Geometric phase

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In classical and quantum mechanics, the geometric phase, Pancharatnam–Berry phase (named after S. Pancharatnam and Sir Michael Berry), Pancharatnam phase or most commonly Berry 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 first discovered in 1956,[1] and rediscovered in 1984.[2] It can be seen in the Aharonov–Bohm effect and in the conical intersection of potential energy surfaces. 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 both may vary as a function of those parameters. The Berry 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 Berry 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 Berry 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 Berry phase. This mechanics analogue of the Berry phase is known as the Hannay angle.

Theory[edit]

In general the geometric phase is given by:

\gamma[C] = i\oint_C \! \langle n,t| \left( \nabla_R |n,t\rangle \right)\,dR \,

where R parametrizes the cyclic adiabatic process. It follows a closed path C in the appropriate parameter space. A recent review on the Berry phase effects on electronic properties was given by Xiao, Chang and Niu.[3] Geometric phase along the closed path C can also be calculated by integrating the Berry curvature over surface enclosed by C.

Examples of geometric phases[edit]

The Foucault pendulum[edit]

One of the easiest examples is the Foucault pendulum. An easy explanation in terms of geometric phases is given by von Bergmann and von Bergmann:[4]

How does the pendulum precess when it is taken around a general path C? For transport along the equator, the pendulum will not precess. [...] Now if C is made up of geodesic segments, the precession will all come from the angles where the segments of the geodesics meet; the total precession is equal to the net deficit angle which in turn equals the solid angle enclosed by C modulo 2π. Finally, we can approximate any loop by a sequence of geodesic segments, so the most general result (on or off the surface of the sphere) is that the net precession is equal to the enclosed solid angle.

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[edit]

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[edit]

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.[5]

Geometric phase defined on attractors[edit]

While Berry's phase was originally defined for linear Hamiltonian systems, it was soon realized by Ning and Haken [6] 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.[7]

Exposure in molecular adiabatic potential surface intersections[edit]

There are several ways to compute the Berry phase in molecules within the Born Oppenheimer framework. One way is through the "non-adiabatic coupling M\times M matrix" defined by

\tau _{ij}^{\mu }=\left\langle  \psi _{i} | \partial ^{\mu }\psi _{j} \right\rangle

where \psi _{i} is the adiabatic electronic wave function, depending on the nuclear parameters R_{\mu }. 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 \Gamma , parameterized by R_{\mu }\left( t \right) where t\in \left[0,1 \right] is a parameter and R_{\mu }\left( t+1 \right)=R_{\mu }\left( t \right). The D-matrix is given by:

D\left[\Gamma  \right]=\hat{P}e^{\oint_{\Gamma }{\tau ^{\mu }dR_{\mu }}}

(here, {\hat{P}} is a path ordering symbol). It can be shown that once M is large enough (i.e. a sufficient number of electronic states is considered) this matrix is diagonal with the diagonal elements equal to e^{i\beta _{j}} where \beta _{j} are the Berry phases associated with the loop for the j adiabatic electronic state.

For time-reversal symmetrical electronic Hamiltonians the Berry phase reflects the number of conical intersections encircled by the loop. More accurately:

e^{i\beta _{j}}=\left( -1 \right)^{N_{j}}

where N_{j} is the number of conical intersections involving the adiabatic state \psi _{j} encircled by the loop \Gamma .

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 Berry phases of a single adiabatic state. In this approach, one takes a number N+1 of points \left( n=0,...,N \right) along the loop R\left( t_{n} \right) with t_{0}=0 and t_{N}=1 then using only the jth adiabatic states \psi _{j}\left[R\left( t_{n} \right) \right] computes the Pancharatnam product of overlaps:

I_{j}\left( \Gamma ,N \right)=\prod\limits_{n=0}^{N-1}{\left\langle  \psi _{j}\left[R\left( t_{n} \right) \right] | \psi _{j}\left[R\left( t_{n+1} \right) \right] \right\rangle }

In the limit N\to \infty one has (See Ryb & Baer 2004 for explanation and some applications):

I_{j}\left( \Gamma ,N \right)\to e^{i\beta _{j}}

Berry's phase and quantization of cyclotron motion[edit]

Electron subjected to magnetic field B moves on a circular (cyclotron) orbit.[1] Classically, any cyclotron radius R_c is acceptable. Quantum-mechanically, only discrete energy levels (Landau levels) are allowed and since R_c is related to electron's energy, this corresponds to quantized values of R_c. The energy quantization condition obtained by solving Schrödinger's equation reads, for example, E=(n+\alpha)\hbar\omega_c, \alpha=1/2 for free electrons (in vacuum) or E=v\sqrt{2(n+\alpha)eB\hbar}, \alpha=0 for electrons in graphene where n=0,1,2,\ldots.[2] 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


  \hbar\oint d\mathbf{r}\cdot \mathbf{k} - e\oint d\mathbf{r}\cdot\mathbf{A} + \hbar\gamma = 2\pi\hbar(n+1/2)

which includes the Berry phase \gamma picked up by the electron while it executes its (real-space) motion along the closed loop of the cyclotron orbit. These well-known facts have recently been nicely summarized by Jiamin Xue.[8] For free electrons, \gamma=0 while \gamma=\pi for electrons in graphene. It turns out that Berry's phase is directly linked to \alpha=1/2 of free electrons and \alpha=0 of electrons in graphene.

See also[edit]

Notes[edit]

^ For simplicity, we consider electrons confined to a plane, such as 2DEG and magnetic field perpendicular to the plane.

^ \omega_c=eB/m is the cyclotron frequency (for free electrons) and v is the Fermi velocity (of electrons in graphene).

Footnotes[edit]

  1. ^ S. Pancharatnam (1956). "Generalized Theory of Interference, and Its Applications. Part I. Coherent Pencils". Proc. Indian Acad. Sci. A 44: 247–262. 
  2. ^ M. V. Berry (1984). "Quantal Phase Factors Accompanying Adiabatic Changes". Proceedings of the Royal Society A 392 (1802): 45–57. Bibcode:1984RSPSA.392...45B. doi:10.1098/rspa.1984.0023. 
  3. ^ Di Xiao et al., Rev. Mod. Phys. 82, 1959 (2010)
  4. ^ Jens von Bergmann and HsingChi von Bergmann (2007). "Foucault pendulum through basic geometry". Am. J. Phys. 75 (10): 888–892. Bibcode:2007AmJPh..75..888V. doi:10.1119/1.2757623. 
  5. ^ N. A. Sinitsyn and I. Nemenman (2007). "The Berry phase and the pump flux in stochastic chemical kinetics". Euro. Phys. Lett. 77 (5): 58001. arXiv:q-bio/0612018. Bibcode:2007EL.....7758001S. doi:10.1209/0295-5075/77/58001. 
  6. ^ C.Z.Ning and H. Haken (1992). "Geometrical phase and amplitude accumulations in dissipative systems with cyclic attractors". Phys. Rev. Lett. 68 (14): 2109–2122. Bibcode:1992PhRvL..68.2109N. doi:10.1103/PhysRevLett.68.2109. 
  7. ^ C.Z.Ning and H. Haken (1992). "The geometric phase in nonlinear dissipative systems". Mod. Phys. Lett.B 6 (25): 1541–1568. Bibcode:1992MPLB....6.1541N. doi:10.1142/S0217984992001265. 
  8. ^ Jiamin Xue: "phase and the unconventional quantum Hall effect in graphene" (2013)

Sources[edit]

  • Cantoni, V.; Mistrangioli, L. (1992). "Three-point phase, symplectic measure, and Berry phase". International Journal of Theoretical Physics 31 (6): 937. doi:10.1007/BF00675086.  edit
  • Baer, M. (1975). "Adiabatic and diabatic representations for atom-molecule collisions: Treatment of the collinear arrangement". Chemical Physics Letters 35: 112. doi:10.1016/0009-2614(75)85599-0.  edit
  • Ryb, I; Baer, R (2004). "Combinatorial invariants and covariants as tools for conical intersections". The Journal of Chemical Physics 121 (21): 10370–5. doi:10.1063/1.1808695. PMID 15549915.  edit