Abstract Recent experiments have successfully realized multi-band non-Abelian topological insulators with parity-time symmetry. Their topological classification transcends the conventional ten-fold classification, necessitating the use of non-Abelian groups, manifesting novel properties that cannot be described using integer topological invariants. The unique non-commutative multiplication of non-Abelian groups, along with the distinct topological classifications in the context of homotopy with or without a fixed base point, makes the identification of different non-Abelian topological phases more nuanced and challenging than in the Abelian case. In this work, we present an unsupervised learning method based on diffusion maps to classify non-Abelian multi-gap topological phases. The automatically adiabatic pathfinding process in our method can correctly sort the samples in the same phase even though they are not connected by adiabatic paths in the sample set. Most importantly, our method can deduce the multiplication table of the non-Abelian topological charges in a data-driven manner without requiring a priori knowledge. Additionally, our algorithm can provide the correct classifications for the samples within both the homotopy with and without a fixed base point. Our results provide insights for future studies on non-Abelian phases using machine learning approaches.
Abstract Recent experiments have successfully realized multi-band non-Abelian topological insulators with parity-time symmetry. Their topological classification transcends the conventional ten-fold classification, necessitating the use of non-Abelian groups, manifesting novel properties that cannot be described using integer topological invariants. The unique non-commutative multiplication of non-Abelian groups, along with the distinct topological classifications in the context of homotopy with or without a fixed base point, makes the identification of different non-Abelian topological phases more nuanced and challenging than in the Abelian case. In this work, we present an unsupervised learning method based on diffusion maps to classify non-Abelian multi-gap topological phases. The automatically adiabatic pathfinding process in our method can correctly sort the samples in the same phase even though they are not connected by adiabatic paths in the sample set. Most importantly, our method can deduce the multiplication table of the non-Abelian topological charges in a data-driven manner without requiring a priori knowledge. Additionally, our algorithm can provide the correct classifications for the samples within both the homotopy with and without a fixed base point. Our results provide insights for future studies on non-Abelian phases using machine learning approaches.
A remarkable breakthrough in topological phase classification is the establishment of the topological periodic table, which is mainly based on the classifying space analysis or K theory, but not based on concrete Hamiltonians that possess finite bands or arise in a lattice. As a result, it is still difficult to identify the topological phase of an arbitrary Hamiltonian; the common practice is, instead, to check the incomplete and still growing list of topological invariants one by one, very often by trial and error. Here, we develop unsupervised classifications of topological gapped systems with symmetries, and demonstrate the data-driven construction of the topological periodic table without a priori knowledge of topological invariants. This unsupervised data-driven strategy can take into account spatial symmetries, and further classify phases that were previously classified as trivial in the past. Our Letter introduces machine learning into topological phase classification and paves the way for intelligent explorations of new phases of topological matter.
The integration of machine learning into fundamental science has opened new avenues for addressing long-standing challenges rooted in mathematical limitations. For instance, while topological invariants are essential for characterizing topological phases of matter, no single invariant is universally applicable. This limitation explains why, over decades of classifying topological phases-primarily in Hermitian systems-many phases initially deemed 'trivial' were later recognized as topological. Recently, the discovery of non-Hermitian band topology has driven substantial efforts in non-Hermitian topological classification, leading to the development of new topological invariants. However, these invariants still fail to capture all non-Hermitian topological features. Here, without relying on any topological invariant, we develop a machine-learning algorithm for the unsupervised classification of symmetry-protected non-Hermitian topological phases. By utilizing random Hamiltonians, we unsupervisedly construct a topological periodic table without requiring advanced mathematical knowledge. Furthermore, based on the learning results, we derive a formula that reveals the impact of parity transformation on periodicity. Our algorithm can also account for boundary effects, enabling the exploration of open-boundary influences on the topological phase diagram. These findings establish an unsupervised approach for classifying symmetry-protected non-Hermitian topological phases, uncover previously unnoticed topological features in non-Hermitian systems, and provide valuable guidance for both theoretical advancements and experimental realizations.
Abstract Floquet engineering provides a unique platform for realizing novel non-equilibrium topological phases. However, the intrinsic multi-gap structure of Floquet systems poses significant challenges for their classification using conventional analytical methods. Here, we propose an unsupervised machine learning framework that employs a kernel defined directly in momentum-time ( k , t ) space, constructed from Floquet–Bloch states. This data-driven approach does not require prior knowledge of topological invariants. By using a compact computational procedure that works directly with the system’s eigenstates, the method can identify topologically distinct phases associated with both the 0 -gap and the π -gap across various symmetry classes (1D AIII, 1D D, and 2D A). Our results demonstrate that key topological characteristics of Floquet systems are directly accessible from their eigenstates in ( k , t ) space, providing a tool for the systematic classification and study of non-equilibrium topological phases without a priori invariant information.
We propose a unified framework, dubbed topological word, for the complete non-Abelian bulk-boundary correspondence in multigap non-Abelian topological insulators. Composed by an ordered sequence of letters, each a non-Abelian charge depicting the gap-resolved topology, the topological word captures both the global non-Abelian topology corresponding to the homotopy classification, and the band-adjacency information. The latter, though crucial for the edge-state pattern across multiple gaps, is often overlooked in previous studies. We confirm our framework using both static models and periodically driven Floquet systems, and discuss its connection and distinction with existing descriptions, such as the phase-band singularities and braiding representations. Intriguingly, topological word continues to provide insight regarding topology and edge states, even as the global non-Abelian topology becomes ill-defined under broken parity-time symmetry.
Classifying topological phases of matter with strong interactions is a notoriously challenging task and has attracted considerable attention in recent years. In this paper, we propose an unsupervised machine learning approach that can classify a wide range of symmetry-protected interacting topological phases directly from the experimental observables and without a priori knowledge. We analytically show that Green's functions, which can be derived from spectral functions that can be measured directly in an experiment, are suitable for serving as the input data for our learning proposal based on the diffusion map. As a concrete example, we consider a one-dimensional interacting topological insulators model and show that, through extensive numerical simulations, our diffusion map approach works as desired. In addition, we put forward a generic scheme to measure the spectral functions in ultracold atomic systems through momentum-resolved Raman spectroscopy. Our work circumvents the costly diagonalization of the system Hamiltonian, and provides a versatile protocol for the straightforward and autonomous identification of interacting topological phases from experimental observables in an unsupervised manner.
Floquet engineering offers an unparalleled platform for realizing novel non-equilibrium topological phases. However, the unique structure of Floquet systems, which includes multiple quasienergy gaps, poses a significant challenge to classification using conventional analytical methods. We propose a novel unsupervised machine learning framework that employs a kernel defined in momentum-time (k,t) space, constructed directly from Floquet-Bloch eigenstates. This approach is intrinsically data-driven and requires no prior knowledge of the underlying topological invariants, providing a fundamental advantage over prior methods that rely on abstract concepts like the micromotion operator or homotopic transformations. Crucially, this work successfully reveals the intrinsic topological characteristics encoded within the Floquet eigenstates themselves. We demonstrate that our method robustly and simultaneously identifies the topological invariants associated with both the 0-gap and the π-gap across various symmetry classes (1D AIII, 1D D, and 2D A), establishing a robust methodology for the systematic classification and discovery of complex non-equilibrium topological matter.