Geometric phase

In classical and quantum mechanics, geometric phase is a phase difference acquired over the course of a cycle, when a system is subjected to cyclic adiabatic processes, which results from the geometrical properties of the parameter space of the Hamiltonian. The phenomenon was independently discovered by T. Kato (1950), S. Pancharatnam (1956), and by H. C. Longuet-Higgins (1958) and later generalized by Sir Michael Berry (1984). 
It is also known as the Pancharatnam–Berry phase, Pancharatnam phase, or Berry phase.
It can be seen in the conical intersection of potential energy surfaces and in the Aharonov–Bohm effect. Geometric phase around the conical intersection involving the ground electronic state of the C6H3F3+ molecular ion is discussed on pages 385-386 of the textbook by Bunker and Jensen.In the case of the Aharonov–Bohm effect, the adiabatic parameter is the magnetic field enclosed by two interference paths, and it is cyclic in the sense that these two paths form a loop. In the case of the conical intersection, the adiabatic parameters are the molecular coordinates. Apart from quantum mechanics, it arises in a variety of other wave systems, such as classical optics. As a rule of thumb, it can occur whenever there are at least two parameters characterizing a wave in the vicinity of some sort of singularity or hole in the topology; two parameters are required because either the set of nonsingular states will not be simply connected, or there will be nonzero holonomy.
Waves are characterized by amplitude and phase, and may vary as a function of those parameters. The geometric phase occurs when both parameters are changed simultaneously but very slowly (adiabatically), and eventually brought back to the initial configuration. In quantum mechanics, this could involve rotations but also translations of particles, which are apparently undone at the end. One might expect that the waves in the system return to the initial state, as characterized by the amplitudes and phases (and accounting for the passage of time). However, if the parameter excursions correspond to a loop instead of a self-retracing back-and-forth variation, then it is possible that the initial and final states differ in their phases. This phase difference is the geometric phase, and its occurrence typically indicates that the system's parameter dependence is singular (its state is undefined) for some combination of parameters.
To measure the geometric phase in a wave system, an interference experiment is required. The Foucault pendulum is an example from classical mechanics that is sometimes used to illustrate the geometric phase. This mechanics analogue of the geometric phase is known as the Hannay angle.
Berry phase in quantum mechanics.
In a quantum system at the n-th eigenstate, an adiabatic evolution of the Hamiltonian sees the system remain in the n-th eigenstate of the Hamiltonian, while also obtaining a phase factor. The phase obtained has a contribution from the state's time evolution and another from the variation of the eigenstate with the changing Hamiltonian. The second term corresponds to the Berry phase and for non-cyclical variations of the Hamiltonian it can be made to vanish by a different choice of the phase associated with the eigenstates of the Hamiltonian at each point in the evolution.
However, if the variation is cyclical, the Berry phase cannot be cancelled; it is invariant and becomes an observable property of the system. By reviewing the proof of the adiabatic theorem given by Max Born and Vladimir Fock, in Zeitschrift für Physik 51, 165 (1928), we could characterize the whole change of the adiabatic process into a phase term. Under the adiabatic approximation, the coefficient of the nth eigenstate under adiabatic process is given by
formula_1
where formula_2 is the Berry's phase with respect of parameter t. Changing the variable t into generalized parameters, we could rewrite the Berry's phase into
where formula_4 parametrizes the cyclic adiabatic process. It follows a closed path formula_5 in the appropriate parameter space. Geometric phase along the closed path formula_5 can also be calculated by integrating the Berry curvature over surface enclosed by formula_5.
Examples of geometric phases.
The Foucault pendulum.
One of the easiest examples is the Foucault pendulum. An easy explanation in terms of geometric phases is given by Wilczek and Shapere 
To put it in different words, there are no inertial forces that could make the pendulum precess, so the precession (relative to the direction of motion of the path along which the pendulum is carried) is entirely due to the turning of this path. Thus the orientation of the pendulum undergoes parallel transport. For the original Foucault pendulum, the path is a circle of latitude, and by the Gauss–Bonnet theorem, the phase shift is given by the enclosed solid angle.
Polarized light in an optical fiber.
A second example is linearly polarized light entering a single-mode optical fiber. Suppose the fiber traces out some path in space and the light exits the fiber in the same direction as it entered. Then compare the initial and final polarizations. In semiclassical approximation the fiber functions as a waveguide and the momentum of the light is at all times tangent to the fiber. The polarization can be thought of as an orientation perpendicular to the momentum. As the fiber traces out its path, the momentum vector of the light traces out a path on the sphere in momentum space. The path is closed since initial and final directions of the light coincide, and the polarization is a vector tangent to the sphere. Going to momentum space is equivalent to taking the Gauss map. There are no forces that could make the polarization turn, just the constraint to remain tangent to the sphere. Thus the polarization undergoes parallel transport and the phase shift is given by the enclosed solid angle (times the spin, which in case of light is 1).
Stochastic pump effect.
A stochastic pump is a classical stochastic system that responds with nonzero, on average, currents to periodic changes of parameters.
The stochastic pump effect can be interpreted in terms of a geometric phase in evolution of the moment generating function of stochastic currents. 
Spin ½.
The geometric phase can be evaluated exactly for a spin-½ particle in a magnetic field.
Geometric phase defined on attractors.
While Berry's formulation was originally defined for linear Hamiltonian systems, it was soon realized by Ning and Haken
 that similar geometric phase can be defined for entirely different systems such as nonlinear dissipative systems that possess certain cyclic attractors. They showed that such cyclic attractors exist in a class of nonlinear dissipative systems with certain symmetries.
Exposure in molecular adiabatic potential surface intersections.
There are several ways to compute the geometric phase in molecules within the Born Oppenheimer framework. One way is through the "non-adiabatic coupling formula_8 matrix" defined by
formula_9
where formula_10 is the adiabatic electronic wave function, depending on the nuclear parameters formula_11. The nonadiabatic coupling can be used to define a loop integral, analogous to a Wilson loop (1974) in field theory, developed independently for molecular framework by M. Baer (1975, 1980, 2000). Given a closed loop formula_12, parameterized by formula_13 where formula_14 is a parameter and formula_15. The D-matrix is given by:
formula_16
(here, formula_17 is a path ordering symbol). It can be shown that once formula_18 is large enough (i.e. a sufficient number of electronic states is considered) this matrix is diagonal with the diagonal elements equal to formula_19 where formula_20 are the geometric phases associated with the loop for the formula_21 adiabatic electronic state.
For time-reversal symmetrical electronic Hamiltonians the geometric phase reflects the number of conical intersections encircled by the loop. More accurately:
formula_22
where formula_23 is the number of conical intersections involving the adiabatic state formula_24 encircled by the loop formula_12.
An alternative to the D-matrix approach would be a direct calculation of the Pancharatnam phase. This is especially useful if one is interested only in the geometric phases of a single adiabatic state. In this approach, one takes a number formula_26 of points formula_27 along the loop formula_28 with formula_29 and formula_30 then using only the jth adiabatic states formula_31 computes the Pancharatnam product of overlaps:
formula_32
In the limit formula_33 one has (See Ryb & Baer 2004 for explanation and some applications):
formula_34
Geometric phase and quantization of cyclotron motion.
An electron subjected to magnetic field formula_35 moves on a circular (cyclotron) orbit. Classically, any cyclotron radius formula_36 is acceptable. Quantum-mechanically, only discrete energy levels (Landau levels) are allowed and since formula_36 is related to electron's energy, this corresponds to quantized values of formula_36. The energy quantization condition obtained by solving Schrödinger's equation reads, for example, formula_39 for free electrons (in vacuum) or formula_40 for electrons in graphene where formula_41. Although the derivation of these results is not difficult, there is an alternative way of deriving them which offers in some respect better physical insight into the Landau level quantization. This alternative way is based on the semiclassical Bohr-Sommerfeld quantization condition
formula_42
which includes the geometric phase formula_43 picked up by the electron while it executes its (real-space) motion along the closed loop of the cyclotron orbit. For free electrons, formula_44 while formula_45 for electrons in graphene. It turns out that the geometric phase is directly linked to formula_46 of free electrons and formula_47 of electrons in graphene.
Notes.
 For simplicity, we consider electrons confined to a plane, such as 2DEG and magnetic field perpendicular to the plane.
 formula_48 is the cyclotron frequency (for free electrons) and formula_49 is the Fermi velocity (of electrons in graphene).