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.

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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.

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Many-Body Physics with Ultracold Gases

V P Maslov and M V Fedoriuk 1981 Holland: D Reidel vi + 301 pp price Dfl125 At last this is the English translation of the Soviet work responsible for the last decade's intense interest shown by pure mathematicians in the connections between waves and rays. Maslov's brilliant original contribution has been the conception and detailed working-out of the association between waves and certain phase–space surfaces – 'Lagrangian manifolds'.

https://iopscience.iop.org/article/10.1088/0031-9112/33/7/027/pdf

Semi-classical Approximation in Quantum Mechanics

Random renormalization in the semiclassical long-time limit of a precessing spin

Wave‐front dislocations are lines in space along which wave fronts end, named by analogy with dislocations in imperfect crystals. Continuous waves and quasimonochromatic pulses (of continuous waves with slowly varying envelopes) are conveniently represented by complex‐valued wavefunctions, the zeros of which correspond to wave‐front dislocations. Pulse dislocations move through the wavefield, and may interact, as the pulse propagates. As examples of realistic model wavefields, the soundfields of a rigid circular piston radiator vibrating in a fixed rigid baffle plane, driven both by a sinusoidal source producing continuous waves and by this source modulated by a Gaussian envelope, are computed exactly and displayed. The birth of dislocations in the nearfield and their propagation with the pulse into the farfield are observed and discussed, and birth and death events in the nearfield are displayed in detail. The behavior of quasimonochromatic pulses and their dislocations may be understood in terms of the carrier wavefield, both qualitatively and, via a perturbation theory, quantitatively.

Wave‐front dislocations in the soundfield of a pulsed circular piston radiator

In the 2 × 2 matrices representing retarders and ideal polarizers, the eigenvectors are orthogonal. An example of the opposite case, where eigenvectors collapse onto one, is matrices M representing crystal plates sandwiched between a crossed polarizer and analyser. For these familiar combinations, M2 = 0, so black sandwiches can be regarded as square roots of zero. Black sandwiches illustrate physics associated with degeneracies of non-Hermitian matrices.

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Black polarization sandwiches are square roots of zero

The wave from an isotropic point source in two and three dimensions is separated into its homogeneous (H) and evanescent (E) parts, with respect to a distinguished direction z, and the far field r θ evaluated asymptotically as a function of polar angle θ. Only in the ‘forward needle’ θ = 0 (three dimensions) and the transverse directions θ = π/2 (two and three dimensions) is the E wave of comparable strength to the H wave. Uniform asymptotic approximations (in terms of Bessel functions and Fresnel integrals) accurately interpolate between these directions of significant evanescence and all other angles (where the amplitude of E relative to H decays as 1/r1/2). The forward needle in three dimensions is analogous to glory scattering.

Michael V Berry

Papers

Semi-classical Approximation in Quantum Mechanics

Random renormalization in the semiclassical long-time limit of a precessing spin

Wave‐front dislocations in the soundfield of a pulsed circular piston radiator

Black polarization sandwiches are square roots of zero

Asymptotics of Evanescence