Berry's phase has been fashionable in many areas of physics and in chemistry and also among mathematicians and mathematical physicists. The mathematical people are attracted to this area because it is related to the beautiful mathematics of bre bundles which underlay gauge theories. In fact this is the most accessible example of a gauge theory for people who know just the elementary facts of no...

We study the Jahn-Teller effect of positive fullerene ions C60 and 1C 60 . The aim is to discover if this case, in analogy with the negative ion, possesses a Berry phase or not, and what are the consequences on dynamical Jahn-Teller quantization. Working in the linear and spherical approximation, we find no Berry phase in 1C 60 , and presence/absence of Berry phase for coupling of one L = 2 hol...

We revisit earlier studies on Berry phases suggested to appear in certain cavity QED settings. It has been especially argued that a nontrivial geometric phase is achievable even in the situation of no cavity photons. We, however, show that such results hinge on imposing the rotating wave approximation (RWA), while without the RWA no Berry phases occur in these schemes. A geometrical interpretat...

Berry phase was originally defined for systems whose states are separated by finite energy gaps. One might naively expect that a system without a gap cannot have a Berry phase. Despite this we ask whether a Berry phase can be observed in a system which has a continuous spectrum because its coupling to the environment has broadened its energy levels. We find that, contrary to the above naive exp...

We obtain the adiabatic Berry phase by defining a generalised gauge potential whose line integral gives the phase holonomy for arbitrary evolutions of parameters. Keeping in mind that for classical integrable systems it is hardly clear how to obtain open-path Hannay angle, we establish a connection between the open-path Berry phase and Hannay angle by using the parametrised coherent state appro...

The effect of the Coriolis force on a moving system can be described as a holonomy with respect to a particular connection known as the Cartan–Hannay–Berry connection. The resulting geometric phase is called the Hannay–Berry phase, and it provides direct information about the imposed motion on the system. This approach assumes that the imposed motion is adiabatic. In this paper we describe the ...

We design an adiabatic quantum algorithm for the counting problem, i.e., approximating the proportion, α, of the marked items in a given database. As the quantum system undergoes a designed cyclic adiabatic evolution, it acquires a Berry phase 2πα. By estimating the Berry phase, we can approximate α, and solve the problem. For an error bound ǫ, the algorithm can solve the problem with cost of o...

Berry phase for a spin–1/2 particle moving in a flat spacetime with torsion is investigated in the context of the Einstein-Cartan-Dirac model. It is shown that if the torsion is due to a dense polarized background, then there is a Berry phase only if the fermion is massless and its momentum is perpendicular to the direction of the background polarization. The order of magnitude of this Berry ph...