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arXiv:gr-qc/0401025

Holonomy, Aharonov-Bohm effect and phonon scattering in superfluids

Claudio Furtado, Alexandre M. de M. Carvalho, L. C. García de Andrade, Fernando Moraes

Published Jan 8, 2004• 5 citations
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Abstract

In this article we discuss the analogy between superfluids and a spinning thick cosmic string. We use the geometrical approach to obtain the geometrical phases for a phonon in the presence of a vortex. We use loop variables for a geometric description of Aharonov-Bohm effect in these systems. We use holonomy transformations to characterize globally the "space-time" of a vortex and in this point of view we study the gravitational analog of the Aharonov-Bohm effect in this system. We demonstrate that in the general case the Aharonov-Bohm effect has a contribution both from the rotational and the translational holonomy. We study also Berrys quantum phase for phonons in this systems.

DOI
10.48550/arxiv.gr-qc/0401025

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In this article we discuss the analogy between superfluids and a spinning thick cosmic string. We use the geometrical approach to obtain the geometrical phases for a phonon in the presence of a vortex. We use loop variables for a geometric description of Aharonov-Bohm effect in these systems. We use holonomy transformations to characterize globally the "space-time" of a vortex and in this point of view we study the gravitational analog of the Aharonov-Bohm effect in this system. We demonstrate that in the general case the Aharonov-Bohm effect has a contribution both from the rotational and the translational holonomy. We study also Berrys quantum phase for phonons in this systems.
I discuss the scattering of phonons by a vortex moving with respect to a superfluid condensate. This allows us to test the compatibility of the scattering-theory derivation of the Iordanskii force with the Galilean invariance of the underlying fluid dynamics. In order to obtain the correct result we must retain O(vs2){O(v}_{s}^{2})O(vs2​) terms in the sound-wave equation, and this reinforces the interpretation, due to Volovik, of the Iordanskii force as an analogue of the gravitational Bohm-Aharonov effect.
A geometric interpretation of Berry’s phase, its Wilczek–Zee non-Abelian generalization, and Aharonov–Bohm effect is given in terms of connections on principal fiber bundles. It is shown that a principal fiber bundle in all cases can be trivial while the connection and its holonomy group are nontrivial. Therefore the main role is played not by topological effects but geometrical ones. The Berry phase is also shown not to be connected to the adiabatic changes of the Hamiltonian parameters. 1
In this work, we make a review of the literature that the so-called Berry’s geometric phase. A recent issue, which has been employed in various areas of physics. For this, we use, in a number of references, in which many were critical to the theoretical foundation of this work. At first, we focus on the study of the time evolution of quantum systems governed by Hamiltonians operators whose time dependence is manifested by a set of parameters called classical fields. The adiabatic approximation, is justified mathematically when demonstrated the adiabatic theorem. Under these conditions, we obtain geometric phases - Berry’s phase - associated with these systems in different representations, each displaying its particularity. Finally, as a particular case of geometric phase, we studi from the viewpoint of duality transformations of Maxwell’s equations, the dual Aharonov-Bohm effect, extracting the geometric Berry’s phase associated with the dynamics of the magnetic monopole in the presence of the electric vector potential
We propose a hydrodynamic analog of the Aharonov-Bohm experiment in which second-sound waves are scattered on a quantized vortex in superfluid helium. The time lag that develops between the two sides of a second-sound plane wave impinging on a vortex from the side would be compared. Estimates for these experiments show that for reasonable experimental conditions the wavefront splitting can be readily measured at reduced temperatures of 10 −6 . The advantages of doing this experiment near the superfluid transition is discussed. This experiment would compliment the three other Aharonov-Bohm–type experiments which involve all four possible combinations of classical/quantum waves and classical/quantum vortices. This technique would provide a non-invasive means of investigating the dynamics of quantum vortex nucleation and their stability as well as addressing the problem of multiply quantized.
The authors discuss certain properties of Berry's phases in two quantum systems with supersymmetrically related Hamiltonians and derive an explicit expression for the difference in their Berry's phases. Moreover, they obtain an expression for a topological quantity which can be interpreted in term of holonomy, and which contains the Berry's phase and is invariant in the two supersymmetrically related quantum systems. They take two examples to illustrate their findings. The first one is the example of the Aharonov-Bohm effect in which the Berry's phases of the two supersymmetrically related quantum systems are equal. The second example is a system with a spin-1/2 particle in a time varying magnetic field. They obtain the stated topological quantity and, in their analysis, they discover that this quantity is corresponding to the essential gauge transformation in such systems.