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

The transmitted field is calculated for a cascade of N biaxial crystals, with their optic axes parallel but with arbitrary orientations about this axis, and arbitrary strengths. The focused pattern is the superposition of 2N − 1 single-crystal concentric conical diffraction patterns, whose ring radii are combinations of those from the individual crystals in the cascade. Explicit expressions are given for the general case of arbitrary incident polarization and for N = 1–4, and for the mean intensity for a cascade with random orientations and strengths and arbitrary N. For circular incident polarization, the ring pattern is rotationally symmetric for all N, but for N > 1 and away from focus the patterns for left and right circular incident polarizations are different.

https://michaelberryphysics.files.wordpress.com/2013/07/berry423.pdf

Conical diffraction from an N-crystal cascade

For time-dependent two-state quantum systems, the transition probability is exponentially small in the adiabatic parameter in , with the exponent determined by a transition point tc in the complex time plane. The authors study the in -independent prefactors associated with different sorts of transition point (which need not correspond to complex degeneracies of the adiabatic energy). Unlike previous approaches the method they use does not make use of special functions. It consists of applying first-order perturbation theory to the Schrodinger equation obtained by transforming to a series of 'superadiabatic' bases clinging ever more closely to the evolving state. If the original matrix elements share a leading singularity (t-tc)r, and their fractional deviation from this is (t-tc)s, the prefactor is 4 sin2(pi s/2(2r+s+2)). This is universal in the sense of being invariant under time reparametrization and quantum changes of frame.

/pdf/universal-transition-prefactors-derived-by-superadiabatic-2if6tl5jep.pdf

Universal transition prefactors derived by superadiabatic renormalization

The moment of conception of the geometric phase can be pinpointed precisely, but related ideas had been formulated before, in various guises. Not less varied were the ramifications that became clear once the concept was identified formally.

https://www.fefox.com/ARTICLES/Berry%20phase%20history%202010.pdf

Geometric phase memories

Transparent overhead-projector foil is an anisotropic material with three different principal refractive indices. Its properties can be demonstrated very simply by sandwiching the foil between crossed polarizers and looking through it at any diffusely lit surface (e.g., the sky). Coloured interference fringes are seen, organized by a pattern of rings centred on two 'bullseyes' in the directions of the two optic axes. The fringes are difference contours of the two refractive indices corresponding to propagation in each direction, and the bullseyes are degeneracies where the refractive-index surfaces intersect conically. Each bullseye is crossed by a black 'fermion brush' reflecting the sign change (geometric phase) of each polarization in a circuit of the optic axis. Simple observations lead to the determination of the three refractive indices, up to an ordering ambiguity.

/pdf/black-plastic-sandwiches-demonstrating-biaxial-optical-4jhtwalms0.pdf

Black plastic sandwiches demonstrating biaxial optical anisotropy

The force on a particle with complex electric polarizability is known to be not derivable from a potential, so its curl is non-zero. This ‘curl force’ is studied in detail for motion near an anisotropic optical vortex of arbitrary strength. Fundamental questions are raised by the fact that although the curl force requires the polarizability to have a non-zero imaginary part, reflecting absorption or scattering (‘dissipation’) in the internal dipole dynamics, the particle motion that it generates is non-dissipative (volume-preserving in the position-velocity state space).

/pdf/physical-curl-forces-dipole-dynamics-near-optical-vortices-euhr3499lg.pdf

Michael V Berry

Papers

Conical diffraction from an N-crystal cascade

Universal transition prefactors derived by superadiabatic renormalization

Geometric phase memories

Black plastic sandwiches demonstrating biaxial optical anisotropy

Physical curl forces: dipole dynamics near optical vortices