Partial Anyon Condensation in the Color Code: A Hamiltonian Approach

Lattice Hamiltonians, which can be tuned between different topological phases, are known as important tools for understanding physical mechanism behind topological phase transitions. In this paper, we introduce a perturbed Color Code Hamiltonian with a rich phase structure which can be well matched to the mechanism of anyon condensation in the Color Code. We consider Color Code model defined on a three-colorable hexagonal lattice and add Ising interactions between spins corresponding to edges of the lattice. We show that Ising interactions play the role of physical factor for condensing anyons in the Color Code. In particular, corresponding to three different colors of edges in the hexagonal lattice, we consider three different coupling parameters. Then, we are able to condense anyons with different colors by tuning power of Ising interactions in the corresponding edges. In particular, we explicitly show that condensation of one type of anyons in the Color Code leads to a phase transition to the Toric Code state. On the other hand, by condensing two types of anyons, we observe a phase transition to a modified version of the Toric Code where partial set of anyons in the Toric Code are condensed and we call it a partially topological phase. Our main method for derivation of the above results is based on a suitable basis transformation on the main Hamiltonian in the sense that our model is mapped onto three decoupled transverse-field Ising models, corresponding to the three colors. We use the above mapping to analyze behavior of string order parameters as non-local indicators of topological order. We introduce three string order parameters that can well characterize different phases of the model. Specifically we give a simple description of the partially condensed phase by using string order parameters.