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arXiv:2106.11468 · Physical Review Research

Unsupervised identification of Floquet topological phase boundaries

Nannan Ma, Jiangbin Gong

Published Mar 28, 2022• 9 citations
arXivHTMLPDF

Abstract

Nonequilibrium topological matter has been a fruitful topic of both theoretical and experimental interest. A great variety of exotic topological phases unavailable in static systems may emerge under nonequilibrium situations. How to locate the borders between different nonequilibrium topological phases is an important issue to facilitate topological characterization and further understand phase transition behaviors. In this paper, we develop an unsupervised machine-learning protocol to distinguish between different Floquet (periodically driven) topological phases possessed by theoretical models, by incorporating the system's dynamics within one driving period, adiabatic deformation in the time dimension, plus the system's symmetry all into our machine-learning algorithm. Results from two rich case studies indicate that machine learning is able to reliably reveal intricate topological phase boundaries and can hence be a powerful tool to analyze and discover previously unknown topological phases afforded by the time dimension. As inspired by previous elegant methods in analyzing Floquet topological phases, one essential ingredient in our machine-learning approach is to directly investigate the system's evolution operators parametrized by time.

DOI
10.1103/physrevresearch.4.013234

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Nonequilibrium topological matter has been a fruitful topic of both theoretical and experimental interest. A great variety of exotic topological phases unavailable in static systems may emerge under nonequilibrium situations. How to locate the borders between different nonequilibrium topological phases is an important issue to facilitate topological characterization and further understand phase transition behaviors. In this paper, we develop an unsupervised machine-learning protocol to distinguish between different Floquet (periodically driven) topological phases possessed by theoretical models, by incorporating the system's dynamics within one driving period, adiabatic deformation in the time dimension, plus the system's symmetry all into our machine-learning algorithm. Results from two rich case studies indicate that machine learning is able to reliably reveal intricate topological phase boundaries and can hence be a powerful tool to analyze and discover previously unknown topological phases afforded by the time dimension. As inspired by previous elegant methods in analyzing Floquet topological phases, one essential ingredient in our machine-learning approach is to directly investigate the system's evolution operators parametrized by time.
Machine-learning driven models have proven to be powerful tools for the identification of phases of matter. In particular, unsupervised methods hold the promise to help discover new phases of matter without the need for any prior theoretical knowledge. While for phases characterized by a broken symmetry, the use of unsupervised methods has proven to be successful, topological phases without a local order parameter seem to be much harder to identify without supervision. Here, we use an unsupervised approach to identify boundaries of the topological phases. We train artificial neural nets to relate configurational data or measurement outcomes to quantities like temperature or tuning parameters in the Hamiltonian. The accuracy of these predictive models can then serve as an indicator for phase transitions. We successfully illustrate this approach on both the classical Ising gauge theory as well as on the quantum ground state of a generalized toric code.
Abstract Machine-learning driven models have proven to be powerful tools for the identification of phases of matter. In particular, unsupervised methods hold the promise to help discover new phases of matter without the need for any prior theoretical knowledge. While for phases characterized by a broken symmetry, the use of unsupervised methods has proven to be successful, topological phases without a local order parameter seem to be much harder to identify without supervision. Here, we use an unsupervised approach to identify boundaries of the topological phases. We train artificial neural nets to relate configurational data or measurement outcomes to quantities like temperature or tuning parameters in the Hamiltonian. The accuracy of these predictive models can then serve as an indicator for phase transitions. We successfully illustrate this approach on both the classical Ising gauge theory as well as on the quantum ground state of a generalized toric code.
The study of topological band structures is an active area of research in condensed matter physics and beyond. Here, we combine recent progress in this field with developments in machine learning, another rising topic of interest. Specifically, we introduce an unsupervised machine learning approach that searches for and retrieves paths of adiabatic deformations between Hamiltonians, thereby clustering them according to their topological properties. The algorithm is general, as it does not rely on a specific parametrization of the Hamiltonian and is readily applicable to any symmetry class. We demonstrate the approach using several different models in both one and two spatial dimensions and for different symmetry classes with and without crystalline symmetries. Accordingly, it is also shown how trivial and topological phases can be diagnosed upon comparing with a generally designated set of trivial atomic insulators.
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.
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\boldsymbol{k},tk,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 000-gap and the πππ-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.
Abstract Identifying phase transitions is one of the key challenges in quantum many-body physics. Recently, machine learning methods have been shown to be an alternative way of localising phase boundaries from noisy and imperfect data without the knowledge of the order parameter. Here, we apply different unsupervised machine learning techniques, including anomaly detection and influence functions, to experimental data from ultracold atoms. In this way, we obtain the topological phase diagram of the Haldane model in a completely unbiased fashion. We show that these methods can successfully be applied to experimental data at finite temperatures and to the data of Floquet systems when post-processing the data to a single micromotion phase. Our work provides a benchmark for the unsupervised detection of new exotic phases in complex many-body systems.