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arXiv:2011.06611 · Physical Review X

Entanglement Order Parameters and Critical Behavior for Topological Phase Transitions and Beyond

Mohsin Iqbal, Norbert Schuch

Published Oct 20, 2021• 25 citations
arXivHTMLPDF

Abstract

Exotic quantum phases lack order parameters that are key to describing phases of matter. By creating entanglement-based order parameters, a new framework offers a way to probe topological and conventional phases.
DOI
10.1103/physrevx.11.041014

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The interplay of the constituents of interacting many-body systems may reveal emergent properties on the macroscopic scale which are not inherent to the individual constituents. These properties are expressed in macroscopic observables \ndescribing the state — denoted as the phase of the system. Continuous phase transition between phases are generically manifested in critical behavior, for example, a divergence of a macroscopic observable. The identification of the present phase of a system and the classification of critical phenomena into universality classes are exciting challenges of condensed-matter physics. \nIn this thesis, we use entanglement entropies as macroscopic quantities for the characterization of phases of quantum matter and critical theories. For groundstates of quantum many-body systems the entanglement entropy is a measure of \nthe amount of entanglement between two subsystems. The generic dependence of the entanglement entropy on the size and shape of the subsystems is contained in the well-known boundary law — stating a scaling of the entanglement entropy with the boundary between the two subsystems. We numerically investigate how the coefficient of this dependence reflects quantum phase transitions in simple spin-half bilayer models. \nSubleading terms to the boundary law such as a logarithmic contribution provide universal numbers for the criticality of field theories. We examine free and interacting theories from an entanglement entropy perspective in order to assess the role of the coefficient of the logarithmic correction induced by corners in the subsystems. Beyond its universality, this coefficient also quantifies degrees of freedom of low-lying excitations in the conformal field theory describing a critical point. \nA constant contribution to the boundary law indicates the presence of so-called topological order in the ground state of a many-body system. This extremely useful property can also be identified in classical counterparts of entanglement entropy which we study at the example of various toric code models. To this endeavor, we have designed Monte Carlo techniques which allow for an efficient numerical computation of the constant contribution. In particular, we analyze via entanglement entropies under which conditions remnants of topological order are present in the quantum system at finite temperature and at perturbations from a magnetic field. The major motivation behind this effort is to use topological order for the robust storage of a quantum information — a basic need for the construction of quantum \ncomputers.
The unrivaled robustness of topologically ordered states of matter against perturbations has immediate applications in quantum computing and quantum metrology, yet their very existence poses a challenge to our understanding of phase transitions. In particular, topological phase transitions cannot be characterized in terms of local order parameters, as it is the case with conventional symmetry-breaking phase transitions. Currently, topological order is mostly discussed in the context of nonlocal topological invariants or indirect signatures like the topological entanglement entropy. However, a comprehensive understanding of what actually constitutes topological order enabling precise quantitative statements is still lacking. Here we show that one can interpret topological order as the ability of a system to perform topological error correction. We find that this operational approach corresponding to a measurable quantity both lays the conceptual foundations for previous classifications of topological order and also leads to a successful classification in the hitherto inaccessible case of topological order in open quantum systems. We demonstrate the existence of topological order in open systems and their phase transitions to topologically trivial states. Our results demonstrate the viability of topological order in nonequilibrium quantum systems and thus substantially broaden the scope of possible technological applications.
Topological order is defined by topological invariants, rather than symmetries and local order parameters. Nonetheless some topological phases can be characterized by string order parameters and entanglement. In this article we study how string order parameters and entanglement spectra behave out-of-equilibrium following quenches in one dimensional topological models with Z\mathbb{Z}Z invariants. Previously it has been suggested that string order parameters could serve as an experimental probe of dynamical quantum phase transitions. Despite the existence of clear zeroes in the order parameters at critical times, we show that in general there is no exact quantitative or qualitative connection with the critical times of dynamical quantum phase transitions. Another possible connection is of dynamical string order parameter zeroes and dynamical crossings at the center of entanglement spectra. Here we see that there can sometimes be a connection, but it is not typical. Again there is no general quantitative or qualitative connection. Each dynamical form of criticality behaves independently, though we do see that critical times tend to be of the same order of magnitude and give an argument for why this is the case. We also find that a string order parameter which labels one topological phase can undergo non-trivial dynamics even following a quench between \emph{other} topological phases. We elucidate where connections can be made, and where they result from a consideration of insufficiently general models. These results cast doubt on the idea of genuine dynamical phases following quenches in such models.
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Order parameters are key to our understanding of phases of matter. Not only do they allow to classify phases, but they also enable the study of phase transitions through their critical exponents which identify the universal long-range physics underlying the transition. Topological phases are exotic quantum phases which are lacking the characterization in terms of order parameters. While probes have been developed to identify such phases, those probes are only qualitative in that they take discrete values, and thus provide no means to study the scaling behavior in the vicinity of phase transitions. In this paper, we develop a framework based on variational tensor networks (infinite Projected Entangled Pair States, or iPEPS) for the quantitative study of topological phase transitions through topological order parameters. These order parameters allow to quantitatively probe the behavior through a phase transition and thus to identify universal signatures of topological phase transitions. We apply our framework to the study of the Toric Code model in different magnetic fields, which along some special lines maps to the (2+1)D Ising model. Our method identifies 3D Ising critical exponents for the entire transition, consistent with those special cases and general belief. However, we in addition also find a novel critical exponent β∗≈0.02\beta^*\approx 0.02β∗≈0.02 for one of our topological order parameters, which we relate to disorder parameters in the 3D Ising model. This shows that our topological order parameters can provide additional means to characterize the universal data at topological phase transitions, and altogether demonstrates the power of this framework to microscopically study topological phase transitions and identify the universal data underlying the transition.