This chapter presents the derivation of Berry's phase. Many classical phenomena have corresponding quantum mechanical analogue. Therefore, it is natural to identify a classical analogue of Berry's phase. The concept of Berry's phase has been used in the study of quantized Hall Effect, the spin statistics properties of quasiparticle excitations, the rotation of photon polarization in helical optical fibers, etc. The chapter discusses various applications of Berry's phase in solid state physics. The Berry's phase in one-dimension is usually called Zak phase and is used for a better understanding of electric polarization in dielectrics, protected edge states and Wannier–Stark ladders and integer quantum Hall effect. The chapter discusses the Aharonov–Bohm and Sagnac effects. The Aharonov–Bohm experiment consists of splitting an electron beam, passing around both sides of a solenoid and then recombining the beam. The Sagnac effect has been studied in resonant microcavities and light propagation in a wave guide also.
This chapter presents the derivation of Berry's phase. Many classical phenomena have corresponding quantum mechanical analogue. Therefore, it is natural to identify a classical analogue of Berry's phase. The concept of Berry's phase has been used in the study of quantized Hall Effect, the spin statistics properties of quasiparticle excitations, the rotation of photon polarization in helical optical fibers, etc. The chapter discusses various applications of Berry's phase in solid state physics. The Berry's phase in one-dimension is usually called Zak phase and is used for a better understanding of electric polarization in dielectrics, protected edge states and Wannier–Stark ladders and integer quantum Hall effect. The chapter discusses the Aharonov–Bohm and Sagnac effects. The Aharonov–Bohm experiment consists of splitting an electron beam, passing around both sides of a solenoid and then recombining the beam. The Sagnac effect has been studied in resonant microcavities and light propagation in a wave guide also.
T HE BERRY PHASE, A CENTRAL CONCEPT IN QUANTUM physics, is rooted in the state space structure of quantum theory and has a profound observable impact within a wide range of phenomena. The remarkably elegant interpretation of this geometric phase as the holonomy of a fiber bundle provides it with a mathematical framework capable of accounting for a broad spectrum of classical and quantum phenomena, such as, for example, the precession of a Foucault pendulum, the parallel transport of a vector on a sphere, the Aharonov-Bohm effect, the gauge theory of molecular physics, the electric polarization and orbital magnetization of solids, and the integer and fractional quantum Hall effects. The Berry phase is the central concept in topological band theory and provides the mechanism responsible for anyonic statistics in correlated many-body systems, a key element that could be used to achieve fault-tolerant quantum computation. In other words, the Berry phase is not just some fashionable idea useful for explaining a few exotic phenomena, but a generic feature of quantum mechanics and an essential ingredient necessary for a coherent understanding of basic quantum phenomena, such as, for example, the electronic properties of materials. It is therefore rather surprising that the full significance of geometric phases was practically ignored for more than fifty years prior to the publication of Berry’s work in 1984 [50]. In this chapter we discuss the basic physics that gives rise to these geometric phases, point out the elements that allow us to distinguish different types of holonomies, summarize the basic concepts relevant to understanding the mathematical structure of geometric phases, and mention a few relevant applications. The concepts and the mathematical framework introduced here will be used in subsequent chapters as the basis for describing the topological properties of condensed matter systems.
We are accustomed to think the phase of single particle states does not matter. After all, the phase cancels out when calculating physical observables. However, the geometric phase can cause interference even in single particle states and can be measured. Berry’s phase is a geometric phase the system accumulates as its time-dependent Hamiltonian is subjected to closed adiabatic excursion in parameter space. In this report, we explore how Berry’s phase manifests itself in various fields of physics, both classical and quantum mechanical. The Hannay angle is a classical analogue to Berry’s phase and they are related by a derivative. The Aharonov-Bohm effect is a manifestation of Berry’s phase. Net rotation of deformable bodies in the language of gauge theory can be translated as a Berry phase. The well-known BornOppenheimer approximation is a molecular Aharonov-Bohm effect and is another manifestation of Berry’s Phase.
This special issue celebrates the discovery of two of the most important aspects of quantum mechanics: the Aharonov–Bohm effect and the Berry phase. The issue includes work presented at two conferences, 50 Years of the Aharonov–Bohm Effect, 11–14 October 2009, Tel Aviv University, Israel, and the Aharonov–Bohm Effect and Berry Phase 50/25 Anniversary, 14–15 December 2009, Bristol University, UK. The Aharonov–Bohm (AB) effect and the Berry phase have many things in common, apart from the happy coincidence that they were discovered 25 years apart, making possible such a joint anniversary. First, they were both discovered in the same place, the H H Wills Physics Laboratory, University of Bristol, UK. Second, they both were there for everybody to see from day one, from the very beginning of quantum mechanics. They were, so to say, right on the surface; there was no need for sophisticated mathematics or involved constructions. Yet it took so long for somebody to see them. The explanation, we dare to speculate, is that they are so unusual, they challenge so much our everyday experience and ingrained intuitions about physics that it was difficult to recognise them. Third, both the AB effect and the Berry phase are quintessentially quantum; they both manifest themselves as topological/geometric quantum phases. The AB effect revolutionized our understanding of the role of potentials in physics. For the first time, it was shown that a particle moving in a field-free region could be affected by a field in a disjoint region. In doing so, it shattered one of the strongest held principles of physics. In the case of the Berry phase, the importance lies in its absolute generality—the topological/geometric phase named after Berry is acquired by all physical systems when the external parameters they depend upon undergo an adiabatic cyclic change. Remarkably this phase depends only on the geometry of the cyclic path in the abstract parameter space, a fact which deepens our appreciation of the underlying geometry of quantum mechanics. The AB effect and the Berry phase are ubiquitous in modern physics and have implications ranging from the most fundamental to the very applied. First and foremost, they give a deeper understanding of the nature of quantum mechanics itself. Furthermore, we find them in fields as varied as cosmology, particle physics, non-abelian gauge theories, condensed matter, chemical and molecular physics, and laser dynamics. Generalizations of the AB effect to non-abelian gauge theories, such as the Wilson and `t Hooft loops, are important tools for studying confinement and spontaneous symmetry breaking. They explain charge quantization, the quantum Hall effect and the Josephson junction and also play a crucial role in electron microscope holography. Without them an understanding of many effects in the new field of mesoscopic physics, where tiny electronic circuits exhibit quantum behaviour, would be impossible. Like good wine, the AB effect and the Berry phase become finer and more appreciated with time. Even though they appear in textbooks and encyclopaedias, the number of researchers thinking and publishing papers on these two effects is increasing annually. The AB effect and the Berry phase keep being observed in new systems and with every day that passes novel applications are routinely found. This special issue is just a small snapshot of the state of the art 50 and 25 years, respectively, from their discoveries.
In the usual Aharonov-Bohm effect, representing an especial case of the Berry phase phenomenon, classical magnetic field within long and thin solenoid (or classical vector potential of the electromagnetic field without this solenoid) causes phase difference and interference shape translation of the quantum propagating (external) electron. In this work we consider a variation of the usual Aharonov-Bohm effect with two solenoids sufficiently close one to the other so that (external) electron cannot propagate between two solenoids but only around both solenoids. Here magnetic field (or classical vector potential of the electromagnetic field) acting at quantum propagating (external) electron represents the quantum mechanical average value or statistical mixture. It is obtained by wave function of single (internal, quantum propagating within some solenoid wire) electron (or homogeneous ensemble of such (internal) electrons) representing a quantum superposition with two practically non-interfering terms. All this implies that phase difference and interference shape translation of the quantum propagating (external) electron represent the quantum mechanical average value or statistical mixture. In this way we obtain a very interesting generalization of the usual Aharonov-Bohm effect and Berry phase concept. As it is well-known in the usual experimental arrangement corresponding to the AharonovBohm effect [1]-[4] (representing an especial case of the Berry phase phenomenon [4], [5]) behind a diaphragm with two slits there is a long and very thin solenoid (placed perpendicularly, i.e. in the z-axis direction, to the x-0-y plane of the propagation of the (external, without solenoid
The phase shift due to the Sagnac Effect, for relativistic matter beams counter-propagating in a rotating interferometer, is deduced on the bases of a formal analogy with the Aharonov-Bohm effect. A procedure outlined by Sakurai, in which non relativistic quantum mechanics and Newtonian physics appear together with some intrinsically relativistic elements, is generalized to a fully relativistic context, using the Cattaneo's splitting technique. This approach leads to an exact derivation, in a self-consistently relativistic way, of the Sagnac effect. Sakurai's result is recovered in the first order approximation.
We provide arguments which demonstrate that the Berry phase effects involving particle interference in the presence of external electric and magnetic fields—namely the Aharonov–Bohm (AB), Aharonov–Casher (AC) and the Röntgen (R) effects—cannot be unambiguously explained (as has been controversially argued) through the use of any classical mechanical force. In fact, in the three scenarios which we consider, the only means to reclaim the standard quantum mechanical results is to use the expressions for the force that fail to satisfy the necessary conditions of gauge invariance. Our arguments explain the foundation of the controversy and confirm the standard idea that the Berry phase effects in question are non-local phenomena which only arise in the application of quantum theory. We lay special emphasis on the use of the concept of a canonical force, this being the quantity which ensures the agreement between the classical and quantum formulations. It is shown that if the Aharonov–Bohm effect is interpreted as being due to a force, through the use of the Bohmian formulation of quantum mechanics, then this force is equal in magnitude to the canonical force.