Ground states of abelian quantum double models

A new preprint (with Matthew Cha and Bruno Nachtergaele) is available on the arXiv: The complete set of infinite volume ground states for Kitaev’s abelian quantum double models. In this paper we consider abelian quantum double models, realised as a local quantum spin system by Kitaev’s description for any finite abelian group $G$.¹ This includes the well-known toric code model, which corresponds to a choice of $G = \mathbb{Z}_2$ for the abelian group.

We consider the model on an infinite $\mathbb{Z}^2$ lattice, with edges between nearest neigbours. Each edge contains a quantum spin system of dimension $\vert G\vert$. Dynamics are implemented by commuting local projectors, acting on stars (the four edged coming out of a vertex), and plaquettes (the four edges making up a square in the lattice). These projectors detect the absence of a charge at a star or plaquette. There is a unique frustration free ground state (in the thermodyamic limit).² However, classifying all ground states is much harder: to find all ground states, one has to find all states that satisfy

Here $\delta(A) = \lim_{\Lambda \to \infty} [H_\Lambda, A]$, with $A$ a local observable, is the generator obtained from the local Hamiltonians. There are many results on the existence and structural properties of the set of ground states of quantum spin models, but surprisingly few results find the complete ground state space of concrete models. In fact, as far as we are aware, our result is the first example in two dimensions.

It turns out that the set of ground states is related to the charge structure of the model. It is possible, using local operators, to create pairs of excitations (or charges) from any state. These excitations are labelled by $(\chi,c)$, where $\chi$ is a character of $G$ and $c \in G$. To find the total charge, one just has to take the product of all the charges. The local operators create a pair of inverse charges at each end, so they do not change the total charge of the system. One can, however, send of one of the charges to infinity, leaving a state with a single charge. These states lead to inequivalent representations or superselection sectors.³

It turns out that these states are ground states, and in fact are essentially all possible ground states. Our main result can be summarised as follows:

Theorem Let $G$ be a finite abelian group. Then each ground state can be written as a convex combination of at most $\vert G \vert^2$ disjoint states. The pure states in each of these sectors are equivalent to a charged stated, whose equivalence classes are labelled by $(\chi, c)$ for $\chi$ a character of $G$, and $c \in G$.

There are two important ideas behind our proof: (i) we can define “local” charge projections, that project onto all states with a specific total charge in a region $\Lambda$, and (ii) a specific choice of boundary conditions, so that states with a single excitation in the bulk are ground states of the local Hamiltonian with these boundary conditions.

Then, by taking any ground state, we can make a sequence of finite volume ground states (with the appropriate boundary conditions), by projecting onto the different charged sectors in a box of size $L \times L$. We show that this limit converges in the weak$*$-topology to a ground state, and that if we take the right convex combination of these states, we get our original state back. Then, to conclude the proof, we can consider a pure state in one of the sectors. By using some topological properties of the model, in particular that there is some freedom in choosing paths necessary to define the local operators that create a pair of charges, it is possible to show that the expectation values of local operators that are sufficiently far away, are indistinguishable from the expectation value in the charged states that can be constructed explicitly. It follows that the states must be equivalent, in the sense that their GNS representations are unitarily equivalent.

Footnotes