Gauging defects in quantum spin systems: A case study

The goal of this work is to build a dynamical theory of defects for quantum spin systems. This is done by explicitly giving an exhaustive case study of a one-dimensional spin chain with $\mathbf{Vec}(\mathbb{Z}/2\mathbb{Z})$ fusion rules, which can easily be extended to more general settings. A kinematic theory for an indefinite number of defects is first introduced exploiting distinguishable Fock space. Dynamics are then incorporated by allowing the defects to become mobile via a microscopic Hamiltonian. This construction is extended to topologically ordered systems by restricting to the ground state eigenspace of Hamiltonians generalizing the golden chain. Technically, this is done by employing generalized tube algebra techniques to model the defects in the chain. We illustrate this approach for the $\mathbf{Vec}(\mathbb{Z}/2\mathbb{Z})$ spin chain, in whose case the resulting dynamical defect model is equivalent to the critical transverse Ising model.

Figure 1

Depiction of different situations where missing qubit defects are inserted on a toric code lattice. In the case of more than one missing link the ground space for the system depends on whether the defects are adjacent to each other.