Topological states of fermionic matter can be induced by means of a suitably engineered dissipative dynamics. Dissipation then does not occur as a perturbation, but rather as the main resource for many-body dynamics, providing a targeted cooling into topological phases starting from arbitrary initial states. We explore the concept of topological order in this setting, developing and applying a general theoretical framework based on the system density matrix that replaces the wave function appropriate for the discussion of Hamiltonian ground-state physics. We identify key analogies and differences to the more conventional Hamiltonian scenario. Differences essentially arise from the fact that the properties of the spectrum and of the state of the system are not as tightly related as in the Hamiltonian context. We provide a symmetry-based topological classification of bulk steady states and identify the classes that are achievable by means of quasi-local dissipative processes driving into superfluid paired states. We also explore the fate of the bulk-edge correspondence in the dissipative setting and demonstrate the emergence of Majorana edge modes. We illustrate our findings in one- and two-dimensional models that are experimentally realistic in the context of cold atoms.
Topological states of fermionic matter can be induced by means of a suitably engineered dissipative dynamics. Dissipation then does not occur as a perturbation, but rather as the main resource for many-body dynamics, providing a targeted cooling into topological phases starting from arbitrary initial states. We explore the concept of topological order in this setting, developing and applying a general theoretical framework based on the system density matrix that replaces the wave function appropriate for the discussion of Hamiltonian ground-state physics. We identify key analogies and differences to the more conventional Hamiltonian scenario. Differences essentially arise from the fact that the properties of the spectrum and of the state of the system are not as tightly related as in the Hamiltonian context. We provide a symmetry-based topological classification of bulk steady states and identify the classes that are achievable by means of quasi-local dissipative processes driving into superfluid paired states. We also explore the fate of the bulk-edge correspondence in the dissipative setting and demonstrate the emergence of Majorana edge modes. We illustrate our findings in one- and two-dimensional models that are experimentally realistic in the context of cold atoms.
Topological states of fermionic matter can be induced by means of a suitably engineered dissipative dynamics. Dissipation then does not occur as a perturbation, but rather as the main resource for many-body dynamics, providing a targeted cooling into topological phases starting from arbitrary initial states. We explore the concept of topological order in this setting, developing and applying a general theoretical framework based on the system density matrix that replaces the wave function appropriate for the discussion of Hamiltonian ground-state physics. We identify key analogies and differences to the more conventional Hamiltonian scenario. Differences essentially arise from the fact that the properties of the spectrum and of the state of the system are not as tightly related as in the Hamiltonian context. We provide a symmetry-based topological classification of bulk steady states and identify the classes that are achievable by means of quasi-local dissipative processes driving into superfluid paired states. We also explore the fate of the bulk-edge correspondence in the dissipative setting and demonstrate the emergence of Majorana edge modes. We illustrate our findings in one- and two-dimensional models that are experimentally realistic in the context of cold atoms.
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.
Robust states emerging at the boundaries of a system are an important hallmark of topological matter. Here, using the Su-Schrieffer-Heeger model and the Kitaev chain as examples, we study the impact of a type of experimentally realizable bond dissipation on topological systems by calculating the steady-state density matrix, and demonstrate that such dissipation applied near the system boundary can assist in preparing topological edge states of the parent Hamiltonian, irrespective of the initial state or filling. This effect stems from the matching between the phase distribution encoded in the topological edge states and the target state prepared through bond dissipation. This work provides new insights into the preparation of topological edge states, particularly in the context of Majorana zero modes.
The Schrödinger equation is nearly a century old, yet we are still in the midst of uncovering the remarkable phenomena emerging in many-body quantum systems. From superconductivity to anyonic quasiparticles, nature consistently surprises with its rich self-organization. To elucidate and grasp this variety, it is paramount to understand the phases of matter that can occur in many-body ground states, as well as their emergent collective excitations. Of particular interest are topological phases of matter, characterized by exotic excitations or edge phenomena. There exist by now several universal frameworks for gapped systems, i.e., those with an energy gap above the ground state. However, in the last decade, a multitude of gapless quantum wires---effectively one-dimensional systems---have been reported to be topologically non-trivial. A framework for their understanding and classification is missing. In addition to ground state order---topological or otherwise---a more complete picture involves the properties of excitations above the ground state. Alas, little is known about excitations beyond the universal low-energy regime. In part, this is due to a lack of analytical and numerical methods able to describe excitations at finite energies, especially in strongly-interacting systems beyond one dimension. In this thesis, we address these issues: firstly, we build a general understanding of topological phases in one dimension, including both gapped and gapless cases. In particular, we unify previously studied examples into a single framework. Secondly, we develop a novel numerical method for obtaining spectral functions in two dimensions---these give direct insight into the properties of excitations and are moreover experimentally measurable. Using this numerical method, we uncover a variety of robust properties of excitations at finite energies. Part I of this thesis concerns gapped and gapless topological phases in one dimension. In Chapter 2, we first treat the case of non-interacting fermions. Therein, we review the known classification of gapped phases before extending it to the gapless case, showing that exponentially-localized Majorana zero modes can still emerge at the edge when the bulk is gapless. Interacting gapped phases are discussed in Chapter 3, with a focus on symmetry-protected topological order. These have already been classified; our contribution is to provide a non-technical review of this classification as well as showing that many paradigmatic model Hamiltonians can be related to one another. Finally, Chapter 4 introduces the notion of symmetry-enriched quantum criticality, which we propose as a framework for classifying gapless phases. The key message is that in the presence of symmetries, a universality class can divide into distinct phases, characterized by the symmetry action on the low-energy scaling operators. This includes gapless topological phases, with examples hiding in plain sight; we clarify their stability and reinterpret previously studied examples. Part II studies the excitations above the ground states of two-dimensional quantum spin models. The main object of our study is the dynamic spin structure factor; this type of spectral function is reviewed in the first part of Chapter 5. The second part of this chapter introduces a novel matrix-product-state-based algorithm to efficiently compute it, opening a new window on the dynamics of two-dimensional quantum systems. We benchmark this numerical method in Chapter 6 on the exactly-solvable Kitaev model---a paradigmatic topological model realizing a quantum spin liquid. By adding non-integrable Heisenberg perturbations, we identify the first unequivocal theoretical realization of a proximate spin liquid: the ground state becomes conventionally ordered, yet the high-energy spectral properties are structurally similar to those of the nearby Kitaev spin liquid. The latter agrees with aspects of recent inelastic neutron scattering experiments on alpha-RuCl3. In Chapter 7, we turn to one of the oldest models in many-body quantum physics: the spin-1/2 Heisenberg antiferromagnet on the square lattice. Despite its venerable history, there is still disagreement about the physical origin of high-energy spectral features which low-order spin wave theory cannot account for. We provide a simple picture for this strongly-interacting-magnon feature by connecting it to a simple Ising limit. Lastly, Chapter 8 discusses the stability of quasiparticles---collective excitations behaving like a single emergent entity, of which magnons are a prime example. These are often known to be stable at the lowest energies and are presumed to decay whenever this is seemingly allowed by energy and momentum conservation. However, we show that strong interactions can prevent this from happening. We numerically confirm this principle of avoided decay in the (slightly-detuned) Heisenberg antiferromagnet on the triangular lattice. Moreover, we can even identify its fingerprints in existing experimental data on Ba3CoSb2O9 and superfluid helium. In this thesis, we thus enlarge our understanding of quantum phases and their excitations. The identification of the key principles of gapless topological phases in one dimension calls for direct analogues in higher dimensions, waiting to be uncovered. With regard to the robust properties of the excitations identified in this thesis, we are hopeful that these can be extended into a theory of quasiparticle properties away from the universal low-energy regime.
The exploration of novel quantum states of matter remains a central theme in condensed matter physics. Among these, topological order stands out for its robust properties against local perturbations and its potential applications in quantum information processing. While considerable progress has been made in understanding topological order in purely bosonic or fermionic systems, the interplay between different quantum statistics in interacting Bose-Fermi mixtures offers a rich platform for discovering new forms of topological order. This paper investigates the theoretical conditions and characteristic signatures for the emergence of topological order in these hybrid quantum systems. We propose that tunable interspecies interactions, combined with appropriate lattice geometries or synthetic gauge fields, can drive Bose-Fermi mixtures into exotic phases exhibiting non-trivial topological properties. Specifically, we explore scenarios where bosonic particles mediate effective interactions between fermions, or where bosonic condensation leads to a modification of the fermionic band structure, thereby inducing topological phases. Our analysis focuses on identifying key experimental observables, such as fractionalized excitations, robust edge modes, and entanglement properties, which can serve as hallmarks of emergent topological order. The theoretical framework incorporates mean-field approaches, variational methods, and numerical simulations to characterize the phase diagram of interacting Bose-Fermi mixtures, providing a pathway for experimental realization and verification of these fascinating quantum states.