Detecting topological order with ribbon operators

We introduce a numerical method for identifying topological order in two-dimensional models based on one-dimensional bulk operators. The idea is to identify approximate symmetries supported on thin strips through the bulk that behave as string operators associated to an anyon model. We can express these ribbon operators in a matrix product form and define a cost function that allows us to efficiently optimize over this ansatz class. We test this method on spin models with Abelian topological order by finding ribbon operators for ${\mathbb{Z}}_d$ quantum double models with local fields and Ising-like terms. In addition, we identify ribbons in the Abelian phase of Kitaev's honeycomb model which serve as the logical operators of the encoded qubit for the quantum error-correcting code. We further identify the topologically encoded qubit in the quantum compass model, and show that despite this qubit, the model does not support topological order. Finally, we discuss how the method supports generalizations for detecting non-Abelian topological order.

Figure 1

The prototypical ribbon operators are the string operators shown above for the toric code. The toric code is defined by $X$-type and $Z$-type four-body interactions on alternate plaquettes of a square lattice, and the (exact) ribbon operators are the stringlike products of $X$ and $Z$ shown above. These ribbon operators commute with the Hamiltonian terms but anticommute with each other, thus identifying a phase with ${\mathbb{Z}}_2$ topological order.

Figure 2

Cost for ribbons in topologically trivial phases the ${\mathrm{QDI}}_2$ model with $\eta ={\mathrm{e}}^{i\varphi }$. A pair of ribbons were optimized on two $w=1$ strips with bond dimension 1 for the paramagnet and 2 in the other cases. We observe two distinct failures to observe topological order. In the case of the paramagnet, a zero-cost ribbon can be found for all $\varphi$. In fact, this can be extended to all $\eta$. In the other cases (closeup in inset plot), there is no $\eta$ for which a nontrivial low-cost ribbon can be obtained. It is possible that this could be due to insufficient width, although we have not observed this to be the case. Note that averaging over many trials (20) to give a typical picture of the algorithms performance. In the case where the variational minimum is used instead, still no signal of topological order is observed.

Figure 3

(a) Cost for a width 1 ribbon found in the ${\mathrm{QDI}}_5$ model of Eq. (18) with $\{J,h,\lambda \}=\{1,0,0\}$ using bond dimension 5 and plotted with $\eta ={\mathrm{e}}^{i\varphi }$. The other ribbon was kept as a string of $Z\mathrm{s}$ as shown in Fig. 1. (b) Cost for ribbons found using alternating minimization on the ${\mathrm{QDI}}_2$ model with $J=1$. In the $h=0$ case a pair of $w=1$, bond dimension 1 ribbons were sought, in the other cases, the pair consisted of a $w=1$ and a $w=2$ ribbon with bond dimensions 1 and 5 respectively. At the exactly solvable points ($h=0$), we see a dramatic cost decrease at $d\mathrm{th}$ roots of unity, signaling both the presence of and type of topological order present in these models. This feature remains even with the addition of an $X$ field or Ising type term which destroys the solvability of the model. (c) Cost of a width 4 ribbon of bond dimension 2 in the ${\mathrm{QDI}}_2$ model. The cost obtained from our numerical optimization is smaller than that of the fixed point string for all (nonzero) values of $h$. This demonstrates the nontrivial nature of the ribbon operators we obtain, even in this perturbative regime.

Figure 4

Cost for ribbons found using alternating minimization on the ${\mathrm{QDI}}_3$ model with $\{J,h,\lambda \}=\{1,0,0\}$. A pair of $w=1$ ribbons with bond dimension 1 were optimized. We observe a signal of topological order only at $\eta ={\mathrm{e}}^{2\pi ik/3}$.

Figure 5

Cost for ribbons found using alternating minimization on the ${\mathrm{QDI}}_2$ model with $\{J,h,\lambda \}=\{1,0.05,0\}$ where $\eta$ runs over points in the complex unit disk. A pair of ribbons with $w=1$ and $w=2$ and bond dimensions 1 and 5 respectively were optimized and the total cost is shown. We observe a signal of topological order at $\eta =-1$. Note that Fig. 3 corresponds to the edge of this disk. The raw data is displayed, but some contours were subjected to gaussian smoothing for clarity in the nearly flat region where the noise in the data and the gradient of the function are comparable in magnitude.

Figure 6

Topologically ordered models should support low-cost ribbons on both the horizontal and the rotated blue regions (or some widening of them). This corresponds to insensitivity of the anyon braid relations to the specific worldlines they trace out. Models with specific lattice symmetries such as the quantum compass model might provide false signatures of topological order if ribbon operators are sought only on nongeneric strips of the lattice.

Figure 7

(a) and (b) show the cost of ribbons with supports oriented as shown in Fig. 6 for two models: (a) the quantum compass model and left bracket on (b) the ${\mathrm{QDI}}_2$ model. The cost is parameterized with $\eta ={\mathrm{e}}^{i\varphi }$, and all ribbons were found using bond dimension 5. (a) Cost of ribbons in the quantum compass model of Eq. (21). Because ribbons in this model are sensitive to their support, we conclude that they do not represent genuine topological order. (b) Cost of ribbons in the ${\mathrm{QDI}}_2$ model of Eq. (18) using $\{J,h,\lambda \}=\{1,0,0\}$. Low-cost ribbons can be found on both unrotated and rotated supports indicating genuine topological order.

Figure 8

Cost of ribbons in the honeycomb model. Strong indications of the ${\mathbb{Z}}_2$ topological order known to be present in this phase are visible. As expected, the cost decreases as the allowed width or bond dimension is increased, however, the cost rapidly saturates with increasing bond dimension. (a) Cost for a bond dimension 5 ribbon on the honeycomb model of Eq. (23) with $J_X=J_Y=J_Z/10$ using $\eta ={\mathrm{e}}^{i\varphi }$. A second ribbon can be obtained analytically, and was fixed as a product of $Z\mathrm{s}$ along the $X$ and $Y$ links of the lattice. The dip at $\varphi =\pi$ is characteristic of ${\mathbb{Z}}_2$ order, and indicates the topological nature of this phase. (b) Cost for a width 3 ribbon at $\eta =-1$ on the honeycomb model with $J_X=J_Y=J_Z/10$ as a function of bond dimension. The cost rapidly saturates to a minimum, justifying our assumption that only small bond dimension is required. (c) As we increase the strength of $J=J_X=J_Y$, we eventually cross a phase transition to a gapless phase at $J=J_Z/2$ (indicated by $\times$). Approaching this point, the signal of $Z_2$ topological order essentially vanishes. Here the strength of $J$ is increased in steps of $1/30$ from $J=0$ ($\circ$) to $J=1$ ($+$). Note that the line marked by $☆$ corresponds to plots $\left(a\right),\left(b\right)$ and a bond dimension of 4 is used for a width 3 ribbon.