Conformal energy currents on the edge of a topological superconductor

The boundary of a two-dimensional topological superconductor can be modeled by a conformal field theory. Here we demonstrate the behaviors of this high level description emerging from a microscopic model at finite temperatures. To achieve that, we analyze the low-energy sector of Kitaev's honeycomb lattice model and probe its energy current. We observe that the scaling of the energy current with temperature reveals the central charge of the conformal field theory, which is in agreement with the Chern number of the bulk. Importantly, these currents can discriminate between distinct topological phases at finite temperatures. We assess the resilience of this measurement of the central charge under coupling disorder, bulk dimerization, and defects at the boundary, thus establishing it as a favorable means of experimentally probing topological superconductors.

Figure 1

Dispersion relation of the edge modes (left) that are exponentially localized on opposite ends of the cylinder (right) moving in opposite directions. These modes are witnessed as two midgap bands that cross at $p=\pi$. The energy gap, $\mathrm{\Delta }$, of the model and the discretization gap, $\delta \varepsilon$, due to the finite size $L$ of the system are shown.

Figure 2

Low and high $T$ behavior of $I\left(T\right)$. Left: In the low-$T$ case increasing system size, $L$, recovers the $T^2$ behavior at lower temperatures. Right: In the high-$T$ case increasing $K\propto \mathrm{\Delta }$ delays the divergence of the curves. The left panel includes the $I_0$ offset, whereas we have removed it from the right panel. In both plots $J=1$. In the left plot $K=0.15$ and in the right plot $L=60$.

Figure 3

Edge current for the no-vortex sector (NV), full-vortex sector (FV), and toric code phases (TC) as a function of $T^2$. The currents are shifted to pass through the origin, clearly revealing the $T^2$ scaling. The central charges correspond to dotted line for $c=0$, dashed line for $c=1/2$, and dashed-dotted line for $c=1$. In these plots we set $K=0.15$. For the NV and FV plots we set all $J=1$, and for the TC plot we take $J=0.1$. Each is plotted for system size $L=60$.

Figure 4

Edge currents indicate a transition between different topological phases at finite temperature. Here we vary $J_x=J_y=J$ for fixed $J_z=1$, corresponding to the path through parameter space shown on the triangle; the critical value of $J$ is then $J^{*}=0.5$. Scaled edge currents $I/T^2$ are plotted against $J$ for different values of $T$. We see a jump through the phase transition that sharpens with decreasing $T$. Inset: $I$ against $T^2$ plotted for the values $J=0.1,0.35,0.65,0.9$ indicated on the triangle. The currents are seen to be robust in the topological phase (red and green crosses) and vanish in the toric code phase (orange and blue crosses). These values of $J$ are indicated on the main plot with vertical lines of the corresponding colors. These plots are for $L=52$ and $K=0.15$.

Figure 5

Robustness of the edge currents to $J$ and $K$ disorder (left) and lattice boundary defects (right). In the left plot we add a random component to either every $J$ term in the Hamiltonian or every $K$ term. We plot curves for $\left|\delta J\right|/J$ and $\left|\delta K\right|/K=0.01$ and 0.1, setting $J=1$ and $K=0.1$, for $L=40$. Each point is averaged over 100 disorder realizations. The right panel shows the effect of removing sites from the boundary. By plotting the current at the location of the defect (green circles) we see that the current shifts down onto the new edge. In both cases the total edge currents remain unaffected.

Figure 6

Finite size scaling analysis of the zero-temperature current in the no-vortex sector. This is carried out by fitting values $I(0,L)$ for $L=10,12,...,52$ to Eq. (8), while varying $J$ and $K$ within the no-vortex phase with $J_z=1$. Left: We plot the extrapolated values $I_0$ as a function of the fermion gap $\mathrm{\Delta }$. The relationship is nonunique. Onto the plot we have added the path of a single $J$ while $K$ increases. Right: The prefactor $\mathrm{\Theta }$ and scaling exponent $\gamma$ are plotted over $J$ and $K$. We see for most parameters $\gamma \approx 2$. Interestingly, $\mathrm{\Theta }$ is not found to have its CFT predicted value $\pi /12c$ (where $c=1/2$) over most of the range investigated.