One-dimensional non-interacting topological insulators with chiral symmetry

We construct microscopical models of one-dimensional non-interacting topological insulators in all of the chiral universality classes. Specifically, we start with a deformation of the Su-Schrieffer-Heeger (SSH) model that breaks time-reversal symmetry, which is in the AIII class. We then couple this model to its time-reversal counterpart in order to build models in the classes BDI, CII, DIII and CI. We find that the ℤ topological index (the winding number) in individual chains is defined only up to a sign. This comes from noticing that changing the sign of the chiral symmetry operator changes the sign of the winding number. The freedom to choose the sign of the chiral symmetry operator on each chain independently allows us to construct two distinct possible chiral symmetry operators when the chains are weakly coupled -- in one case, the total winding number is given by the sum of the winding number of individual chains while in the second case, the difference is taken. We find that the chiral models that belong to ℤ classes, AIII, BDI and CII are topologically equivalent, so they can be adiabatically deformed into one another so long as the chiral symmetry is preserved. We study the properties of the edge states in the constructed models and prove that topologically protected edge states must all be localised on the same sublattice (on any given edge). We also discuss the role of particle-hole symmetry on the protection of edge states and explain how it manages to protect edge states in ℤ2 classes, where the integer invariant vanishes and chiral symmetry alone does not protect the edge states anymore. We discuss applications of our results to the case of an arbitrary number of coupled chains, construct possible chiral symmetry operators for the multiple chain case, and briefly discuss the generalisation to any odd number of dimensions

Employer-Provided Health Insurance as a Potential Deterrent to Entrepreneurship

The phenomenon of job-lock in the United States may be caused by a major non-portable fringe benefit provided by employers: health insurance. Would-be entrepreneurs and other self-employed individuals may not be achieving their full potential due to being “locked” in their wage-employment. With data from the Survey of Consumer Finances in years 2004, 2007, and 2009, this study explores this effect, whether it exists, and whether it is lessened by worse health status. Amongst married households, there is evidence that husbands are 9.2% more likely to be entrepreneurs if their spouses have employer coverage, whereas wives are not. Somewhat surprisingly, this effect is not associated with health care demand. Amongst non-married individuals, employer coverage restricts transitions into self-employment by 3.6%. Both of these results provide evidence for job lock, and have loose implications on how universal healthcare may free individuals to pursue entrepreneurship