Anomalies and Renormalization of Impure States in Quantum Theories

In a Hamiltonian approach to anomalies parity and time reversal symmetries can be restored by introducing suitable impure (or mixed) states. However, the expectation values of observables such as the Hamiltonian diverges in such impure states. Here we show that such divergent expectation values can be treated within a renormalization group framework, leading to a set of $\beta$-functions in the moduli space of the operators representing the observables. This leads to well defined expectation values of the Hamiltonian in a phase where the impure state restores the $P$ and $T$ symmetry. We also show that this RG procedure leads to a mass gap in the spectrum. Such a framework may be relevant for long wavelength descriptions of condensed matter systems such as the quantum spin Hall effect.