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arXiv:0709.2717 · Physical Review B

Autocorrelations and thermal fragility of anyonic loops in topologically quantum ordered systems

Zohar Nussinov, Gerardo Ortíz

Published Feb 14, 2008• 180 citations
arXivHTMLPDF

Abstract

Are systems that display topological quantum order (TQO), and have a gap to excitations, hardware fault-tolerant at finite temperatures? We show that in models that display low $d$-dimensional gaugelike symmetries, such as Kitaev's and its generalizations, the expectation value of topological symmetry operators vanishes at any nonzero temperature, a phenomenon that we coined thermal fragility. The autocorrelation time for the nonlocal topological quantities in these systems may remain finite even in the thermodynamic limit. We provide explicit expressions for the autocorrelation functions in Kitaev's toric code model. If temperatures far below the gap may be achieved then these autocorrelation times, albeit finite, can be made large. The physical engine behind the loss of correlations at large spatial and/or temporal distance is the proliferation of topological defects at any finite temperature as a result of a dimensional reduction. This raises an important question: How may we best quantify the degree of protection of quantum information in a topologically ordered system at finite temperature?

DOI
10.1103/physrevb.77.064302

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Are systems that display topological quantum order (TQO), and have a gap to excitations, hardware fault-tolerant at finite temperatures? We show that in models that display low ddd-dimensional gaugelike symmetries, such as Kitaev's and its generalizations, the expectation value of topological symmetry operators vanishes at any nonzero temperature, a phenomenon that we coined thermal fragility. The autocorrelation time for the nonlocal topological quantities in these systems may remain finite even in the thermodynamic limit. We provide explicit expressions for the autocorrelation functions in Kitaev's toric code model. If temperatures far below the gap may be achieved then these autocorrelation times, albeit finite, can be made large. The physical engine behind the loss of correlations at large spatial and/or temporal distance is the proliferation of topological defects at any finite temperature as a result of a dimensional reduction. This raises an important question: How may we best quantify the degree of protection of quantum information in a topologically ordered system at finite temperature?
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We identify a three-dimensional system that exhibits long-range entanglement at sufficiently small but nonzero temperature-it therefore constitutes a quantum topological order at finite temperature. The model of interest is known as the fermionic toric code, a variant of the usual 3D toric code, which admits emergent fermionic pointlike excitations. The fermionic toric code, importantly, possesses an anomalous two-form symmetry, associated with the spacelike Wilson loops of the fermionic excitations. We argue that it is this symmetry that imbues low-temperature thermal states with a novel topological order and long-range entanglement. Based on the current classification of three-dimensional topological orders, we expect that the low-temperature thermal states of the fermionic toric code belong to a phase of matter that, in the context of equilibrium phases of matter, only exists at nonzero temperatures. We conjecture that further examples of topological orders at nonzero temperatures are given by discrete gauge theories with anomalous two-form symmetries. Our Letter therefore opens the door to studying quantum topological order at nonzero temperature in physically realistic dimensions.
Topological systems are an exciting frontier of condensed matter physics, as well as having deep relationships to quantum computing. Progress continues to be made both towards building practical quantum computers and experimentally identifying topological features in real materials. However, these attempts have to contend with the messiness of the real world including the effects of finite temperatures. In quantum computing thermal errors are studied through the framework of quantum error correction. This allows a characterisation of how damaging these errors are, and has provided many general results on the limits of how well given quantum codes can endure thermal errors. In the first part of this thesis we study the typical response of one of the best known quantum codes (the two-dimensional toric code) when subjected to thermal noise. This typical behaviour is compared to the limiting bounds and we note differences that arise due to entropic effects. Topological features are typically properties of the ground state of a system. In some cases this means finite temperatures can be damaging, washing out the topological properties with excitations above the ground state. But in other cases it has been shown that topological systems can have their own unique finite temperature behaviours, which can be used to identify them. In the second part of the thesis we study the topological model Kitaev's honeycomb. Fixing certain conserved quantities restricts the model to special sectors, some of which carry topological phases. We begin by investigating an intrinsically finite temperature property of these topological phases, thermal edge currents. At too high temperatures, close to the bulk gap, the topological behaviour of these currents breaks down. However, we show that at lower temperatures they can be identified and shown to have the expected properties such as robustness to perturbations. Following this we investigate the thermal states of the full model. Our approach to this allows us to probe the topological order of the model in a way that has not been done before and we identify a number of interesting new features of the finite temperature physics.
In this work the topological order at finite temperature in two-dimensional color code is studied. The topological entropy is used to measure the behavior of the topological order. Topological order in color code arises from the colored string-net structures. By imposing the hard constrained limit the exact solution of the entanglement entropy becomes possible. For finite size systems, by raising the temperature, one type of string-net structure is thermalized and the associative topological entropy vanishes. In the thermodynamic limit the underlying topological order is fragile even at very low temperatures. Taking first the thermodynamic limit and then the zero-temperature limit and vice versa does not commute, and their difference is related only to the topology of regions. The contribution of the colors and symmetry of the model in the topological entropy is also discussed. It is shown how the gauge symmetry of the color code underlies the topological entropy.
In this work the topological order at finite temperature in two-dimensional color code is studied. The topological entropy is used to measure the behavior of the topological order. Topological order in color code arises from the colored string-net structures. By imposing the hard constrained limit the exact solution of the entanglement entropy becomes possible. For finite size systems, by raising the temperature, one type of string-net structure is thermalized and the associative topological entropy vanishes. In the thermodynamic limit the underlying topological order is fragile even at very low temperatures. Taking first the thermodynamic limit and then the zero-temperature limit and vice versa does not commute, and their difference is related only to the topology of regions. The contribution of the colors and symmetry of the model in the topological entropy is also discussed. It is shown how the gauge symmetry of the color code underlies the topological entropy.
As a prototype model of topological quantum memory, two-dimensional toric code is genuinely immune to generic local static perturbations, but fragile at finite temperature and also after non-equilibrium time evolution at zero temperature. We show that dynamical localization induced by disorder makes the time evolution a local unitary transformation at all times, which keeps topological order robust after a quantum quench. We verify this conclusion by investigating the Wilson loop expectation value and topological entanglement entropy. Our results suggest that the two dimensional topological quantum memory can be dynamically robust at zero temperature.