Condensation of achiral simple currents in topological lattice models: Hamiltonian study of topological symmetry breaking

We describe a family of phase transitions connecting phases of differing nontrivial topological order by explicitly constructing Hamiltonians of the Levin-Wen [Levin and Wen, Phys. Rev. B 71, 045110 (2005)] type which can be tuned between two solvable points, each of which realizes a different topologically ordered phase. We show that the low-energy degrees of freedom near the phase transition can be mapped onto those of a Potts model, and we discuss the stability of the resulting phase diagram to small perturbations about the model. We further explain how the excitations in the condensed phase are formed from those in the original topological theory, some of which are split into multiple components by condensation, and we discuss the implications of our results for understanding the nature of general achiral topological phases in 2 $+$ 1 dimensions in terms of doubled Chern-Simons theories.

Figure 1

Creating a pair of simple-current vortices on adjacent plaquettes. The operator $V_{e_{12}}^{†}\left({\phi }^n\right)$ acts on the edge-separating plaquettes $P_1$ and $P_2$, assigning a phase $e^{2\pi i\frac{n{q}_i}{Q}}$ to any component of the wave function in which this edge is labeled by $i$. If the plaquettes start without flux as shown in the figure, the result is a pair of vortices of flux ${\phi }^n$ on $P_1$ and ${\phi }^{Q-n}$ on $P_2$.

Figure 2

Effective mapping from the low-energy sector of the string-net model to the Potts model, illustrated here for the Ising case. (a) In the low-energy sector, we retain only magnetic excitations of flux ${\phi }^n$, as all other excitations remain gapped throughout the phase transition. Up to an index specifying the topological ground state sector, the relevant states are specified uniquely by assigning a Potts spin $n\in \{0\cdots Q-1\}$ to each plaquette. In this figure, an arrow pointing right (left) indicates $n=0$ (1) on that plaquette. (In the spin model we identify these with $S_x=1$ and $-1$, respectively.) (b) The terms in the second line of the Hamiltonian Eq. (7) act nontrivially on these states: the plaquette term controls the energetic cost of creating a flux, which we identify with the Potts transverse field. The flux-creation term ${\widehat{V}}_e$ gives a ferromagnetic Potts interaction in the basis of Eq. (20), indicated here by arrows pointing up (down) to denote the states $l=0$ ($l=1$) on that plaquette (or $S_z=1$ and $-1$). The eigenvalue of ${\widehat{V}}_e$ is 1 if the edge $e$ is labeled by a representation $i$ with $q_i=0$, and 0 otherwise. This indicates that any edge label $i$ with $q_i\ne 0$ (mod $Q$) signals a domain wall in the Potts model.

Figure 3

Mapping intersecting Wilson line operators to ${\mathbb{Z}}_Q$ domain walls. (a) We specify a set of Wilson lines by their contour on the lattice and their linking. Here the red line crosses once over and once under the blue line, so that the Wilson lines are linked. (b) Up to constants, which are scale independent and determined by the topological order, the effect of a pair of linked Wilson lines is to flip all ${\mathbb{Z}}_Q$ spins encircled by each Wilson line by an appropriate amount. (Here we show this for $Q=2$, where there is only one type of domain wall.) Spins encircled by two domain walls will be flipped twice.

Figure 4

Quasiparticle creation operators in the Levin-Wen model are “string” operators that act on states to create excitations at each of the string’s end points. (a) These strings can carry electric flux, in which case the string operator raises or lowers the label on each edge it runs parallel to, as shown. In this figure a string operator with label $s$ acts on an edge carrying flux $i$ to form an edge with flux $i\times s$. (b) Strings can also carry magnetic flux, in which case the operator assigns a label-dependent phase to the wave function each time the string crosses an edge separating two plaquettes. In the doubled Chern-Simons theory that we begin with, all quasiparticle strings are composed of composites of right (electric only) and left (composite electric and magnetic) components. In this figure a magnetic string operator labeled $s\times \overline{s}$ acts on an edge $i$. The net electric flux is thus the combined flux from the two $s$ operators, which in general may take on multiple values. Thus the vortex string operator is the sum of the string operator ${\sum }_{X,Y\in s\times \overline{s}}$. For each $X,Y$ in the superposition the operator acts on the edge by raising its flux by $X$ and incurring a phase, as shown on the right. This operation comes with an overall numerical prefactor not shown here. The chief importance of this prefactor is to ensure that only diagrams in which $X$ and $Y$ raise the label $i$ by the same amount occur in the superposition.

Figure 5

The $\sigma$ vortex in the doubled Ising anyon theory consists of a pair of electric sources (one right- and one left-handed). Its action on an edge is given by a phase (depicted here by the red ring) every time the string crosses between plaquettes, together with an electric component that raises the edge label by Id or $\psi$. This gives the four possibilities shown here each time the string operator crosses between plaquettes. The labels $(\mathrm{Id},\mathrm{Id})$, $(\psi ,\mathrm{Id})$, $(\mathrm{Id},\psi )$, and $(\psi ,\psi )$ denote the associated electric flux on the upper and lower sides of the crossing.

Figure 6

String operators creating topological excitations can slide freely over vertices without altering the state of the system. String operators with this property are said to obey the hexagon relation[36]; the corresponding excitations are topological because their energy is independent of the trajectory that the string takes through the lattice. String operators that are not topological leave behind a trail of excitations. After condensation, some achiral particles may split, meaning that subsets of the possible edge labels associated with a particle $a_L\times a_R$ separately obey the hexagon relation. This happens because the trail of excitations that these string operators leave behind are vortices which have condensed.