Low-energy effective theory of the toric code model in a parallel magnetic field

We determine analytically the phase diagram of the toric code model in a parallel magnetic field which displays three distinct regions. Our study relies on two high-order perturbative expansions in the strong- and weak-field limits, as well as a large-spin analysis. Calculations in the topological phase establish a quasiparticle picture for the anyonic excitations. We obtain two second-order transition lines that merge with a first-order line, giving rise to a multicritical point as recently suggested by numerical simulations. We compute the values of the corresponding critical fields and exponents that drive the closure of the gap. We also give the one-particle dispersions of the anyonic quasiparticles inside the topological phase.

Figure 1

(Color online) A piece of the square lattice on which the TCM is defined. Spins (dots) are located on the bond and interact if they share either a vertex $\left(s\right)$ or a plaquette $\left(p\right)$.

Figure 2

(Color online) One-flux (up) and one-charge (down) dispersions for $h_z=0.1$, $h_x=0.05$, and $J=1/2$ in the reciprocal lattice. In this case $\left(h_x<h_z\right)$, the gap is given by the minimum of the one-charge dispersion [$\Delta \simeq 0.54$ as can be checked from Eq. (9)].

Figure 3

(Color online) Phase diagram in the plane $\left(h_x,h_z\right)$ for $J=1/2$. Second-order transition lines are full (red), the first-order transition line is dashed (blue), the multicritical point is a full circle, and the critical point is represented as an empty circle. Phases I, II, and III are discussed in the text.