Self-duality and bound states of the toric code model in a transverse field

We investigate the effect of a transverse magnetic field on the toric code model. We show that this problem can be mapped onto the Xu-Moore model and thus onto the quantum compass model, which are known to be self-dual. We analyze the low-energy spectrum by means of perturbative continuous unitary transformations and determine accurately the energy gaps of various symmetry sectors. Our results are in very good agreement with exact diagonalization data for all values of the parameters except at the self-dual point where level crossings are responsible for a first-order phase transition between a topological phase and a polarized phase. Interestingly, bound states of two and four quasiparticles with fermionic and bosonic statistics emerge, and display dispersion relations of reduced dimensionality.

Figure 1

(Color online) Original square lattice on which plaquettes $p$ and stars $s$ are defined. Big red (small black) dots define the lattice $\Lambda$ $\left(\tilde{\Lambda }\right)$. Here, we show the lattice with $N=18$ spins and (implicit) periodic boundary conditions. Contour ${\mathcal{C}}_1$ $\left({\mathcal{C}}_2\right)$ is one of the diagonal (antidiagonal) cycles used to define conserved parity operators.

Figure 2

(Color online) Ground-state energy per spin $e_0$ obtained from PCUT and ED. The vertical dashed line marks the transition at $\theta =\pi /4$. Inset: magnetization $m_y=-{\partial }_{h_y}{e}_0$.

Figure 3

(Color online) Comparison between PCUT and ED results $\left(N=32\right)$ for the 1QP gap ${\Delta }_1$ and lowest 2QP gaps ${\Delta }_2^{\text{a},\text{b},\text{c}}$. Pictograms give an illustration of the four corresponding states (with $\pi /4$-tilted lattice compared to Fig. 1). Crosses denote particles and filled circles empty sites.

Figure 4

(Color online) Lowest 4QP gap ${\Delta }_4$ obtained from PCUT and ED. Pictograms illustrate four 4QP configurations, with different relative positions.