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.
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.
Symmetry-protected topological (SPT) phases are short-range entangled phases of matter with a non-local order parameter which are preserved under a local symmetry group. Here, by using unsupervised learning algorithm, namely the diffusion maps, we demonstrate that can differentiate between symmetry broken phases and topologically ordered phases, and between non-trivial topological phases in different classes. In particular, we show that the phase transitions associated with these phases can be detected in different bosonic and fermionic models in one dimension. This includes the interacting SSH model, the AKLT model and its variants, and weakly interacting fermionic models. Our approach serves as an inexpensive computational method for detecting topological phases transitions associated with SPT systems which can be also applied to experimental data obtained from quantum simulators.
Experimental quantum simulators have become large and complex enough that discovering new physics from the huge amount of measurement data can be quite challenging, especially when little theoretical understanding of the simulated model is available. Unsupervised machine learning methods are particularly promising in overcoming this challenge. For the specific task of learning quantum phase transitions, unsupervised machine learning methods have primarily been developed for phase transitions characterized by simple order parameters, typically linear in the measured observables. However, such methods often fail for more complicated phase transitions, such as those involving incommensurate phases, valence-bond solids, topological order, and many-body localization. We show that the diffusion map method, which performs nonlinear dimensionality reduction and spectral clustering of the measurement data, has significant potential for learning such complex phase transitions unsupervised. This method may work for measurements of local observables in a single basis and is thus readily applicable to many experimental quantum simulators as a versatile tool for learning various quantum phases and phase transitions.
Symmetry-protected topological (SPT) phases are short-range entangled phases of matter with a nonlocal order parameter which are preserved under a local symmetry group. Here, by using an unsupervised learning algorithm, namely, diffusion maps, we demonstrate that we can detect phase transitions between symmetry-broken and topologically ordered phases, and between nontrivial topological phases in different classes. In particular, we show that the phase transitions associated with these phases can be detected in different bosonic and fermionic models in one dimension. This includes the interacting Su-Schrieffer-Heeger model, the Affleck-Kennedy-Lieb-Tasaki model and its variants, and weakly interacting fermionic models. Our approach serves as an inexpensive computational method for detecting topological phase transitions associated with SPT systems which can also be applied to experimental data obtained from quantum simulators.
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 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.