Abelian Gauge Potentials on Cubic Lattices

Abstract

The study of the dynamics of a quantum particle in a magnetic field is a fascinating subject of perduring interest both in physics and mathematics literature. The quantization of energy levels giving rise to the Landau levels is at the basis of our understanding of integer and fractional quantum Hall effects [1, 2, 3, 4, 5] and its higher-dimensional counterparts and generalizations, including topological insulators [6,7,8]. The use of vector potentials in quantum mechanics is associated in itself to very interesting consequences, such as the purely quantum mechanical interference in the Aharonov-Bohm effect [9]. On the other side, the mathematical formalism for a particle in a magnetic field has been developed and refined along the time, based on the rigorous definition of Schrödinger operators with magnetic fields [10]. An important role is played by the construction of families of observables in the presence of gauge fields relying on the progresses in gauge covariant

Exploring helical phases of matter in bosonic ladders

“…In both cases, suitable pairs of Raman lasers allow us to engineer an effective long-time dynamics of the system described by a low-energy tight-binding Hamiltonian characterized by position-dependent phases in the hopping amplitudes of the atoms (see, for example, Refs. [46][47][48][49]). The resulting artificial gauge flux χ in each plaquette depends on the recoil momentum of the Raman lasers and can be varied by tuning the angle between them (see, for example, the experiments [50][51][52]).…”

Gauge Theories with Ultracold Atoms

“…In Section 3, several examples of techniques to engineer complex phases on the hopping parameters are discussed. Reviews can be found in [21][22][23][24].…”

Exact diagonalization of cubic lattice models in commensurate Abelian magnetic fluxes and translational invariant non-Abelian potentials

“…, L − 1. The Hamiltonian (2) is then finally rewritten in the above defined MBZ and in the basis of the n s sublattices as in equation (17), and three n s × n s matrices are derived similarly to what has beed done in the previous section for the three matrices T AB ĵ in equation (18). Similar results can be obtained using a different optimal gauge choice through the mapping onto a onedimensional model in momentum space [38].…”

“…(36) has the same symmetries of the one in Eq. (17). As a consequence, its spectrum divides in 2n generally non-degenerate subbands.…”

“…Beside the successes in realizing such models in solid state devices, the recent experimental developments in the field of ultracold atomic gases [13][14][15] opened new scenarios to realize Abelian and non-Abelian gauge potentials in optical lattices, for instance imposing laser-assisted tunneling amplitudes to trapped atoms [15,16] (see as well the recent review [17] in the book [18]). The possibility of implementing tunable gauge potentials became a paradigmatic example of the tools that can be exploited in the experimental design of novel quantum phases of matter and is providing a remarkable arena of challenging mathematical developments.…”