Persistent homology of ${\mathbb{Z}}_2$ gauge theories

Topologically ordered phases of matter display a number of unique characteristics, including ground states that can be interpreted as patterns of closed strings in the case of general ${\mathbb{Z}}_2$ string liquids. In this paper, we consider the problem of detecting and distinguishing closed strings in Ising spin configurations sampled from the classical ${\mathbb{Z}}_2$ gauge theory. We address this using the framework of persistent homology, which computes the size and frequency of general loop structures in spin configurations via the formation of geometric complexes. Implemented numerically on finite-size lattices, we show that the first Betti number of the Vietoris-Rips complexes achieves a high density at low temperatures in the ${\mathbb{Z}}_2$ gauge theory. In addition, it displays a clear signal at the finite-temperature deconfinement transition of the three-dimensional theory. We argue that persistent homology should be capable of interpreting prominent loop structures that occur in a variety of systems, making it a useful tool in theoretical and experimental searches for topological order.

Figure 1

Point cloud $\mathbb{X}$ and associated VR complex ${\mathcal{V}}_{\ell }\left(\mathbb{X}\right)$. At a radius of $r=\ell$, connections (1-simplices) form between all neighboring points. The resulting complex is homeomorphic to a circle, and ultimately has a first Betti number of $b_1=1$.

Figure 2

Illustration of persistent homology applied to a set of data points (blue dots). For any radius $2r<\ell$, no circles overlap, and hence no connections form. At a radius of $2r=\ell$, all points within a distance of $\ell$ from each other form connections. The corresponding graph has two holes, labeled $A$ and $B$. Since these holes form exactly at $2r=\ell$, they are said to be born at $2r=\ell$. At a radius of $2r=\ell \sqrt{2}$, diagonal connections of length $\ell \sqrt{2}$ form, and the resulting simplices (triangles) are filled in (pink). Since hole $B$ is no longer present, it is said to have died at $2r=\ell \sqrt{2}$. Similarly, at a radius of $r=\ell$, one final connection forms, and two more simplices are filled in. Since hole $A$ is no longer present, it is said to have died at $r=\ell$. To measure the significance of each hole, one can compute the persistence as death minus birth. These are given by $p_A=2\ell -\ell =\ell$ and $p_B=\ell \sqrt{2}-\ell \approx 0.4\ell$ for holes $A$ and $B$, respectively. Since hole $A$ has a greater persistence than hole $B$, it is considered to be a more significant topological feature in the data

Figure 3

Examples of closed strings in ground-state configurations of the ${\mathbb{Z}}_2$ gauge theory, where blue dots depict down spins. The application of a single gauge transformation to a configuration consisting solely of up spins generates a closed string of minimal size. Arrows depict steps in which single adjacent gauge transformations are applied, which stretches this loop to greater sizes. In the context of persistent homology, this corresponds to generating loops of greater persistence. The first configuration corresponds to an $H_1$ homology that is born at $r=\ell /\sqrt{2}$ and dies at $r=\ell$. The remaining configurations correspond to $H_1$ homologies that are born at $r=\ell$ and die at $r=\{\ell \sqrt{2},\ell \sqrt{5/2},\ell \sqrt{9/2}\}$, respectively.

Figure 4

Top (bottom): Illustration of VR complex corresponding to ${\mathit{\sigma }}^{\prime }$ (${\mathit{\sigma }}^{″}$), which is generated by applying one (two) gauge transformations to a configuration consisting solely of up spins. Blue points depict down spins. Red structures are hollow and beige structures are filled. The top VR complex of ${\mathit{\sigma }}^{\prime }$ holds generally for $\ell /\sqrt{2}\le r<\ell$. The bottom VR complex of ${\mathit{\sigma }}^{″}$ holds generally for $\ell \le r<\ell \sqrt{3/2}$. $\cong$ denotes homotopic equivalence.

Figure 5

Average first Betti number of $r=\ell$ VR complexes of configurations in the ${\mathbb{Z}}_2$ gauge theory with $D=2$. $\langle b_1\rangle$ is averaged over 2000 samples for each temperature. Standard errors bars defined as $99\%$ Gaussian confidence intervals.

Figure 6

Average first Betti number of $r=\ell$ VR complexes of configurations in the ${\mathbb{Z}}_2$ gauge theory with $D=3$. $\langle b_1\rangle$ is averaged over 1000 samples for each temperature. Dashed line indicates theoretical value of critical temperature. Location of the phase transition is correctly identified by a change in the concavity of $\langle b_1\rangle$. Statistical error bars are not visible.

Figure 7

Average frequency of $H_1$ homologies with birth and death values $r_b=\ell$ and $r_d=\ell \sqrt{3/2}$. Based on the same Monte Carlo configurations used in Fig. 6.

Figure 8

Histogram of number of Monte Carlo samples $f$ for $b_1$ obtained under the VR complex construction of samples from the $3D{\mathbb{Z}}_2$ gauge theory, with $L=18$ and $T=0.7$. Based on the same Monte Carlo samples used to generate Fig. 6.

Figure 9

Two manifolds $\mathcal{A}$ and $\mathcal{B}$ and two curves ${\gamma }_1$ and ${\gamma }_2$. In $\mathcal{A}$, ${\gamma }_1$ and ${\gamma }_2$ are homotopically equivalent, since they can be smoothly deformed into each other. However, in $\mathcal{B}$, ${\gamma }_1$ and ${\gamma }_2$ are not homotopically equivalent, since ${\gamma }_2$ winds around a hole while ${\gamma }_1$ does not. Generally, there exists a homotopy class for each winding number around any given hole.

Figure 10

Two graphs $\mathcal{A}$ and $\mathcal{B}$. 0-cells (points) and 1-cells (lines) are labeled by Greek and Roman letters, respectively. $\mathcal{B}$ differs from $\mathcal{A}$ by the presence of an additional 2-cell, labeled $A$, which fills one of the cycles.

Figure 11

Illustration of first four simplices. The faces of any $n$-simplex are $n-1$ simplices. Both the 2-simplex and 3-simplex are completely filled.