reasonable. More over, its original construction was made by looking for this gauge independence. The aim of this paper, however, is to prove that the quantum metric tensor does depend on the gauge. In addition, a real gauge invariant quantum metric tensor is introduced. In this paper, the gauge dependence is explicitly shown by computing the quantum metric tensor of the Landau problem in different gauges. Then, a real gauge independent metric tensor is proposed and computed for the same Landau problem. Since the gauge dependence has not been observed before, the results of this paper might lead to a new study of topics that were believed to be completely understood.

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