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Physical review. B./Physical review. B

Unsupervised learning of interacting topological and symmetry-breaking phase transitions

En-Jui Kuo, Hossein Dehghani

Published Jun 27, 2022• 9 citations
Publisher

Abstract

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.

DOI
10.1103/physrevb.105.235136

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Progress of Theoretical and Experimental Physics•Mar 16, 2022•1 citation
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.
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.
We generalize the hidden symmetry-breaking picture of symmetry-protected topological (SPT) order developed by Kennedy and Tasaki in the context of the Haldane phase. Our generalization applies to a wide class of SPT phases in one-dimensional spin chains, protected by an on-site representation of a finite Abelian group. This generalization takes the form of a nonlocal unitary map that relates local symmetry-respecting Hamiltonians in an SPT phase to local Hamiltonians in a symmetry-broken phase. Using this unitary, we establish a relation between the two-point correlation functions that characterize fully symmetry-broken phases with the string-order correlation functions that characterize the SPT phases, therefore establishing the perspective in these systems that SPT phases are characterized by hidden symmetry breaking. Our generalization is also applied to systems with continuous symmetries, including SO(2k+1)\mathrm{SO}(2k+1)SO(2k+1) and SU(k)\mathrm{SU}(k)SU(k).
The discovery of topological features of quantum states plays a central role in modern condensed matter physics and various artificial systems. Due to the absence of local order parameters, the detection of topological phase transitions remains a challenge. Machine learning may provide effective methods for identifying topological features. In this work, we show that manifold learning can successfully retrieve topological quantum phase transitions in momentum space. Our results show that the Chebyshev distance (CD) between two data points can successfully capture the main features of topological phase transitions, while the widely used Euclidean distance in general fails. The similarity matrix built upon the CD thus naturally exhibits uncorrelated cluster structures, corresponding to distinct sectors in the topological phase diagram. Then a diffusion map is applied to implement dimensionality reduction and to learn about topological quantum phase transitions in an unsupervised manner. Our demonstrations on the Su-Schrieffer-Heeger (SSH) model, the 2D Qi-Wu-Zhang (QWZ) model, and the quenched SSH model show the capability of generic unsupervised learning, when equipped with a suitable distance metric, in exploring topological quantum phase transitions.
The discovery of topological features of quantum states plays an important role in modern condensed matter physics and various artificial systems. Due to the absence of local order parameters, the detection of topological quantum phase transitions remains a challenge. Machine learning may provide effective methods for identifying topological features. In this work we show that the unsupervised manifold learning can successfully retrieve topological quantum phase transitions in momentum and real space. Our results show that the Chebyshev distance between two data points sharpens the characteristic features of topological quantum phase transitions in momentum space, while the widely used Euclidean distance is in general suboptimal. Then a diffusion map or isometric map can be applied to implement the dimensionality reduction, and to learn about topological quantum phase transitions in an unsupervised manner. We demonstrate this method on the prototypical Su-Schrieffer-Heeger (SSH) model, the Qi-Wu-Zhang (QWZ) model, and the quenched SSH model in momentum space, and further provide implications and demonstrations for learning in real space, where the topological invariants could be unknown or hard to compute. The interpretable good performance of our approach shows the capability of manifold learning, when equipped with a suitable distance metric, in exploring topological quantum phase transitions.
Symmetry-protected topological (SPT) phases exhibit nontrivial order if symmetry is respected but are adiabatically connected to the trivial product phase if symmetry is not respected. However, unlike the symmetry-breaking phase, there is no local order parameter for SPT phases. Here we employ a tensor-network method to compute the topological invariants characterized by the simulated modular SSS and TTT matrices to study transitions in a few families of two-dimensional (2D) wave functions which are ZN{\mathbb{Z}}_{N}ZN​ (N=2and3)(N=2\phantom{\rule{4.pt}{0ex}}\text{and}\phantom{\rule{4.pt}{0ex}}3)(N=2and3) symmetric. We find that in addition to the topologically ordered phases, the modular matrices can be used to identify nontrivial SPT phases and detect transitions between different SPT phases as well as between symmetric and symmetry-breaking phases. Therefore modular matrices can be used to characterize various types of gapped phases in a unifying way.
One problem of paramount importance in physics over various fields such as elementary particle physics, condensed matter physics, and statistical mechanics has been to make classification of phases of matter, including nontrivial structures of the vacuum. The triumphant history of theory on the phase classification and the phase transition can be traced back to Landau, which is based on the spontaneous breaking of symmetry. It is known, however, that other kinds of phases exist that cannot be characterized by conventional arguments of symmetry breaking, namely, topological phases. Topological phases are gapped phases at zero temperature without symmetry breaking. Examples of the topological phases include a quantum phase in a fractional quantum Hall system, where the ground state exhibits degeneracy, and an anyon, a quasi-particle with nontrivial statistical nature, neither boson nor fermion, emerges as a low-energy excitation. In contemporary physics of the 21st century, it has been revealed that, among various topological states, there exists a nontrivial ground state that does not have ground state degeneracy and yet can be distinguished from a trivial state. This nontrivial ground state, i.e., a topological insulator, has been recognized first in free electron systems [1–3] and later extended to more general quantum many-body systems with interactions including bosonic (spin) systems. Now, such a generalized situation is called a symmetry protected topological (SPT) phase [4–7]. From the point of a modern view of the SPT phase, nontrivial states, such as the Haldane state proposed in antiferromagnetic quantum spin systems in the 1980’s [8] and the ground state in the Affleck-Kennedy-Lieb-Tasaki model [9], belong to the SPT phase. A remarkable feature of the SPT phase is that, if the system has a surface, gapless surface states appear and they are stable against disturbances which do not break the symmetry. One way to understand these surface states is the quantum anomaly on the surface. That is, as long as the surface is concerned, the quantum anomaly arises, but the whole system has no quantum anomaly, and the symmetry is preserved due to cancellation from the bulk contribution. This canceling mechanism of the quantum anomaly has been long known as the anomaly inflow in elementary particle physics [10]. In fact, the classification of the SPT phases can be identified as the classification of the quantum anomaly through the bulk-boundary correspondence. This type of the quantum anomaly closely related to the SPT phase can provide us with a useful theoretical device to constrain phases that can or cannot be realized under the ’t Hooft anomaly matching condition [11–13]. It is quite difficult to scrutinize nonperturbatively interacting systems, but the ’t Hooft anomaly matching condition can be applied to those systems, and possible phases can be constrained. Nowadays, our understanding of the SPT phases has been so developed that applications of the anomaly matching condition can cover systems with finite groups and/or extending objects [14–16]. In this way, the concept of the topological phases has been essential by now in a wide range of physics, particularly in elementary particle physics, condensed matter physics, and statistical mechanics, as mentioned in the beginning. This special section is a collection of papers related to the topological aspects of physics including the topological phases. We would like to express our sincere gratitude to all the authors who made excellent contributions.