I have recently started working on a new project, building upon some previous work I did with Professor Jarah Evslin and Andy Royston. In the previous works,we discussed the question: If a vacuum sector Hamiltonian is regularized by an energy cutoff, how is the one-kink sector Hamiltonian regularized? We find that it is not regularized by an energy cutoff, indeed normal modes of all energies are present in the kink Hamiltonian, but rather the decomposition of the field into normal mode operators yields coefficients which lie on a constrained surface that forces them to become small for energies above the cutoff. This explains the old observation that an energy cutoff of the kink Hamiltonian leads to an incorrect one-loop kink mass. We impose that the regularized kink sector Hamiltonian is unitarily equivalent to the regularized vacuum sector Hamiltonian. This condition implies that the two regularized Hamiltonians have the same spectrum and so guarantees that the kink Hamiltonian yields the correct kink mass.

In our new project, we will try to generalized this method to models beyond $1+1$ dimensional $\phi^4$ model! At the same time, perhaps develop a better understanding of how renormalization works between different sectors from a categorical viewpoint. My guess is that it can be considered as a groupoid fibered at different topological sectors.

A detailed introduction to the old problem of finding the kink-mass correction can be found here, it is a note I put together a long time ago.