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The geometry of Hamiltonian's eigenstates is encoded in the quantum geometric tensor (QGT). It contains both the Berry curvature, central to the description of topological matter and the quantum metric. So far the full QGT has been measured…

The importance of the quantum metric in flat-band systems has been noticed recently in many contexts such as the superfluid stiffness, the dc electrical conductivity, and ideal Chern insulators. Both the quantum metric of degenerate and…

On the background of the Born-Oppenheimer (adiabatic) approximation, we investigate the geometrical and topological structure in the theory of quantum gravity by using the path integral method and halfclassical approximation. As we know,…

The physical interpretation and eventual fate of gravitational singularities in a theory surpassing classical general relativity are puzzling questions that have generated a great deal of interest among various quantum gravity approaches.…

Understanding the relationship between quantum geometry and topological invariants is a central problem in the study of topological states. In this work, we establish the relationship between the quantum metric and the Euler curvature in…

The past few years have seen a revived interest in quantum geometrical characterizations of band structures due to the rapid development of topological insulators and semi-metals. Although the metric tensor has been connected to many…

We use the idea of the symmetry between the spacetime coordinates x^\mu and the energy-momentum p^\mu in quantum theory to construct a momentum space quantum gravity geometry with a metric s_{\mu\nu} and a curvature P^\lambda_{\mu\nu\rho}.…

Dynamics of quasi-particles in graphene with an impurity and a sharp edge is considered with the kp perturbation theory that allows an unified approach without usage of any models. Dirac and Weyl equations are derived by the above-mentioned…

The graphene membrane irradiated by weak activating alternative electric field in terahertz range was considered. The quantum approach based on the time-dependent density matrix method was used. The exact solution was obtained for graphene…

The results of the simulations by Monte Carlo method of graphene with structural defects are presented. The calculations are performed within an effective quantum field theory with non-compact $3\hm + 1$--dimensional Abelian gauge field and…

In this paper, we investigate the Schr\"odinger equation in a three-dimensional helically twisted space characterized by a non-trivial torsion parameter. By applying exact separation of variables, we derive the radial equation governing the…

The quantum geometry in the momentum space of semiconductors and insulators, described by the quantum metric of the valence band Bloch state, has been an intriguing issue owing to its connection to various material properties. Because the…

In a recent work [arXiv:2307.13489 [gr-qc]], we have described spherically symmetric and static quantum black holes as deformations of the classical Schwarzschild metric that depend on the physical distance to the horizon. We have developed…

Using the effective field theory of quantum gravity at second order in curvature, we calculate quantum corrections to the metric of gravastars and the closely related dark energy stars. We find that the quantum corrections in the exterior…

Topological properties of quantum system is directly associated with the wave function. Based on the decomposition theory of gauge potential, a new comprehension of topological quantum mechanics is discussed. One shows that a topological…

The quantum metric and Berry curvature capture essential properties of non-trivial Bloch states and underpin many fascinating phenomena. However, it becomes increasingly evident that a more comprehensive understanding of quantum state…

Hamiltonian and eigenstate problem is formulated for a bilayer graphene in terms of Clifford's geometric algebra \textit{Cl}$_{3,1}$. It is shown that such approach allows to perform analytical calculations in a simple way if geometrical…

We prove a `Whitney' presentation, and a `Coulomb branch' presentation, for the torus equivariant quantum K theory of the Grassmann manifold $\mathrm{Gr}(k;n)$, inspired from physics, and stated in an earlier paper. The first presentation…

We analyse the electronic and optical properties of graphene quantum dots (GQD) using accurate \textit{ab initio} many-body $GW$ and Bethe-Salpeter calculations. We show that most pristine GQD, including structures with irregular shapes,…

3-dimensional gravity coupled to Maxwell (or Klein-Gordon) fields is exactly soluble under the assumption of axi-symmetry. The solution is used to probe several quantum gravity issues. In particular, it is shown that the quantum…