Adiabatic Preparation of Topological Order

Topological order characterizes those phases of matter that defy a description in terms of symmetry and cannot be distinguished in terms of local order parameters. Here we show that a system of $n$ spins forming a lattice on a Riemann surface can undergo a second order quantum phase transition between a spin-polarized phase and a string-net condensed phase. This is an example of a quantum phase transition between magnetic and topological order. We furthermore show how to prepare the topologically ordered phase through adiabatic evolution in a time that is upper bounded by $O(\sqrt{n})$. This provides a physically plausible method for constructing and initializing a topological quantum memory.

Figure 1

The square lattice on a torus. Closed $x$ strings [solid (red) lines] commute with the star operator $A_s$ because they have an even number of links in common. The elementary closed $x$ string is the plaquette $B_p$. Dashed $z$ strings connect plaquettes $p$. The two incontractible $x$ strings are denoted $t_{1,2}$. The dual string variable [dashed (blue) lines] connects the reference lines $t_1$, $t_2$ with the vertices $p$ in the dual lattice. Thus the dual operator ${\mu }_{\to (\uparrow )}^z(p)$ anticommutes with the plaquette variable ${\mu }^x(p)\equiv B_p$.