The bulk electric polarization $P$ of one-dimensional crystalline insulators is defined modulo a polarization quantum $P_q$. The latter is a measurable quantity that depends on the number $n_s$ of sites per unit cell. For two-band models, $n_s=1$ or
$2$ and $P_q=g/n_s$ ($g=1$ or $2$ being the spin degeneracy). For inversion-symmetric crystals either $P=0$ or $P_q/2$ mod $P_q$. Depending on the position of the two inversion centers with respect to the ions, three situations arise: bond, site or mixed inversion. As representative two-band examples of these three cases, we study the Su-Schrieffer-Heeger (SSH), charge density wave (CDW) and Shockley models. SSH has a unique phase with $P=0$ mod $g/2$, CDW has a unique phase with $P=g/4$ mod $g/2$, and Shockley has two distinct phases with $P=0$ or $g/2$ mod $g$. In all three cases, as long as inversion symmetry is present, chiral symmetry is found to be irrelevant for $P$. As a generalization of SSH and CDW, we analytically compute $P$ for the RM model and illustrate the role of the unusual $P_q=g/2$ on edge and soliton fractional charges and on adiabatic pumping.
This paper provides a pedagogical introduction to recent developments in geometrical and topological band theory following the discovery of graphene and topological insulators. Amusingly, many of these developments have a connection to contributions
in high-energy physics by Dirac. The review starts by a presentation of the Dirac magnetic monopole, goes on with the Berry phase in a two-level system and the geometrical/topological band theory for Bloch electrons in crystals. Next, specific examples of tight-binding models giving rise to latti
We study the Wegner-Wilson loops in the string-net model of Levin and Wen in the presence of a string tension. The latter is responsible for a phase transition from a topological deconfined phase (weak tension) to a trivial confined phase (strong ten
sion). We analyze the behavior of all Wegner-Wilson loops in both limiting cases for an arbitrary input theory of the string-net model. Using a fluxon picture, we compute perturbatively the first contributions to a perimeter law in the topological phase as a function of the quantum dimensions. In the trivial phase, we find that Wegner-Wilson loops obey a modified area law, in agreement with a recent mean-field approach.
In two dimensions, topological phases of free Majorana fermions coupled to a $mathbb{Z}_2$ gauge field are known to be classified according to the Chern number $ u in mathbb{Z}$. Its value mod 16 specifies the type of anyonic excitations. In this pap
er, we investigate triangular vortex configurations (and their dual) in the Kitaev honeycomb model and show that fourteen of these sixteen phases can be obtained by adding a time-reversal symmetry-breaking term. Missing phases are $ u=pm 7$. More generally, we prove that any periodic vortex configuration with an odd number of vortices per geometric unit cell can only host even Chern numbers whereas odd Chern numbers can be found in other cases.
We study the energy spectrum of a two-dimensional electron in the presence of both a perpendicular magnetic field and a potential. In the limit where the potential is small compared to the Landau level spacing, we show that the broadening of Landau l
evels is simply expressed in terms of the structure factor of the potential. For potentials that are either periodic or random, we recover known results. Interestingly, for potentials with a dense Fourier spectrum made of Bragg peaks (as found, e.g., in quasicrystals), we find an algebraic broadening with the magnetic field characterized by the hyperuniformity exponent of the potential. Furthermore, if the potential is self-similar such that its structure factor has a discrete scale invariance, the broadening displays log-periodic oscillations together with an algebraic envelope.
The electrodynamics of a two-dimensional gas of massless fermions in graphene is studied by a collisionless hydrodynamic approach. A low-energy dispersion relation for the collective modes (plasmons) is derived both in the absence and in the presence
of a perpendicular magnetic field. The results for graphene are compared to those for a standard two-dimensional gas of massive electrons. We further compare the results within the classical hydrodynamic approach to the full quantum mechanical calculation in the random phase approximation. The low-energy dispersion relation is shown to be a good approximation at small wave vectors. The limitations of this approach at higher order is also discussed.
We review different scenarios for the motion and merging of Dirac points in two dimensional crystals. These different types of merging can be classified according to a winding number (a topological Berry phase) attached to each Dirac point. For each
scenario, we calculate the Landau level spectrum and show that it can be quantitatively described by a semiclassical quantization rule for the constant energy areas. This quantization depends on how many Dirac points are enclosed by these areas. We also emphasize that different scenarios are characterized by different numbers of topologically protected zero energy Landau levels
We present a theoretical description of Bernstein modes that arise as a result of the coupling between plasmon-like collective excitations (upper-hybrid mode) and inter-Landau-level excitations, in graphene in a perpendicular magnetic field. These mo
des, which are apparent as avoided level crossings in the spectral function obtained in the random-phase approximation, are described to great accuracy in a phenomenological model. Bernstein modes, which may be measured in inelastic light-scattering experiments or in photo-conductivity spectroscopy, are a manifestation of the Coulomb interaction between the electrons and may be used for a high-precision measurement of the upper-hybrid mode at small non-zero wave vectors.
We study collective electronic excitations in graphene in the integer quantum Hall regime, concentrating mainly on excitations with spin reversal such as spin-flip and spin-wave excitations. We show that these excitations are correctly accounted for
in the time-dependent Hartree-Fock and strong magnetic field approximations, in contrast to spin-conserving (magneto-exciton) modes which involve a strong Landau-level mixing at non-zero wave vectors. The collective excitations are discussed in view of prominent theorems, such as Kohns and Larmors. Whereas the latter remains valid in graphene and yields insight into the understanding of spin-dependent modes, Kohns theorem does not apply to relativistic electrons in graphene. We finally calculate the exchange correction to the chemical potential in the weak magnetic field limit.
The particle-hole excitation spectrum for doped graphene is calculated from the dynamical polarizability. We study the zero and finite magnetic field cases and compare them to the standard two-dimensional electron gas. The effects of electron-electro
n interaction are included within the random phase approximation. From the obtained polarizability, we study the screening effects and the collective excitations (plasmon, magneto-excitons, upper-hybrid mode and linear magneto-plasmons). We stress the differences with the usual 2DEG.
