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

TL;DR

The paper addresses how phase transitions between distinct topological orders in 2+1 dimensions can be realized and understood. By deforming exactly solvable Levin-Wen lattice Hamiltonians to condense an achiral simple current with $\bZ_Q$ symmetry, the authors map the critical behavior to a $Q$-state transverse-field Potts model, enabling controlled analysis of universality and excitation fate. The condensed phase exhibits confinement, identification, and sometimes splitting of excitations, with a second solvable point providing a concrete description of the final spectrum; the results connect these transitions to doubled Chern-Simons theories and, in many cases, to Drinfeld doubles. The work offers a general framework to relate topological symmetry breaking to Landau-like critical behavior and provides concrete constructions and examples (e.g., $SU(2)_2$) that illuminate the spectrum and operator content of the condensed phases.

Abstract

We describe a family of phase transitions connecting phases of differing non-trivial topological order by explicitly constructing Hamiltonians of the Levin-Wen[PRB 71, 045110] 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.

Figures (6)

• Figure 1: Creating a pair of simple-current vortices on adjacent plaquettes. The operator $V^\dag_{e_{12}} (\phi^n)$ 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 $n \in \{ 0 . . . Q -1 \}$ 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. (\ref{['Eq_HFull']}) act non-trivially 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 $\hat{V}_e$ gives a ferromagnetic Potts interaction in the basis of Eq. (\ref{['Eq_PottsBasis']}), 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 $\hat{V}_e$ is $1$ if the edge $e$ is labeled by a representation $i$ with $q_i =0$, and $0$ otherwise -- indicating that any edge label $i$ with $q_i \neq 0$ (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: Quasi-particle creation operators in the Levin-Wen model are 'string' operators which act on states so as to create excitations at each of the string's endpoints. (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 quasi-particle 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 pre-factor not shown here. The chief importance of this pre-factor 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 which 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 $(Id,Id), (\psi, Id), (Id, \psi)$, and $(\psi, \psi)$ denote the associated electric flux on the upper and lower sides of the crossing.