Exact description of the boundary theory of the Kitaev toric code with open boundary conditions

In this paper, we consider the Kitaev toric code with specific open boundary conditions. Such a physical system has a highly degenerate ground state determined by the degrees of freedom localized at the boundaries. We can write down an explicit expression for the ground state of this model. Based on this, the entanglement properties of the model are studied for two types of bipartition: one, where subsystem $A$ is completely contained in $B$; and the second, where the boundary of the system is shared between $A$ and $B$. In the former configuration, the entanglement entropy is the same as for the periodic boundary condition case, which means that the bulk is completely decoupled from the boundary on distances larger than the correlation length. In the latter, deviations from the torus configuration appear due to the edge states and lead to an increase of the entropy. We then determine an effective theory for the boundary of the system. In the case where we apply a small magnetic field as a perturbation, the degrees of freedom on the boundary acquire a dispersion relation. The system can there be described by a Hamiltonian of the Ising type with a generic spin-exchange term.

Figure 1

In this figure, we present the boundary conditions we are going to use. In (a), only star operators act on the boundary of the system, while in (b), spins on the boundary are constrained only by plaquette operators. The latter configuration, (b), is going to be studied in our paper. Despite the fact that these two look quite different, they are connected by a gauge transformation.

Figure 2

Representation of the operator ${\mathcal{W}}_i$ presented in Eq. (9) as a small path connecting two nearest–neighbor sites. We have a freedom of choice for the orientation of the boundary, which is a one–dimensional system with periodic boundary conditions, so it can be mapped into a circle.

Figure 3

In this figure, we present all the plaquette configurations present in the ground state. The red dots represent spins up, while the plain line represents a spin down. As long as we always have an even amount of spins up, the eigenvalue of the plaquette operator is always positive.

Figure 4

In this figure, we present how the star operator ${\widehat{A}}_s$ acts on some spin configurations. One can see how it is possible to come back to the initial state applying the same star operator twice.

Figure 5

In this figure, we want to present a typical boundary state of our model, where we use the boundary conditions of Fig. 1. One can see how a spin flip along a line connecting the two boundaries creates a vertex-free state where all the plaquettes have eigenvalue $+1$. In the bulk, the star operators act modifying the path connecting the two points; see, for example, (b), but they do not modify the points on the boundary.

Figure 6

A generic boundary state which forms a particular basis element of the ground state. Because of the action of star operators, the path connecting the boundary points is not important, so we can choose by convention the one closest to the boundary. Even if this state looks extremely complicated, this and all the others can be constructed starting from the building block presented in Eq. (9).

Figure 7

Here we give a graphical representation of Eq. (10). The ${\mathcal{W}}_i$ operators can be multiplied to create all the possible configurations on the boundary system.

Figure 8

Left panel: Partition of the full system into two subsystems $A$ and $B$ sharing the boundary of the full system with OBCs. Right panel: Degrees of freedom of a star, in this case, shared between the $A$ and $B$ subsystems.

Figure 9

Macroscopic tunneling effect on the boundary theory due to the magnetic field. This perturbation allows the excitations on the boundary to travel with dispersion relations that can be computed using perturbation theory at the leading order in $h_x/J_m$. For small magnetic field, the transitions are just nearest–neighbor because the energy needed is exponential in the number of spin flips. As the intensity of the perturbation is increased, the tunneling can affect more distant sites, meaning that we have a “nonlocal” Hamiltonian.

Figure 10

Pictorial representation of the tunneling process of the boundary theory induced by the magnetic field ${\widehat{H}}_I$. For this scheme, it is also possible to get the leading order in the same process at distance $R$, Eq. (48).

Figure 11

The dispersion relation of the boundary theory for different values of $\lambda$ compared to a pure cosine. To have a better scale, we traced the dispersion relations in unit of $2h_x{\lambda }^2$. In the upper panel: dispersion relation for $h=0.01$ and $J_m=1$ compared to a pure cosine. It is possible to see that the difference between these two is almost zero, meaning that our theory can be understood using a nearest-neighbor model. In the lower panel, we have the same dispersion relation for $h=0.2$ and $j_m=1$, $\lambda =-0.1$, compared again to a cosine function. The deviation from a pure nearest–neighbor hopping is more evident in this case and a more “long-range” physics is taking place.