Skyrme and Wigner crystals in graphene

At low-energy, the band structure of graphene can be approximated by two degenerate valleys $(K,K^{\prime})$ about which the electronic spectra of the valence and conduction bands have linear dispersion relations. An electronic state in this band spectrum is a linear superposition of states from the $A$ and $B$ sublattices of the honeycomb lattice of graphene. In a quantizing magnetic field, the band spectrum is split into Landau levels with level N=0 having zero weight on the $B(A)$ sublattice for the $$ valley. Treating the valley index as a pseudospin and assuming the real spins to be fully polarized, we compute the energy of Wigner and Skyrme crystals in the Hartree-Fock approximation. We show that Skyrme crystals have lower energy than Wigner crystals \textit{i.e.} crystals with no pseudospin texture in some range of filling factor $\nu$ around integer fillings. The collective mode spectrum of the valley-skyrmion crystal has three linearly-dispersing Goldstone modes in addition to the usual phonon mode while a Wigner crystal has only one extra Goldstone mode with a quadratic dispersion. We comment on how these modes should be affected by disorder and how, in principle, a microwave absorption experiment could distinguish between Wigner and Skyrme crystals.Comment: 14 pages with 11 figure

Mountain maple and balsam fir early response to partial and clear-cut harvesting under aspen stands of northern Quebec

This study is a component of the Sylviculture et am

A Seismic Investigation of the Crust and Moho on a Line Perpendicular to the Grenville Front

A major structural feature 50 to 75 km in width has been outlined along the Moho in an area just south of the Grenville Front from a delay-time analysis of Pn waves arriving from a number of different azimuths. The crust in this anomalous region was found to be from 5–10 km thicker than in the surrounding area. The low frequency portion of the gravity low in central Quebec can be attributed to this feature. The fact that the front is deep-seated suggests that it may mark the position of an ancient fault connected tectonically in origin to the mid-continent gravity high and Lake Superior syncline.During the summer of 1968, the University of Western Ontario participated with the Dominion Observatory of Canada and other research groups in a crustal survey of eastern Quebec. Shots of various sizes were fired by the Dominion Observatory at six different locations in central Quebec. Two of the locations were in the Superior (structural) Province, two in the Grenville (structural) Province, and two in the vicinity of the Grenville Front. The University of Western Ontario recording stations were located at distances from 60 to 800 km southwest of the shots and lay along (a) a 550-km long line running perpendicular to the Grenville Front from Lake Evans in the Superior Province to a point just south of Lake St. John in the Grenville Province, (b) a 100-km long line running parallel to the Grenville Front from Chibougamau to the northeastern end of Lake Mistassini. Six portable seismic recorders (two of them 3-component) were moved every day. A total of 45 locations were occupied, with observations being made in a fan-like manner.The apparent crust and upper mantle velocities as determined by a least squares analysis of all the data are 6.43 ±.03 km/s (43 observations) and 8.15 ±.03 km/s (70 observations) respectively. From an analysis involving combined seismic and gravitational observations along with synthetic seismograms, the following structure emerges. The mean velocity of the upper crust is 6.31 km/s, but because of the lateral variations in its structure the true velocity varies from 6.1 to 6.6 km/s, the higher values occurring just south of the Grenville Front. No clear evidence for layering could be found throughout the crust, but both seismic and gravitational observations indicated that the mean velocity of the lower portion of the crust was 6.55 km/s with this value increasing to at least 7 km/s just above the Moho. The apparent velocities of smaller regional samples of Pn observations varied erratically as a result of topography on the Moho. However values obtained from arrays oriented parallel to the Grenville Front were significantly higher, indicating that the upper mantle may be behaving in an anisotropic manner. </jats:p