Statistical Topological Insulators

We define a class of insulators with gapless surface states protected from localization due to the statistical properties of a disordered ensemble, namely due to the ensemble's invariance under a certain symmetry. We show that these insulators are topological, and are protected by a $\mathbb{Z}_2$ invariant. Finally, we prove that every topological insulator gives rise to an infinite number of classes of statistical topological insulators in higher dimensions. Our conclusions are confirmed by numerical simulations.Comment: 6 pages, 1 table, 5 figures, this is the final, published versio

Investigating Heating and Cooling in the BCS & B55 Cluster Samples

We study clusters in the BCS cluster sample which are observed by Chandra and are more distant than redshift, z>0.1. We select from this subsample the clusters which have both a short central cooling time and a central temperature drop, and also those with a central radio source. Six of the clusters have clear bubbles near the centre. We calculate the heating by these bubbles and express it as the ratio r_heat/r_cool=1.34+/-0.20. This result is used to calculate the average size of bubbles expected in all clusters with central radio sources. In three cases the predicted bubble sizes approximately match the observed radio lobe dimensions. We combine this cluster sample with the B55 sample studied in earlier work to increase the total sample size and redshift range. This extended sample contains 71 clusters in the redshift range 0<z<0.4. The average distance out to which the bubbles offset the X-ray cooling in the combined sample is at least r_heat/r_cool=0.92+/-0.11. The distribution of central cooling times for the combined sample shows no clusters with clear bubbles and t_cool>1.2Gyr. An investigation of the evolution of cluster parameters within the redshift range of the combined samples does not show any clear variation with redshift.Comment: 12 pages, 9 figures, accepted for publication in MNRA