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.
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.
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.
One of the important characteristics of topological phases of matter is the topology of the underlying manifold on which they are defined. In this paper, we present the sensitivity of such phases of matter to the underlying topology, by studying the phase transitions induced due to the change in the boundary conditions. We claim that these phase transitions are accompanied by broken symmetries in the excitation space and to gain further insight we analyze various signatures like the ground state degeneracy, topological entanglement entropy while introducing the open-loop operator whose expectation value effectively captures the phase transition. Further, we extend the analysis to an open quantum setup by defining effective collapse operators, the dynamics of which cool the system to distinct steady states both of which are topologically ordered. We show that the phase transition between such steady states is effectively captured by the expectation value of the open-loop operator.
Topological phases of matter offer a promising platform for quantum computation and quantum error correction. Nevertheless, unlike its counterpart in pure states, descriptions of topological order in mixed states remain relatively underexplored. Our work gives two definitions for replica topological order in mixed states, which involve n copies of density matrices of the mixed state. Our framework categorizes topological orders in mixed states as either quantum, classical, or trivial, depending on the type of information that can be encoded. For the case of the toric code model in the presence of decoherence, we associate for each phase a quantum channel and describes the structure of the code space. We show that in the quantum-topological phase, there exists a postselection-based error correction protocol that recovers the quantum information, while in the classical-topological phase, the quantum information has decohere and cannot be fully recovered. We accomplish this by describing the mixed state as a projected entangled pairs state (PEPS) and identifying the symmetry-protected topological order of its boundary state to the bulk topology. Using this formalism, we enumerate all the possible mixed state phases which result from applying a local decoherence channel to the toric code. In addition to the classical-topological phases that have been previously reported on, we also find mixed states exhibiting chiral topological order. We discuss the extent that our findings can be extrapolated to $n\ensuremath{\rightarrow}1$ limit.
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 phases of matter, characterized by robust edge states and quantized bulk invariants, have revolutionized condensed matter physics. Identifying and classifying their associated phase transitions, however, remains a significant challenge, particularly when conventional order parameters are absent. This paper introduces and explores the utility of determinantal quantities as powerful signatures for detecting and characterizing topological phase transitions. We demonstrate how determinants derived from the Bloch Hamiltonian, Green's functions, or related matrices can exhibit distinct, often singular, behaviors at critical points separating different topological phases. Through theoretical analysis and illustrative examples from various topological models, including the Su-Schrieffer-Heeger model, Haldane model, and Weyl semimetals, we show that these determinantal signatures provide a direct and robust link to the underlying topological invariants and spectral properties, such as band gap closures and inversions. The methodology offers a complementary and potentially more direct approach to unraveling the intricate nature of topological phase transitions, opening new avenues for both theoretical understanding and experimental detection in a wide range of quantum materials.
Topological phases of matter have been the focus of intense research for the past decades. Their intrinsic topological nature exhibits perturbation-resilient effects in macroscopic materials. These properties have applications that span sensing, electronics, spintronics, lasers, and quantum computing. They have also inspired the development of new mathematical models to study their underlying topological quantum field theory. In this work, we visit three aspects of topological phases of matter. First, we use tensor-network tools, along with the aid of Purdue's high-performance computing clusters, to simulate symmetry-protected topological phases in 1D. We update works on the realization of the Haldane phase in Cu2O excitons. In addition, we related the group-cohomological classification of these phases to their order parameters. Second, we design 2D topological insulators with higher Chern numbers. This is achieved through distant physical interactions in accordance with algebraic geometry arguments. Third, we study 2D topological quantum error-correcting codes. In particular, we construct the boundaries, domain walls, and 0D defects of Abelian twisted quantum doubles leveraging anyon condensation. We discuss these codes' advantages in correcting errors for quantum computing. We also give microscopic and macroscopic methods for computing their ground state degeneracy. These distinct but related studies illustrate the power of using physical arguments and mathematical proofs in the simulation, design, and computing aspects of topological phases.