Surface magnetization and critical behavior of a hierarchical quantum Ising chain

We study the surface magnetization and critical behavior of a hierarchical quantum Ising chain. The surface magnetization exponent ${\mathrm{\beta }}_{\mathit{s}}$ is calculated exactly by using scaling arguments. It is found that, although the size of the fluctuations in the exchange couplings may be bounded or unbounded depending on the value of the hierarchical parameter r, the surface critical behavior stays nonuniversal with ${\mathrm{\beta }}_{\mathit{s}}$ varying continuously with the value of r. This result is rather different from the cases where the aperiodicity of the couplings is generated by substitution rules and different types of critical behavior are observed for bounded and unbounded fluctuations in the couplings.