Changing anyonic ground degeneracy with engineered gauge fields

For systems of lattice anyons such as Majoranas and parafermions, the unconventional quantum statistics determines a set of global symmetries (e.g., fermion parity for Majoranas) admitting no relevant perturbations. Any operator that breaks these symmetries explicitly would violate locality if added to the Hamiltonian. As a consequence, the associated quasidegeneracy of topologically nontrivial phases is protected, at least partially, by locality via the symmetries singled out by quantum statistics. We show that it is possible to bypass this type of protection by way of specifically engineered gauge fields, in order to modify the topological structure of the edge of the system without destroying the topological order completely. To illustrate our ideas in a concrete setting, we focus on the ${\mathbb{Z}}_6$ parafermion chain. Starting in the topological phase of the chain (sixfold ground degeneracy), we show that a gauge field with restricted dynamics acts as a relevant perturbation, driving a transition to a phase with threefold degeneracy and ${\mathbb{Z}}_3$ parafermion edge modes. The transition from the ${\mathbb{Z}}_3$ to the topologically trivial phase occurs on a critical line in the three-state Potts universality class. We also investigate numerically the emergence of Majorana edge modes when the ${\mathbb{Z}}_6$ chain is coupled to a differently restricted gauge field.

Figure 1

Lowest energy levels of the Hamiltonian (62) modeling a ${\mathbb{Z}}_6$ parafermionic ring chain as a function of the enclosed flux $\mathrm{\Phi }$. The ${\mathbb{Z}}_6$ parafermions are coupled to ${\mathbb{Z}}_6$ gauge fields whose dynamics is restricted to ${\mathbb{Z}}_2$. (a) For static gauge fields with $\kappa =0$, the spectrum shows the $12\pi$ flux periodicity expected for ${\mathbb{Z}}_6$ parafermions. (b) For dynamic gauge fields with $\kappa =0.2J$, the flux periodicity is reduced from $12\pi$ to a flux periodicity of $6\pi$ consistent with effective ${\mathbb{Z}}_3$ parafermions.

Figure 2

Behavior of the Binder cumulants $U_L=1-\langle m^4\rangle /3{\langle m^2\rangle }^2$, defined in terms of the magnetization $m={\sum }_{i=1}^N(v_i+v_i^{†})/2N$, for systems of different sizes $N$ along the line $h=\kappa$ in vicinity of the crossing points.

Figure 3

Behavior of the susceptibility $\chi$ along the line $h=\kappa$ for different system sizes $N$. Data points are not indicated for better visual clarity.

Figure 4

Data collapse of the susceptibility data shown in Fig. 3 with $\nu =0.832, \gamma /\nu =1.731$ determined from a minimization of the residuals Eq. (68) in an interval of size $\mathrm{\Delta }z=\pm 0.4$ around the susceptibility maximum. The critical point $h_c$ is fixed at the value obtained using Binder cumulants. Away from the maximum, no complete data collapse is obtained. To the right of the maximum, the curves for different $N$ approach the maximum $N_{\text{max}}=128$ curve monotonically from below, whereas to the left of the maximum, they approach the $N_{\text{max}}$ curve monotonically from above. This systematic structure of the deviations suggests contributions due to irrelevant scaling fields, possibly associated with the symmetry-breaking boundary operators in the Hamiltonian $H_q$, as a possible explanation. Data points are not indicated for better visual clarity; the data point spacing is roughly $\mathrm{\Delta }z=0.1$ for all curves.

Figure 5

Data collapse of the Binder cumulants $U_L$ obtained using $h_c, \nu$ determined from Binder cumulants and susceptibility. Data points are not indicated for better visual clarity; the data point spacing is roughly $\mathrm{\Delta }z=0.1$ for all curves. Insets show close-ups of the curves in the marked regions, showing that the distance to the $N_{\text{max}}=128$ curve systematically decreases as $N$ is increased.

Figure 6

Data collapse of the susceptibility curves obtained for the model (51) along the line $\kappa =0$ with $p=2$ which corresponds to an Ising model with symmetry-breaking boundary conditions, using the exactly known critical point $h_c/J=1$ and the critical exponents $\nu =1, \gamma =7/4$. The collapse shows a structure similar to the collapse of the susceptibility data obtained along the line $\kappa =h$ for $p=6$ that was shown in Fig. 4, with deviations from perfect data collapse away from the susceptibility peak.

Figure 7

Location of the critical line for the transition from the nontrivial phase (to the left of the line) to the trivial phase (to the right of the line) for the $\mathbb{Z}(6,2)$ and $\mathbb{Z}(6,3)$ model. The critical line is determined from the crossing points of the Binder cumulants $U_L$ for system sizes $(N_1,N_2)=(32,64)$ and the Binder cumulants are calculated numerically using the DMRG algorithm with internal bond dimension $D=64$. Data points are marked as small crosses and linearly interpolated for visual guidance. The large cross indicates the position of the point where we have determined the critical exponents $\nu =0.832, \gamma /\nu =1.731$ consistent with a ${\mathbb{Z}}_3$ phase [43] by means of a more detailed finite-size scaling analysis. The inset shows a close-up of the region around $\kappa =0$, showing that in both cases, the critical lines of the models end at the same point at $\kappa =0$.