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

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 1980 - … TIME CRYSTAL Frank Wilczek

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

Berry Phase Geometric Phase Adiabatic Anholonomy

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:

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

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

Adiabatic Anholonomy

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

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.

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.

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.

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

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

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

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

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

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

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:

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:

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.

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.

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

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

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

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

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

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!

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

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

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

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

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” https://arxiv.org/pdf/1104.3099.pdf