Quantum matter involves the study of entanglement patterns in the ground states of many-body \nsystems. Of significant interest have recently been topological states of matter, \nwhich exhibit characteristics only described globally. As such they are robust to local deformations. \nIn this thesis, we study inter-correlations of many-body states through the entanglement spectrum, \nobtained by a bipartition of both topological and non-topological systems. \n \nIn particular, we introduce two novel diagnostics which operate on entanglement spectra. \nFor topological phases supporting edge states on open boundaries \nwe take a quantum-information inspired approach by invoking \nthe monogamy relations obeyed by multi-partite systems. \nWithin a strictly single-particle framework, \nwe establish a correspondence between highly entangled mode and the existence of edge states. \nIn the many-body context, we introduce the interaction distance \nof a mixed state. \nExclusively via the entanglement spectrum it \ndetermines how close a free-fermion state lies and what the emergent free quasiparticles are. \n \nWe apply these two measures to diagnose the properties of a variety of free and interacting fermionic topological systems and reinterpret their properties from a fresh point of view. \nOur case studies revolve around Kitaev's honeycomb model, \nwhich supports both short-range and long-range topological order, \nconstituting it thus relevant to both the monogamy qualifier and the interaction distance. \nThe possibility to diagnose whether a model has zero interacting distance or if it supports maximally entangled states provides central and compact information about the behaviour of complex quantum systems.
Quantum matter involves the study of entanglement patterns in the ground states of many-body \nsystems. Of significant interest have recently been topological states of matter, \nwhich exhibit characteristics only described globally. As such they are robust to local deformations. \nIn this thesis, we study inter-correlations of many-body states through the entanglement spectrum, \nobtained by a bipartition of both topological and non-topological systems. \n \nIn particular, we introduce two novel diagnostics which operate on entanglement spectra. \nFor topological phases supporting edge states on open boundaries \nwe take a quantum-information inspired approach by invoking \nthe monogamy relations obeyed by multi-partite systems. \nWithin a strictly single-particle framework, \nwe establish a correspondence between highly entangled mode and the existence of edge states. \nIn the many-body context, we introduce the interaction distance \nof a mixed state. \nExclusively via the entanglement spectrum it \ndetermines how close a free-fermion state lies and what the emergent free quasiparticles are. \n \nWe apply these two measures to diagnose the properties of a variety of free and interacting fermionic topological systems and reinterpret their properties from a fresh point of view. \nOur case studies revolve around Kitaev's honeycomb model, \nwhich supports both short-range and long-range topological order, \nconstituting it thus relevant to both the monogamy qualifier and the interaction distance. \nThe possibility to diagnose whether a model has zero interacting distance or if it supports maximally entangled states provides central and compact information about the behaviour of complex quantum systems.
Topological phases of matter possess intricate correlation patterns typically probed by entanglement entropies or entanglement spectra. In this Letter, we propose an alternative approach to assessing topologically induced edge states in free and interacting fermionic systems. We do so by focussing on the fermionic covariance matrix. This matrix is often tractable either analytically or numerically, and it precisely captures the relevant correlations of the system. By invoking the concept of monogamy of entanglement, we show that highly entangled states supported across a system bipartition are largely disentangled from the rest of the system, thus, usually appearing as gapless edge states. We then define an entanglement qualifier that identifies the presence of topological edge states based purely on correlations present in the ground states. We demonstrate the versatility of this qualifier by applying it to various free and interacting fermionic topological systems.
Topological phases are characterized by their entanglement properties, which is manifest in a direct relation between entanglement spectra and edge states discovered by Li and Haldane. We propose to leverage the power of synthetic quantum systems for measuring entanglement via the Entanglement Hamiltonian to probe this relationship experimentally. This is made possible by exploiting the quasi-local structure of Entanglement Hamiltonians. The feasibility of this proposal is illustrated for two paradigmatic examples realizable with current technology, an integer quantum Hall state of non-interacting fermions on a 2D lattice and a symmetry protected topological state of interacting fermions on a 1D chain. Our results pave the road towards an experimental identification of topological order in strongly correlated quantum many-body systems.
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.
Phases of gapped quantum liquids are topologically ordered and have very interesting physical features that are completely robust against any local perturbation that do not close the bulk energy gap. These universal properties are hidden in the ground states of these systems, as different patterns of many-body long-range entanglement. In this thesis we study the universal properties of gapped quantum liquids from various perspectives. We propose the notion of Universal Wavefunction Overlap as a way of extracting almost complete information about the underlying entanglement structure in a system with topological order. We propose an efficient numerical methods to use these universal wavefunction overlaps as topological order parameters and demonstrate their usefulness with concrete numerical computations. In 2 + 1D these overlaps correspond to known quantities and contain information about anyonic particle excitations. We show that in 3 + 1D, these overlaps contain information about linked multi-string braiding processes, in particular three-string braiding. \n \nIn the second part of this thesis, we study boundary physics of systems with topological order. We investigate the correspondence between edge and entanglement spectra for non-chiral topological systems in general and with the presence of extra symmetries and dualities. We also show that by local deformations of the fixed-point wavefunction on non- chiral topological orders, all possible edge theories can be extracted from its entanglement Hamiltonian. Finally we introduce the notion of fermionic gapped boundaries and see how the phase diagram of the simplest topological orders get significantly enriched.
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.
Identifying entanglement-based order parameters characterizing topological systems, in particular topological superconductors and topological insulators, has remained a major challenge for the physics of quantum matter in the last two decades. Here we show that the end-to-end, long-distance, bipartite squashed entanglement between the edges of a many-body system, defined in terms of the edge-to-edge quantum conditional mutual information, is the natural nonlocal order parameter for topological superconductors in one dimension as well as in quasi-one-dimensional geometries. For the Kitaev chain in the entire topological phase, the edge squashed entanglement is quantized to ln(2)/2, half the maximal Bell-state entanglement, and vanishes in the trivial phase. Such a topological squashed entanglement exhibits the correct scaling at the quantum phase transition, is stable in the presence of interactions, and is robust against disorder and local perturbations. Edge quantum conditional mutual information and edge squashed entanglement defined with respect to different multipartitions discriminate topological superconductors from symmetry breaking magnets, as shown by comparing the fermionic Kitaev chain and the spin-1/2 Ising model in transverse field. For systems featuring multiple topological phases with different numbers of edge modes, like the quasi-1D Kitaev ladder, topological squashed entanglement counts the number of Majorana excitations and distinguishes the different topological phases of the system. In fact, we show that the edge quantum conditional mutual information and the edge squashed entanglement remain valid detectors of topological superconductivity even for systems, like the Kitaev tie with long-range hopping, featuring geometrical frustration and a suppressed bulk-edge correspondence.