This article reviews the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlled by application of external electric and magnetic fields, or by altering sample geometry and/or topology. The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers. Edge (surface) states in graphene depend on the edge termination (zigzag or armchair) and affect the physical properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.

/pdf/the-electronic-properties-of-graphene-2siyzsnf5w.pdf

The electronic properties of graphene

A quantal system in an eigenstate, slowly transported round a circuit C by varying parameters R in its Hamiltonian Ĥ(R), will acquire a geometrical phase factor exp{iγ(C)} in addition to the familiar dynamical phase factor. An explicit general formula for γ(C) is derived in terms of the spectrum and eigenstates of Ĥ(R) over a surface spanning C. If C lies near a degeneracy of Ĥ, γ(C) takes a simple form which includes as a special case the sign change of eigenfunctions of real symmetric matrices round a degeneracy. As an illustration γ(C) is calculated for spinning particles in slowly-changing magnetic fields; although the sign reversal of spinors on rotation is a special case, the effect is predicted to occur for bosons as well as fermions, and a method for observing it is proposed. It is shown that the Aharonov-Bohm effect can be interpreted as a geometrical phase factor.

Quantal phase factors accompanying adiabatic changes

Quantum cryptography could well be the first application of quantum mechanics at the individual quanta level. The very fast progress in both theory and experiments over the recent years are reviewed, with emphasis on open questions and technological issues.

Quantum Cryptography

Conventional optical components rely on gradual phase shifts accumulated during light propagation to shape light beams. New degrees of freedom are attained by introducing abrupt phase changes over the scale of the wavelength. A two-dimensional array of optical resonators with spatially varying phase response and subwavelength separation can imprint such phase discontinuities on propagating light as it traverses the interface between two media. Anomalous reflection and refraction phenomena are observed in this regime in optically thin arrays of metallic antennas on silicon with a linear phase variation along the interface, which are in excellent agreement with generalized laws derived from Fermat’s principle. Phase discontinuities provide great flexibility in the design of light beams, as illustrated by the generation of optical vortices through use of planar designer metallic interfaces.

http://www.zenogaburro.it/papers/LightPropagation.pdf

Light Propagation with Phase Discontinuities: Generalized Laws of Reflection and Refraction

This paper reviews recent experimental and theoretical progress concerning many-body phenomena in dilute, ultracold gases. It focuses on effects beyond standard weak-coupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, or lowest-Landau-level physics in quasi-two-dimensional gases in fast rotation. Strong correlations in fermionic gases are discussed in optical lattices or near-Feshbach resonances in the BCS-BEC crossover.

/pdf/many-body-physics-with-ultracold-gases-552s10zfdn.pdf

Many-Body Physics with Ultracold Gases

In the helicity representation, the Poynting vector (current) for a monochromatic optical field, when calculated using either the electric or the magnetic field, separates into right-handed and left-handed contributions, with no cross-helicity contributions. Cross-helicity terms do appear in the orbital and spin contributions to the current. But when the electric and magnetic formulas are averaged ('electric-magnetic democracy'), these terms cancel, restoring the separation into right-handed and left-handed currents for orbital and spin separately.

Note on the helicity decomposition of spin and orbital optical currents

Review of Would-be worlds: how simulation is changing the frontiers of science by Casti, J

Topology in Ordered Phases

Quantum chaology, prime numbers and Riemann's zeta function

On a ring threaded by a flux line moves a charged quantum particle whose rotation is hindered by an angle-dependent potential. When the potential is rigidly rotated through , the particle (in the nth eigenstate of the potential on the ring) acquires a geometric phase . A general formula is , where is the dimensionless quantum flux and is a Schrodinger current associated with the state. Properties of are obtained in terms of the transmission coefficient round the ring. vanishes when the box is impenetrable or when is integer or half-integer. Energy levels form bands, with playing the role of Bloch pseudomomentum. For unhindered semiclassical states above the barrier, a WKB theory gives the geometric phase and hence the (previously calculated) classical Hannay angle . This does not vanish when is integer or half-integer, but these cases correspond to band edges where the semiclassical states are degenerate and differ greatly from the true asymptotic states. The theory is illustrated by the exact calculation of for a model where the potential is a delta function.

/pdf/aharonov-bohm-geometric-phases-for-rotated-rotators-2jcurtbc30.pdf

Michael V Berry

Papers

Note on the helicity decomposition of spin and orbital optical currents

Review of Would-be worlds: how simulation is changing the frontiers of science by Casti, J

Topology in Ordered Phases

Quantum chaology, prime numbers and Riemann's zeta function

Aharonov - Bohm geometric phases for rotated rotators