Unsupervised Learning of Topological Non-Abelian Braiding in Non-Hermitian Bands

TL;DR

A machine learning algorithm is presented for the unsupervised identification of non-Abelian braiding within multiple complex-energy bands and a winding matrix is introduced as a topological invariant for characterizing braiding topology.

Abstract

The topological classification of energy bands has laid the groundwork for the discovery of various topological phases of matter in recent decades. While this classification has traditionally focused on real-energy bands, recent studies have revealed the intriguing topology of complex-energy, or non-Hermitian bands. For example, the spectral winding of complex-energy bands can from unique topological structures like braids, holding promise for advancing quantum computing. However, discussions of complex-energy braids have been largely limited to the Abelian braid group $\mathbb{B}_2$ for its relative simplicity, while identifying topological non-Abelian braiding is still difficult since it has no universal topological invariant for characterization. Here, we present a machine learning algorithm for the unsupervised identification of non-Abelian braiding of multiple complex-energy bands. The consistency with Artin's well-known topological equivalence conditions in braiding is demonstrated. Inspired by the results from unsupervised learning, we also introduce a winding matrix as a topological invariant in charactering the braiding topology and unveiling the bulk-edge correspondence of non-Abelian braided non-Hermitian bands. Finally, we extend our approach to identify non-Abelian braiding topology in 2D/3D exceptional semimetals and successfully address the unknotting problem in an unsupervised manner.

Figures (4)

• Figure 1: Non-Abelian braiding topology in non-Hermitian bands. (a) Abelian braiding between two bands $\{ E_1, E_2 \}$ in a 1D non-Hermitian system. (b) Topologically equivalent deformation of the braiding in (a) while keeping $E_1$ as a constant (i.e., $0$). After the deformation, the braiding can be characterized by the winding of $E_2$ with the reference energy as $E_1$. (c) Topologically equivalent deformation of the braiding in (a) while keeping $E_2$ as a constant. The braiding can be characterized by the winding of $E_1$ with the reference energy as $E_2$. (d) Non-Abelian braiding between three bands $\{ E_1, E_2, E_3\}$ in a 1D non-Hermitian system. (e) Topologically equivalent deformation of the braiding in (d) while keeping $E_1$ as a constant. The braiding can be characterized by the winding of $\{E_2, E_3\}$ with the reference energy as $E_1$. (f) Topologically equivalent deformation of the braiding in (d) while keeping $E_2$ as a constant. The braiding can be characterized by the winding of $\{E_1, E_3\}$ with the reference energy as $E_2$. (g) Topologically equivalent deformation of the braiding in (d) while keeping $E_3$ as a constant. The braiding can be characterized by the winding of $\{E_1, E_2\}$ with the reference energy as $E_3$.
• Figure 2: Unsupervised identification of topological non-Abelian braids. We generate all combinations by randomly selecting 3 braid operations in different braid groups $\mathbb{B}_2$, $\mathbb{B}_3$, $\mathbb{B}_4$, $\mathbb{B}_5$ and $\mathbb{B}_6$. After performing unsupervised learning algorithm, we plot two braids in typical clusters: (1) cluster 1 in $\mathbb{B}_2$ ($c=2$); (2) cluster 2 in $\mathbb{B}_3$ ($c=12$); (3) cluster 3 in $\mathbb{B}_4$ ($c=12$); (4) cluster 4 in $\mathbb{B}_5$ ($c=34$); (5) cluster 5 in $\mathbb{B}_6$ ($c=16$). From the plotted braids in the different clusters (i.e., in the same phases), we can see that our algorithm can be consistent with the Artin's topological equivalent conditions in the braiding, which guarantees the success of our algorithm in classifying the topological phases among braids. More details can be found in the Supplementary Material Note1.
• Figure 3: 2D expcetional topological semimetal with non-Abelian braiding topology. The topology around the central exceptional nodal point is described by the band topology on the sphere $S^1$ denoted as $\Gamma$ (shown in an arrowed blue circle). The bands on $S^1$ (denoted as the red lines) possess the braiding topology, which is described as the braid words that are shown on the top of each figure.
• Figure 4: Solving the unknotted problem unsupervisedly. (a) The braids can be closed due to the periodicity of BZ. Here, we show an unknotted bandstructure ($\sigma_3\sigma_2\sigma_1$). After connecting the bands, we can continuously deform the resultant structure into an unknot. (b) Some unknots identified by our algorithm. We firstly generate all braid words of unknots for $N$ bands and form a group $\mathcal{S}$. Here, we set $N=4$. Then, we randomly generate the braids that belong to $\mathbb{B}_4$. We then perform the similarity function to measure the difference between the generated samples and the samples in $\mathcal{S}$. The generated braid that is in the same phase as any in $\mathcal{S}$ is an unknot.