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.
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.
The fundamental understanding of phases and their transitions has been a central theme in condensed matter physics. Until recently, it was largely believed that the Landau symmetry breaking principle was effective in distinguishing different phases of matter, with broken symmetries signaling the phase transition. But with the discovery of topological phases which are beyond the Landau symmetry breaking principle, the identification and classification of quantum phases at absolute zero has opened up new unexplored avenues thus leading to exciting theoretical discoveries further propelling technological advancement. Topological phases of matter are characterized by the notion of topological order and in this work we aim to explore and understand topological phases by introducing novel signatures which characterize topological order. The robustness of these phases to external perturbation makes them an ideal candidate to store and manipulate quantum information thus making them an unique and interesting prospect for realizing quantum computers. There have been several signatures to characterize intrinsic topological order, for instance the invariance of the topologically ordered state under local operators, the dependence of ground state degeneracy on the underlying manifold and its robustness to external perturbation, topological entanglement entropy related to the quantum dimension of the supers-selection sectors, the inability to construct the topologically ordered state from a product state via constant depth unitary transformations. With toric code as the toy model, we analyze the robustness of topological order on a manifold supporting open boundaries by computing some of the above signatures which effectively detect a topological to trivial phase transition. We then probe the existence of a quantum criticality between distinct topological phases obtained by varying the underlying manifold. In these scenarios, most of the above signatures turn out to be ineffective in detecting the distinct phases leading to the introduction of an non- local order parameter whose construction is facilitated by the phenomenon of anyon condensation. The signatures for quantitatively and qualitatively characterizing intrinsic topological order being highly scenario dependent and also with its definition for mixed states being elusive we introduce an operational definition based on concepts of topological error correction. We define a state to be topologically ordered if the errors in the state can be corrected by an error correction circuit of finite depth. To concretize the notion of topological to trivial phase transition in an open setting we turn to nonequilibrium phenomenon, for example: Directed Percolation, with the change in percolation rate driving a dynamical phase transition between absorbing and active states with the former being topologically ordered while the latter being topologically trivial. Additionally, we explore the notion of topological phase transitions between distinct topological phases obtained by varying underlying topology in an open setting, analogous to the closed setting discussed earlier. To summarize, we have introduced various mixed states which exhibit topological order and also an operational definition to quantify topological order applicable across multitude of scenarios. We extend the above operational definition to quantify and detect quantum phase transitions in the case of Symmetry Protected Topological (SPT) phases. To further validate the above notion, we consider the perturbed variants of the Su-Schrieffer–Heeger (SSH) models and detect quantum phase transitions to a high accuracy by employing the techniques from the framework of tensor networks. It is significant to note the distinction of the error correction algorithms applied earlier in the case of intrinsic topological order were independent of symmetry constraints while in the current scenario we impose additional symmetry constraints to accurately detect the phase transition. In addition, we also devise error correction strategies with respect to topologically trivial states to detect quantum phase transitions which do not involve topological phases. This gives rise to a very fundamental question on whether error correction statistics with a well defined error correction algorithm, not necessarily optimal, are capable of detecting a equivalence classes of phases and thereby acting as a reliable probe to effectively detect topological/quantum phase transitions? From theoretical and numerical end of the spectrum we shift gears to explore possible experimental platforms with an aim to realize some of the quantum many-body phenomenon discussed earlier. While there have been several innovative experimental avenues to realize the above, one such promising candidate has been ultracold polar molecules setups that offer additional degrees of freedom due to the ro-vibrational degrees of freedom. Based on the chemical reaction between atoms and molecules which results in a quantum Zeno-based blockade, we devise several optimal strategies to efficiently detect molecules using atom as probe, we further extend the above technique to entangle the internal states of molecules and atoms. In addition, we also present optimal strategies for dissipative state engineering using the atom-molecule interactions.
Topological order represents a profound state of matter characterized by exotic properties robust against local perturbations, such as fractionalized excitations and ground state degeneracy dependent on system topology. While extensively studied in closed quantum systems, the detection and characterization of topological order in open quantum systems, which inevitably interact with their environment, remain a significant challenge. Environmental coupling typically introduces decoherence and dissipation, potentially obscuring or even destroying fragile quantum correlations. This paper investigates the concept of "entropic fingerprints" as a robust diagnostic tool for identifying topological order in systems subjected to environmental influences. We explore how various entanglement measures, particularly the topological entanglement entropy and related information-theoretic quantities, can be adapted and reinterpreted in the context of non-equilibrium steady states or dynamically evolving open systems. Through a theoretical framework employing Lindblad master equations and tensor network methods, we demonstrate that universal signatures of topological order, though potentially modified, can persist in the steady-state entanglement properties of open systems. Our findings suggest that specific patterns in the bipartite entanglement entropy, conditional entropies, and mutual information, robust against a finite range of environmental parameters, can serve as reliable indicators of underlying topological phases. This research not only offers a pathway for experimentally probing topological materials in realistic, noisy environments but also deepens our fundamental understanding of how quantum information is preserved or degraded under open system dynamics.
Topological quantum phase transitions are characterized by changes in global topological invariants. These invariants classify many-body systems beyond the conventional paradigm of local order parameters describing spontaneous symmetry breaking. For noninteracting electrons, it is well understood that such transitions are continuous and always accompanied by a gap closing in the energy spectrum, given that the symmetries protecting the topological phase are maintained. Here, we demonstrate that a sufficiently strong electron-electron interaction can fundamentally change the situation: we discover a topological quantum phase transition of first-order character in the genuine thermodynamic sense that occurs without a gap closing. Our theoretical study reveals the existence of a quantum critical endpoint associated with an orbital instability on the transition line between a 2D topological insulator and a trivial band insulator. Remarkably, this phenomenon entails unambiguous signatures related to the orbital occupations that can be detected experimentally.
We analyze the computational aspects of detecting topological order in a quantum many-body system. We contrast the widely used topological entanglement entropy with a recently introduced operational definition for topological order based on error correction properties, finding exponential scaling with the system size for the former and polynomial scaling for the latter. We exemplify our approach for a variant of the paradigmatic toric code model with mobile particles, finding that the error correction method allows to treat substantially larger system sizes. In particular, the phase diagram of the model can be successfully computed using error correction, while the topological entanglement entropy is too severely limited by finite size effects to obtain conclusive results. While we mainly focus on one-dimensional systems whose ground states can be expressed in terms of matrix product states, our strategy can be readily generalized to higher dimensions and systems out of equilibrium, even allowing for an efficient detection of topological order in current quantum simulation experiments.
In this work, we present some new understanding of topological order, including three main aspects. (1) It was believed that classifying topological orders corresponds to classifying gapped quantum states. We show that such a statement is not precise. We introduce the concept of gapped quantum liquid as a special kind of gapped quantum states that can ``dissolve'' any product states on additional sites. Topologically ordered states actually correspond to gapped quantum liquids with stable ground-state degeneracy. Symmetry-breaking states for on-site symmetry are also gapped quantum liquids, but with unstable ground-state degeneracy. (2) We point out that the universality classes of generalized local unitary (gLU) transformations (without any symmetry) contain both topologically ordered states and symmetry-breaking states. This allows us to use a gLU invariant---topological entanglement entropy---to probe the symmetry-breaking properties hidden in the exact ground state of a finite system, which does not break any symmetry. This method can probe symmetry- breaking orders even without knowing the symmetry and the associated order parameters. (3) The universality classes of topological orders and symmetry-breaking orders can be distinguished by stochastic local (SL) transformations (i.e., local invertible transformations): small SL transformations can convert the symmetry-breaking classes to the trivial class of product states with finite probability of success, while the topological-order classes are stable against any small SL transformations, demonstrating a phenomenon of emergence of unitarity. This allows us to give a definition of long-range entanglement based on SL transformations, under which only topologically ordered states are long-range entangled.
Abstract We analyze the computational aspects of detecting topological order in a quantum many-body system. We contrast the widely used topological entanglement entropy with a recently introduced operational definition for topological order based on error correction properties, finding exponential scaling with the system size for the former and polynomial scaling for the latter. We exemplify our approach for a variant of the paradigmatic toric code model with mobile particles, finding that the error correction method allows to treat substantially larger system sizes. In particular, the phase diagram of the model can be successfully computed using error correction, while the topological entanglement entropy is too severely limited by finite size effects to obtain conclusive results. While we mainly focus on one-dimensional systems whose ground states can be expressed in terms of matrix product states, our strategy can be readily generalized to higher dimensions and systems out of equilibrium, even allowing for an efficient detection of topological order in current quantum simulation experiments.