Topologically-ordered phases of matter at nonzero temperature are conjectured to exhibit universal patterns of long-range entanglement, which can be detected using the entanglement negativity, a mixed-state entanglement measure. In this paper, we show that the entanglement negativity in certain topological orders can be understood through the properties of an emergent symmetry-protected topological (SPT) order that is localized on the entanglement bipartition. This connection leads to an understanding of (i) universal contributions to the entanglement negativity, which diagnose finite-temperature topological order and (ii) the behavior of the entanglement negativity across certain phase transitions in which thermal fluctuations eventually destroy long-range entanglement across the bipartition surface. Within this correspondence, the universal patterns of entanglement in the finite-temperature topological order are related to the stability of an emergent SPT order against a symmetry-breaking field. SPT orders protected by higher-form symmetries---which arise, for example, in the description of the entanglement negativity for ${\mathbb{Z}}_{2}$ topological order in $d=4$ spatial dimensions---remain robust even in the presence of a weak symmetry-breaking perturbation, leading to long-range entanglement at nonzero temperature for certain topological orders.Topologically-ordered phases of matter at nonzero temperature are conjectured to exhibit universal patterns of long-range entanglement, which can be detected using the entanglement negativity, a mixed-state entanglement measure. In this paper, we show that the entanglement negativity in certain topological orders can be understood through the properties of an emergent symmetry-protected topological (SPT) order that is localized on the entanglement bipartition. This connection leads to an understanding of (i) universal contributions to the entanglement negativity, which diagnose finite-temperature topological order and (ii) the behavior of the entanglement negativity across certain phase transitions in which thermal fluctuations eventually destroy long-range entanglement across the bipartition surface. Within this correspondence, the universal patterns of entanglement in the finite-temperature topological order are related to the stability of an emergent SPT order against a symmetry-breaking field. SPT orders protected by higher-form symmetries---which arise, for example, in the description of the entanglement negativity for ${\mathbb{Z}}_{2}$ topological order in $d=4$ spatial dimensions---remain robust even in the presence of a weak symmetry-breaking perturbation, leading to long-range entanglement at nonzero temperature for certain topological orders.

arXiv:2201.07792Physical Review ResearchJul 17, 202319 citations

Entropic Barriers and the Kinetic Suppression of Topological Defects

Yi-Lin Tsao, Zhu-Xi Luo

Many quantum phases, from topological orders to superfluids, are destabilized at finite temperature by the proliferation and motion of topological defects such as anyons or vortices. Conventional protection mechanisms rely on energetic gaps and fail once thermal fluctuations exceed the gap scale. Here we examine a complementary mechanism of entropic protection, in which defect nucleation is suppressed by coupling to mesoscopic auxiliary reservoirs of dimension $M$ , generating an effective free-energy barrier that increases with temperature. In the Ising chain, this produces a characteristic three-regime evolution of the correlation length as a function of temperature - linear growth, entropy-controlled plateau, and eventual breakdown - indicating a general modification of defect behavior. Focusing on two spatial dimensions, where true finite-temperature topological order is forbidden in the thermodynamic limit, we show that entropic protection can nevertheless strongly enhance stabilization at finite system size, the regime directly relevant for quantum memory and experiments. Owing to the topological character of the defects, creation and transport are independently suppressed, yielding a double parametric reduction of logical errors in the entropic toric code and enhanced coherence when the framework is extended to Berezinskii-Kosterlitz-Thouless transitions. Entropic barriers thus provide a passive and scalable route to stabilizing quantum phases in experimentally relevant regimes. We propose an experimental setup for entropic toric code using dual species Rydberg arrays with dressing.Many quantum phases, from topological orders to superfluids, are destabilized at finite temperature by the proliferation and motion of topological defects such as anyons or vortices. Conventional protection mechanisms rely on energetic gaps and fail once thermal fluctuations exceed the gap scale. Here we examine a complementary mechanism of entropic protection, in which defect nucleation is suppressed by coupling to mesoscopic auxiliary reservoirs of dimension $M$, generating an effective free-energy barrier that increases with temperature. In the Ising chain, this produces a characteristic three-regime evolution of the correlation length as a function of temperature - linear growth, entropy-controlled plateau, and eventual breakdown - indicating a general modification of defect behavior. Focusing on two spatial dimensions, where true finite-temperature topological order is forbidden in the thermodynamic limit, we show that entropic protection can nevertheless strongly enhance stabilization at finite system size, the regime directly relevant for quantum memory and experiments. Owing to the topological character of the defects, creation and transport are independently suppressed, yielding a double parametric reduction of logical errors in the entropic toric code and enhanced coherence when the framework is extended to Berezinskii-Kosterlitz-Thouless transitions. Entropic barriers thus provide a passive and scalable route to stabilizing quantum phases in experimentally relevant regimes. We propose an experimental setup for entropic toric code using dual species Rydberg arrays with dressing.

quant-pharXiv:2602.16777Feb 20, 20260 citations

Topological Quantum Order from Symmetry and the Role of Temperature

Gerardo Ortíz

Bulletin of the American Physical SocietyMar 16, 20090 citations

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

Zohar Nussinov, Gerardo Ortíz

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?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?

arXiv:0709.2717Physical Review BFeb 14, 2008180 citations

{\mathbb{Z}}_{2}

topological order in

d=4

spatial dimensions---remain robust even in the presence of a weak symmetry-breaking perturbation, leading to long-range entanglement at nonzero temperature for certain topological orders.

M

, generating an effective free-energy barrier that increases with temperature. In the Ising chain, this produces a characteristic three-regime evolution of the correlation length as a function of temperature - linear growth, entropy-controlled plateau, and eventual breakdown - indicating a general modification of defect behavior. Focusing on two spatial dimensions, where true finite-temperature topological order is forbidden in the thermodynamic limit, we show that entropic protection can nevertheless strongly enhance stabilization at finite system size, the regime directly relevant for quantum memory and experiments. Owing to the topological character of the defects, creation and transport are independently suppressed, yielding a double parametric reduction of logical errors in the entropic toric code and enhanced coherence when the framework is extended to Berezinskii-Kosterlitz-Thouless transitions. Entropic barriers thus provide a passive and scalable route to stabilizing quantum phases in experimentally relevant regimes. We propose an experimental setup for entropic toric code using dual species Rydberg arrays with dressing.

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?