Many magnetic materials are predicted to exhibit bosonic topological edge modes in their excitation spectra, because of the nontrivial topology of their magnon, triplon, or other quasiparticle band structures. However, there is a discrepancy between theory prediction and experimental observation, which suggests some underlying mechanism that intrinsically suppresses the expected experimental signatures, like the thermal Hall current. Many-body interactions that are not accounted for in the noninteracting quasiparticle picture are most often identified as the reason for the absence of the topological edge modes. Here we report persistent bosonic edge modes at the boundaries of a ladder quantum paramagnet with gapped triplon excitations in the presence of the full many-body interaction. We use tensor network methods to resolve topological edge modes in the time-dependent spin-spin correlations and the dynamical structure factor, which is directly accessible experimentally. We further show that signatures of these edge modes survive even when the noninteracting quasiparticle theory breaks down; we discuss the topological phase diagram of the model, demonstrate the fractionalization of its low-lying excitations, and propose potential material candidates.
Many magnetic materials are predicted to exhibit bosonic topological edge modes in their excitation spectra, because of the nontrivial topology of their magnon, triplon, or other quasiparticle band structures. However, there is a discrepancy between theory prediction and experimental observation, which suggests some underlying mechanism that intrinsically suppresses the expected experimental signatures, like the thermal Hall current. Many-body interactions that are not accounted for in the noninteracting quasiparticle picture are most often identified as the reason for the absence of the topological edge modes. Here we report persistent bosonic edge modes at the boundaries of a ladder quantum paramagnet with gapped triplon excitations in the presence of the full many-body interaction. We use tensor network methods to resolve topological edge modes in the time-dependent spin-spin correlations and the dynamical structure factor, which is directly accessible experimentally. We further show that signatures of these edge modes survive even when the noninteracting quasiparticle theory breaks down; we discuss the topological phase diagram of the model, demonstrate the fractionalization of its low-lying excitations, and propose potential material candidates.
It has recently been found that bosonic excitations of ordered media, such as phonons or spinons, can exhibit topologically nontrivial band structures. Of particular interest are magnon and triplon excitations in quantum magnets, as they can easily be manipulated by an applied field. Here, we study triplon excitations in an S=1/2 quantum spin ladder and show that they exhibit nontrivial topology, even in the quantum-disordered paramagnetic phase. Our analysis reveals that the paramagnetic phase actually consists of two separate regions with topologically distinct triplon excitations. We demonstrate that the topological transition between these two regions can be tuned by an external magnetic field. The winding number that characterizes the topology of the triplons is derived and evaluated. By bulk-boundary correspondence, we find that the nonzero winding number implies the presence of localized triplon end states. Experimental signatures and possible physical realizations of the topological paramagnetic phase are discussed.
The concept of free fermion topology has been generalized to d-dimensional phases that exhibit (d−n)-dimensional boundary modes, such as zero-dimensional (0D) corner excitations. Motivated by recent extensions of these ideas to magnetic systems, we consider 2D quantum paramagnets formed by interacting spin dimers with dispersive triplet excitations. We propose two examples of such dimer models, where the spin-gapped bosonic triplon excitations are shown to host bands with nontrivial higher-order topology. We demonstrate this using real-space Bogoliubov--de Gennes calculations that reveal the existence of mid-bandgap corner triplon modes as a signature of higher-order bulk topology. We provide an understanding of the higher-order topology in these systems via a computation of bulk topological invariants as well as the construction of edge theories, and study their phase transitions as we tune parameters in the model Hamiltonians. We also discuss possible experimental approaches for detecting the emergent corner triplon modes.
The concept of free fermion topology has been generalized to d-dimensional phases that exhibit $(d\ensuremath{-}n)$-dimensional boundary modes, such as zero-dimensional (0D) corner excitations. Motivated by recent extensions of these ideas to magnetic systems, we consider 2D quantum paramagnets formed by interacting spin dimers with dispersive triplet excitations. We propose two examples of such dimer models, where the spin-gapped bosonic triplon excitations are shown to host bands with nontrivial higher-order topology. We demonstrate this using real-space Bogoliubov--de Gennes calculations that reveal the existence of near-mid band gap corner triplon modes as a signature of higher-order bulk topology. We provide an understanding of the higher-order topology in these systems via computation of bulk topological invariants as well as the construction of edge theories and study their phase transitions as we tune parameters in the model Hamiltonians. We also discuss possible experimental approaches for detecting the emergent corner triplon modes.
Many strongly correlated electron systems fall outside the conventional quasiparticle paradigm. Their low-energy fermionic excitations need not be adiabatically connected to the bare electrons or appear as poles of the physical electron single-particle Green's function. This thesis investigates to what extent an effective quasiparticle description can nevertheless survive, and which emergent degrees of freedom provide the appropriate low-energy structure. First, the interaction-driven evolution from a quantum spin Hall insulator to a Mott insulator is studied using the dynamical cluster approximation. Dispersive zeros of the single-particle Green's function emerge near the Mott transition and retain a nontrivial topological character. When symmetry breaking is allowed, a nontopological excitonic insulator is found to intrude between quantum spin Hall and Mott insulators. Encoding both poles and zeros within a low-energy quasiparticle Hamiltonian provides a continuous one-body description of the three phases. On the Mott side, its gap evolution supports interpreting the soft excitonic mode as a bound state of quasiparticle and quasihole excitations tied to the Green's function zeros, rather than to the widely separated Hubbard bands. Second, the intrinsic anomalous Hall conductivity of a topological metal is analyzed within a multiband extension of Landau Fermi-liquid theory. Residual interactions among quasiparticles at the Fermi surface dress the dynamic current vertex and generate corrections to the Berry-curvature contribution obtained from the quasiparticle bands alone. Our result supports recent claims that the correct expressions for topological observables include vertex corrections besides the topological invariants built just upon the Green's function. It also demonstrates that such corrections are naturally accounted for by Landau Fermi-liquid theory. Extracting the relevant ingredients for an effective description of material-relevant correlated systems requires computational tools that remain reliable at low temperatures. To this end, a deterministic strong-coupling impurity solver for dynamical mean-field theory is discussed. By providing static, dynamical, and thermodynamic observables without stochastic sampling, it offers a promising route toward multiorbital and spin-orbit coupled problems where conventional Monte Carlo methods currently struggle. Finally, the auxiliary quasiparticle framework underlying the ghost-Gutzwiller Ansatz is investigated. On a model for correlated quantum spin Hall insulators, we show that the topology of the interacting state is encoded in the auxiliary one-body Hamiltonian. The resulting band-structure reveals topological Hubbard bands whose topological character can be tuned through a finite magnetization. When applied to the single-band t-J model, the same framework yields a fractionalized Fermi liquid (FL*). In this phase, dispersive neutral spinons, with vanishing physical-electron spectral weight, coexist with conventional quasiparticles forming a small hole-like Fermi surface, thus violating Luttinger's theorem. The corresponding temperature--doping phase diagram contains a low-doping FL*, a d-wave superconducting dome, and an overdoped conventional Fermi liquid, thereby reproducing key qualitative features of cuprate phenomenology.
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.
Topological superconductors are believed to host exotic quasiparticle excitations known as Majorana zero modes (MZMs), with much of the evidence based on BCS mean-field theory. The direct application of mean-field arguments is tenuous in finite, isolated systems relevant in some experiments. Here, we develop a new correlation-based method for identifying MZMs in interacting, number-conserving systems. Using the density matrix renormalization group, we study fermion number-conserving models with long-range interactions, which under periodic boundary conditions exhibit robust topological and nontopological superconductivity, tuned by the strength of interaction [Ortiz et al., Phys. Rev. Lett. 113, 267002 (2014)PRLTAO0031-900710.1103/PhysRevLett.113.267002]. We find evidence that, on the topological side, Majorana edge modes appear in open chains, manifesting as the vanishing of the energy splitting between odd- and even-parity ground states with increasing system size. Additionally, off-diagonal two-point correlation functions show nonlocal, parity-dependent edge effects. These correlations reveal the spatial structure of Majorana modes in the many-body wave function. We show that the correlation diagnostic applies broadly, including to short-range interacting models, where topological superconductivity is more fragile due to the absence of a bulk excitation gap.