Tight bosonic analogs of free-fermionic symmetry-protected topological phases, and their associated edge-localized excitations, have long evaded the grasp of condensed-matter and AMO physics. In this paper, building on our initial exploration [Phys. Rev. Lett. 127, 245701 (2021)], we identify a broad class of quadratic bosonic systems subject to Markovian dissipation that realize tight bosonic analogs of the Majorana and Dirac edge modes characteristic of topological superconductors and insulators, respectively. To this end, we establish a general framework for topological metastability for these systems, by leveraging pseudospectral theory as the appropriate mathematical tool for capturing the nonnormality of the Lindbladian generator. The resulting dynamical paradigm, which is characterized by both a sharp separation between transient and asymptotic dynamics and a nontrivial topological invariant, is shown to host edge-localized modes, which we dub Majorana and Dirac bosons. Generically, such modes consist of one conserved mode and a canonically conjugate generator of an approximate phase-space translation symmetry of the dynamics. The general theory is exemplified through several representative models exhibiting the full range of exotic boundary physics that topologically metastable systems can engender. In particular, we explore the extent to which Noether's theorem is violated in this dissipative setting and the way in which certain symmetries can nontrivially modify the edge modes. Notably, we also demonstrate the possibility of anomalous parity dynamics for a bosonic cat state prepared in a topologically metastable system, whereby an equal distribution between even and odd parity sectors is sustained over a long transient. For both Majorana and Dirac bosons, observable multitime signatures in the form of anomalously long-lived quantum correlations and divergent zero-frequency power spectral peaks are proposed and discussed in detail. Our results point to a paradigm for symmetry-protected topological physics in free bosons, embedded deeply in the long-lived transient regimes of metastable dynamics.
Tight bosonic analogs of free-fermionic symmetry-protected topological phases, and their associated edge-localized excitations, have long evaded the grasp of condensed-matter and AMO physics. In this paper, building on our initial exploration [Phys. Rev. Lett. 127, 245701 (2021)], we identify a broad class of quadratic bosonic systems subject to Markovian dissipation that realize tight bosonic analogs of the Majorana and Dirac edge modes characteristic of topological superconductors and insulators, respectively. To this end, we establish a general framework for topological metastability for these systems, by leveraging pseudospectral theory as the appropriate mathematical tool for capturing the nonnormality of the Lindbladian generator. The resulting dynamical paradigm, which is characterized by both a sharp separation between transient and asymptotic dynamics and a nontrivial topological invariant, is shown to host edge-localized modes, which we dub Majorana and Dirac bosons. Generically, such modes consist of one conserved mode and a canonically conjugate generator of an approximate phase-space translation symmetry of the dynamics. The general theory is exemplified through several representative models exhibiting the full range of exotic boundary physics that topologically metastable systems can engender. In particular, we explore the extent to which Noether's theorem is violated in this dissipative setting and the way in which certain symmetries can nontrivially modify the edge modes. Notably, we also demonstrate the possibility of anomalous parity dynamics for a bosonic cat state prepared in a topologically metastable system, whereby an equal distribution between even and odd parity sectors is sustained over a long transient. For both Majorana and Dirac bosons, observable multitime signatures in the form of anomalously long-lived quantum correlations and divergent zero-frequency power spectral peaks are proposed and discussed in detail. Our results point to a paradigm for symmetry-protected topological physics in free bosons, embedded deeply in the long-lived transient regimes of metastable dynamics.
Majorana bosons, that is, tight bosonic analogs of the Majorana fermionic quasiparticles of condensed-matter physics, are forbidden for gapped free bosonic matter within a standard Hamiltonian scenario. We show how the interplay between dynamical metastability and nontrivial bulk topology makes their emergence possible in noninteracting bosonic chains undergoing Markovian dissipation. This leads to a distinctive form of topological metastability, whereby a conserved Majorana boson localized on one edge is paired, in general, with a symmetry generator localized on the opposite edge. We argue that Majorana bosons are robust against disorder and identifiable by signatures in the zero-frequency steady-state power spectrum. Our results suggest that symmetry-protected topological phases for free bosons may arise in transient metastable regimes, which persist over practical timescales.
Majorana bosons, that is, tight bosonic analogues of the Majorana fermionic quasi-particles of condensed-matter physics, are forbidden for gapped free bosonic matter within a standard Hamiltonian scenario. We show how the interplay between dynamical metastability and non-trivial bulk topology makes their emergence possible in non-interacting bosonic chains undergoing Markovian dissipation. This leads to a distinctive form of topological metastability whereby, consistent with the breakdown of Noether's theorem for open systems, a conserved Majorana boson localized on one edge is paired with a distinct symmetry generator localized on the opposite edge. We argue that Majorana bosons are robust against disorder and identifiable by signatures in the zero-frequency steady-state power spectrum. Our results suggest that symmetry-protected topological phases for free bosons may arise in transient metastable regimes, which are still stable over practical time scales.
Understanding the interplay between symmetries and phases lies at the heart of characterizing states of matter. The conventional Landau's paradigm associates local ordered states with symmetry breaking, but this paradigm fails to include gapped topological phases, which can only be distinguished by the global topological invariant and the present of gapless edge modes at open boundaries. In this thesis, variety of states of matter is enriched from two aspects: a new paradigm of phases that incorporates both symmetry broken phase and topological phase, and exotic topological phases that are realized in different degree of freedoms of the same system. The two aspects are explored by studying corresponding one dimensional fermionic models. Besides gapped phases, exotic quasiparticle interferences of gapless topological systems are also investigated. \n \nThe first chapter serves as a theoretical basis for the next two chapters by giving an introduction to topological phases in one dimension. In particular, a bosonization method is introduced to describe topological degeneracy and edge zero modes, which is favoured by the ability to deal with interactions as well as to uncover the phenomena of spin-charge separation. \n \nIn the second chapter a new paradigm of phases is proposed and investigated that incorporates both symmetry broken phases and symmetry-protected topological phases. Through exact-solvable models bond spin density wave(BSDW) states are proposed as candidates for this new paradigm, which are ZZ2 symmetry breaking due to spontaneous dimerization and yet each is topological nontrivial with edge zero modes of fractional fermions. More general lattice models are constructed, and it is shown that BSDW exists in a considerable regime of parameters for a family of lattice models, from both mean-field analysis and bosonization. \n \nIn the third chapter, we give an effective field description for spin-charge separated edge modes consisting of both Majorana modes and fractional fermions, and demonstrate how to harvest both simultaneously in a spinless fermionic ladder with particle-hole symmetry and single-chain fermion parity. Further, the idea of spin-charge separated edge modes is applied to reinterpret the prominent Haldane phase of spin-1 chain in its equivalent spin-frac12 fermionic model. \n \nIn contrast to the above two theoretically oriented topics that aim to add variety to topological phases, the third topic establishes a bridge between abstract concepts for topological systems to remarkable features that can be detected in experiments. The last chapter reveals how topological charges for topological Fermi points, through exhibiting exotic quasiparticle interference in the presence of impurities, could be visualized in STS measurements. \nThe generalized joint density of states(GJDOS) is applied to address effects from the geometry of band dispersion and topological charges separately. Both analytical results and numerical simulation suggest distinct and robust quasiparticle interference features as indicators for topological charges that can be accessed by FT-STS experiments.
We provide a framework for understanding dynamical metastability in open many-body systems of free bosons, whereby the dynamical stability properties of the system in the infinite-size (thermodynamic) limit may sharply differ from those of any finite-size truncation, and anomalous transient dynamics may arise. By leveraging pseudospectral techniques, we trace the discrepancy between asymptotic and transient dynamics to the non-normality of the underlying quadratic bosonic Lindbladian (QBL) generator and show that two distinct flavors of dynamical metastability can arise. QBLs exhibiting type I dynamical metastability, previously discussed in the context of anomalous transient amplification [Phys. Rev. Lett. 127, 245701 (2021)], are dynamically unstable in the infinite-size limit yet stable once open boundaries are imposed. In contrast, type II dynamically metastable QBLs, which we uncover in this work, are dynamically stable for infinite size but become unstable under open boundary conditions for arbitrary finite system size. We exhibit representative models for both types of metastability in the dissipative, as well as the closed-system (Hamiltonian) settings, and analyze distinctive behavior they can engender. We show that dynamical metastability manifests itself in the generation of entanglement entropy by way of a transient which reflects the stability phase of the infinite (rather than the actual finite) system and, as a result, is directly tied to the emergence of supervolume entanglement scaling in type I systems. Finally, we demonstrate how, even in Hermitian, and especially in highly non-normal regimes, the spectral properties of an infinite-size QBL are reflected in the linear response functions of the corresponding finite QBLs by way of resonant pseudospectral modes.
A new understanding of criticality in systems described by quadratic bosonic Hamiltonians (QBHs) ties the emergence of long-range correlations to boundaries of dynamical, not thermodynamical, stability in parameter space. This separation occurs because the solution of the Heisenberg equations of motion is determined by an auxiliary pseudo-Hermitian dynamical system. The boundary points of a region of dynamical stability can be either exceptional points, generically associated with long-range correlations, or Krein collisions, where correlations can be either long- or short-range. We investigate the interplay of this landscape of possibilities with band topology and boundary physics, by relying on both specific examples and general arguments. The examples stem from a two-parameter, thermodynamically unstable family of QBHs obtained from the bosonic Su-Schrieffer-Heeger model by breaking particle conservation while preserving a chiral pseudo-symmetry. As a function of an interpolation parameter, distinct regions emerge within a fixed dynamical-stability phase, which prove to be topologically trivial and nontrivial, respectively. The topological phase transition is a line of Krein collisions, which coincides with the closing of a band gap at zero and causes the localization length of the topologically mandated boundary zero modes to diverge before disappearing. We show that the chiral pseudo-symmetry induces, despite the broken particle-number symmetry, enough structure on its associated dynamical matrices to support a topological classification and a bulk-boundary correspondence, independently of dynamical stability. This strongly suggests that, for non-interacting bosons, topological physics extracted from basic index theory is insensitive to dynamical stability and, a posteriori, criticality.
The simplest continuum model of a one-dimensional noninteracting superconducting fermionic symmetry-protected topological (SPT) phase is studied in great detail using analytical methods. In a first step, we present a full exact diagonalization of the fermionic Bogoliubov--de Gennes Hamiltonian for a system of finite length and with open boundaries. In particular, we derive exact analytical expressions for the Majorana zero modes emerging in the topologically nontrivial phase, revealing their spatial localization, their transformation properties under symmetry operations, and the exact finite-size energy splitting of the associated quasidegenerate ground states. We then proceed to analyze the model via exact operator bosonization in both open and closed geometries. In the closed wire geometry, we demonstrate fermion parity switching from twisting boundary conditions in the topologically nontrivial phase. For the open wire, on the other hand, we first take a semiclassical approach employing the Mathieu equation to study the two quasidegenerate ground states as well as their energy splitting at finite system sizes. We then finally derive the exact forms of the Majorana zero modes in the bosonic language using vertex-algebra techniques. These modes are verified to be in exact agreement with the results obtained from the fermionic description. The complementary viewpoints provided by the fermionic and bosonic formulations of the superconducting SPT phase are reconciled, allowing us to provide a complete and exact account of how Majorana zero modes manifest in a bosonized description of an SPT phase.