1. Berry Phases in Physics
Matthew Spitzer memorial lecture series in physics
Tatiana Seletskaia
Physics Department, CSU East Bay

2. Phenomena predicted by theoreticians in the twentieth century
OR INVENTION
PREDICTED
EXPERIMENTALLY CONFIRMED
LASER
Albert Einstein
1917
Theodore Maiman
1960
POSITRON
Paul Dirac
1928
Carl David Anderson
1932
GRAPHENE
P. R. Wallace
1947
Geim, Novoselov
2004
COMPUTER
Turing
1935
John Bardeen
HIGGS BOSON
Peter Higgs
1964
LHC
2012
KITAEV FERMION
Alexei Kitaev
2000
V. Mourik et al
QUANTUM
Benioff, Manin, Simon …
TIME CRYSTAL
Frank Wilczek

3. What we call a Berry phase?
Geometric Phase or Adiabatic Anholonomy

4. Berry Phase
Geometric Phase Adiabatic Anholonomy

5. Phase Phase is an angle that describes position of the object
during its periodic motion
Phase of the harmonic oscillator and of the pendulum depend on the cyclic velocity:

6. Geometric Phase
Motion along the circle – Parallel transport Phase is an angle that describes orientation of the vector after its parallel transport along the closed contour
On the plane initial and final vectors coincide
On the curved space the vectors differ

7. Geometric Phase
Geometric phase depends only on the geometry of the contour It does not depend on the velocity!!!

8. Adiabatic Anholonomy

9. Holonomy
Holography – recording of the light field, a point has information about the entire wavefront

10. Holonomy
Holonomy - holonomic function – solution of the linear differential equation – continuous smooth functions
At every point of the function all derivatives are known. Thus, every point has the information about the properties of the function at the entire range.

11. Holonomy
Holonomy - holonomic function – solution of the linear differential equation – continuous smooth functions
At every point of the function all derivatives are known. Thus, every point has the information about the properties of the function at the entire range.

12. Holonomy
Holonomy - holonomic function – solution of the linear differential equation – continuous smooth functions.
At every point of the function all derivatives are known. Thus, every point has the information about the properties of the function at the entire range.

13. Anholonomy – Non-integrability
A vector field with finite number of dimensions is defined in the space with finite number of dimensions
In the curved space a differential of the vector is not a vector!
Christoffel symbols
Connection defines parallel transport
of the vector in the curved space
Covariant derivative – affine connection

14. Adiabatic Process
A process in which external conditions change slowly No heat is gained or lost by the system Entropy is constant Adiabatic process is reversible Examples from thermodynamics: Sound wave, heat engine

15. Adiabatic Anholonomy
A vector does not to return to its initial state after being transported along a closed loop

16. Examples of Berry Phases
Foucault Pendulum
Polarized Light in Optical Fibers
Aharonov-Bohm Experiment
Modern Polarization Theory

17. Foucault Pendulum
A pendulum that changes its plane of swinging due to the rotation of the Earth around its axis
A trajectory of Foucault pendulum as it is seen from the above
Original Foucault pendulum was build in 1851

18. Foucault Pendulum - gravity force - tension - coriolis force
Newton’s equation for Foucault pendulum: ϕϕ
For small oscillations equations are linear. They can be reduced to one equation: ϕ
solution of the differential equation
ϕ-latitude

19. Foucault Pendulum Dynamical part: swinging of the pendulum
Geometric part: rotation of the plane
Berry phases arise when we can decompose motion into the slow and fast parts
Phase shift after one
complete rotation:

20. Foucault Pendulum Geometric approach to the problem
Consider local basis on the surface of the sphere Vector u is along parallel, vector v is along meridian
n = u+iv u and v – local basis vectors
Berry phase for one rotation –
solid angle:

21. Foucault Pendulum
How Berry phase will change if rotational frequency of the planet increases two times?
Answer: Berry phase will be the same because coordinates of the pendulum depend on the product If frequency increases two times, Coriolis force responsible for precession will increase two times too. In general Berry phase does not depend on the velocity of the parallel transport of the vector.

22. Polarized light
Polarized wave: electric and magnetic field oscillate in one direction Incoherent and polarized light is a random mixture of polarized waves with different polarization, wave-length and phase.

23. Polarized light
Poincare sphere: the points of the sphere show all the possible polarization states of the light
Electric field of the wave propagating in z-direction is described by two complex vectors dx and dy: ψ
There are two free parameters
that are angles and Θ φ

24. Polarized light Pancharatnam work:
Interference phenomena in absorbing biaxial crystals
Question:
How to find the phase difference between two waves with the unit intensity and with different elliptical polarization?
Polarized light travels through biaxial crystal
The optical axis are at the outmost right and left sides

25. Polarized light
Light polarization of the two states: |A> and |B> The intensity of light:
How to find δ?
δ is the area of the triangle with the sides being geodesics between the three states
Interference satisfies the requirements of the quantum adiabatic theorem ψ

26. Polarized Light in Optical Fibers
In the single mode optical fiber light can travel only one way. If the fiber is in the form of the semi-circle, let’s find the change in polarization.
Momentum sphere

27. Aharonov-Bohm Effect Electrons are passing through two slits
Diffraction pattern is formed on the screen in the absence of the solenoid

28. Aharonov-Bohm Effect
A solenoid with the current is placed just behind the slits
Uniform magnetic field is inside the solenoid
Magnetic field outside is equal to zero
But vector potential of the magnetic field is non-zero outside!

29. Aharonov-Bohm Effect
Diffraction pattern of the electrons is shifted as it is shown below! Vector potential can produce direct physics effect at the quantum scale

30. Aharonov-Bohm Effect
Vector potential changes the phase of the electron’s wave function:
Vector potential changes the phase of the wavefunction

31. Polarization Theory
Macroscopic polarization of the dielectric: Atomic polarizibility – each atom has a dipole moment: How to define a dipole moment in periodic structure?

32. Polarization Theory Localized charges -> Adiabatic flow
Polarization is adiabatic flow of current occurring as a crystal is modified or deformed
It is Berry phase of the Block wavefunctions
Application:
Ferroelectric materials – spontaneous polarization that can be reversed by external electric field
Experiments measure polarization change:
dimensionless parameter playing role of the time

33. Berry Phases in Different Areas of Physics
Quantum Hall Effect: At low temperature and strong magnetic field filling number can be either integer or fraction Astrophysical Research: “Twisting of Light around Rotating Black Hole”