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, 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.
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.
This paper develops a rigorous C*-algebraic framework for understanding and classifying topological quantum phases through the lens of spectral entanglement. Topological quantum phases represent a frontier in condensed matter physics, characterized by robust properties that are insensitive to local perturbations and often manifest exotic excitations. Concurrently, entanglement, a quintessential quantum mechanical phenomenon, serves as a crucial resource for quantum information processing and a powerful diagnostic tool for quantum phases of matter. By integrating the formalisms of C*-algebras and spectral theory, we propose a robust mathematical language to describe the algebraic structure of observables in systems exhibiting topological order. Our approach leverages the spectral properties of density matrices and entanglement Hamiltonians within this algebraic setting to define and quantify spectral entanglement, revealing its intrinsic connection to topological invariants. We demonstrate how this framework provides a unified perspective on the interplay between local algebras of observables, their spectral characteristics, and the non-local correlations that define topological order. The paper outlines theoretical constructs for computing entanglement measures and topological indices directly from the C*-algebraic properties, thereby offering new avenues for classifying these phases and exploring their underlying quantum information content with potential implications for fault-tolerant quantum computing and novel electronic devices.
The discovery of topological features of quantum states plays a central role in modern condensed matter physics and various artificial systems. Due to the absence of local order parameters, the detection of topological phase transitions remains a challenge. Machine learning may provide effective methods for identifying topological features. In this work, we show that manifold learning can successfully retrieve topological quantum phase transitions in momentum space. Our results show that the Chebyshev distance (CD) between two data points can successfully capture the main features of topological phase transitions, while the widely used Euclidean distance in general fails. The similarity matrix built upon the CD thus naturally exhibits uncorrelated cluster structures, corresponding to distinct sectors in the topological phase diagram. Then a diffusion map is applied to implement dimensionality reduction and to learn about topological quantum phase transitions in an unsupervised manner. Our demonstrations on the Su-Schrieffer-Heeger (SSH) model, the 2D Qi-Wu-Zhang (QWZ) model, and the quenched SSH model show the capability of generic unsupervised learning, when equipped with a suitable distance metric, in exploring topological quantum phase transitions.
Understanding correlation effects in topological phases of matter is at the forefront of current research in condensed matter physics. Here we try to clarify some subtleties in studying topological behaviors of interacting Weyl semimetals. It is well known that there exist two topological invariants defined to identify their topological character. One is the many-body Chern number, which can be directly linked to the Hall conductivity and thus to the two-particle correlations. The other is the topological index constructed from the single-particle Green's functions. Because the information on Green's functions is achieved more easily than the many-body wave functions, usually only the topological index is employed in the literature. However, the approach based on the single-particle Green's function can break down in the strongly correlated phase. For illustration, an exactly solvable two-orbital model with momentum-local two-body interactions is discussed, in which both topological invariants can be calculated analytically. We find that the topological index calculated from the Green's function formalism can be nonzero even for a nontopological strongly correlated phase with a vanishing many-body Chern number. In addition, we stress that the physical surface states implied by nonzero many-body Chern numbers should be the edge modes of particle-hole collective excitations, rather than those of a quasiparticle nature derived from the Green's function formalism. Our observations thus demonstrate the limitation of the validity of the Green's function formalism in the investigations of interacting topological materials.
The detection of topological phases of matter has become a central issue in recent years. Conventionally, the realization of a specific topological phase in condensed matter physics relies on probing the underlying surface band dispersion or quantum transport signature of a real material, which may be imperfect or even absent. On the other hand, quantum simulation offers an alternative approach to directly measure the topological invariant on a universal quantum computer. However, experimentally demonstrating high-dimensional topological phases remains a challenge due to the technical limitations of current experimental platforms. Here, we investigate the three-dimensional topological insulators in the AIII (chiral unitary) symmetry class, which yet lack experimental realization. Using the nuclear magnetic resonance system, we experimentally demonstrate their topological properties, where a dynamical quenching approach is adopted and the dynamical bulk-boundary correspondence in the momentum space is observed. As a result, the topological invariants are measured with high precision on the band-inversion surface, exhibiting robustness to the decoherence effect. Our Letter paves the way toward the quantum simulation of topological phases of matter in higher dimensions and more complex systems through controllable quantum phases transitions.
This paper explores the rigorous formulation of entanglement spectra within the framework of C*-algebras, proposing their utility as robust topological invariants for classifying quantum phases of matter. Entanglement spectra, derived from the eigenvalues of reduced density matrices, have emerged as powerful diagnostic tools for characterizing topological order, often revealing hidden patterns beyond conventional symmetry-breaking paradigms. By embedding quantum systems and their subsystem decompositions into the abstract mathematical structure of C*-algebras, we provide a generalized and mathematically rigorous approach to define and analyze these spectra. This C*-algebraic perspective allows for a unified treatment of finite and infinite quantum systems, including those arising in quantum field theory, and naturally incorporates the algebraic notion of locality and quantum states. We demonstrate how specific features of the entanglement spectrum, such as degeneracy patterns or gaps, can be viewed as invariants under C*-algebraic deformations that preserve the underlying topological order. This framework suggests a deep connection between the algebraic properties of observables, the structure of entanglement, and the classification of topological phases, potentially linking to established mathematical tools like K-theory. The work establishes a theoretical foundation for leveraging entanglement spectra as fundamental topological indicators within a highly abstract yet physically relevant algebraic setting.