Spectroscopy of a topological phase

Dynamical correlation functions of the toric code in a uniform magnetic field are studied inside the topological phase, in the small-field limit. Such an experimentally measurable quantity displays rich field-dependent features that can be understood via the interplay of the kinetics and the interaction of the anyonic excitations. In particular, it is sensitive to the two-quasiparticle bound states that are present in the spectrum for a wide range of magnetic fields. Interestingly, such collective modes can even constitute the lowest-energy excitations of the system.

Figure 1

The lattice and the unit vectors are shown in (a). Normalized 2QP densities of states (DOSs) as a function of the momentum $\mathbf{K}$ for $h_x=0$ and (b) $(h_y,h_z)=(0,0.06)$, (c) $(h_y,h_z)=(0.2,0)$, and (d) $(h_y,h_z)=(0.2,0.06)$. Dashed (red) lines are the boundaries of the continuum computed from the single-quasicharge dispersion. Since bound states have a vanishing weight in the DOSs, they are explicitly highlighted by gray (cyan) lines.

Figure 2

$\mathcal{S}(\mathbf{K},\omega )$ in the 2QP approximation. Upper and lower panels correspond to $h_y=0$ and $h_y=0.2$, respectively. From left to right $(h_x,h_z)=$ $(0,0.06)$, $(0.03,0.06)$, $(0.06,0.06)$, $(0.06,0.03)$, and $(0.06,0)$. Boundaries of quasicharge and quasiflux continua are represented by dashed (red) and dotted (green) lines, respectively.

Figure 3

Normalized DOS (left) and $\mathcal{S}(\mathbf{K},\omega )$ (right) for $(h_x,h_y,h_z)=(0.06,0.3,0.06)$ (see Fig. 1 for conventions).

Figure 4

A typical hopping process with an initial state (left) that consists in two quasicharges ($\tau =c$), separated by a vector $\mathbf{d}=(1,0)$ and a final state (right) that consists in two quasifluxes (${\tau }^{\prime }=f$), separated by a vector ${\mathbf{d}}^{\prime }=(1,1)$. This hopping induces a shift $\mathbf{m}=(1/2,0)$ of the center of mass. The corresponding hopping amplitude is $-(i/2)h_x{h}_y$ (see Table 1).

Figure 5

Two (among many) states generated by the action of the effective observable ${\sigma }_{\mathbf{r},\mathrm{eff}}^z$ on the white site. Left: $\tau =f$, $\mathbf{d}=(1,0)$, $\mathbf{m}=(0,0)$, and the amplitude is $-(i/4)h_y$. Right: $\tau =c$, $\mathbf{d}=(2,1)$, $\mathbf{m}=(1,0)$, and the amplitude is $(5/8)h_z^2$.