Most topological properties of quantum systems, such as the braiding of anyons or protected edge modes, are easiest to define in systems with an energy gap. While some topological phenomena persist in the presence of gapless modes, this paper explores such properties that are only possible without a gap. These systems are here dubbed intrinsically gapless topological phases. The authors study a simple model -- a hole-doped one-dimensional antiferromagnet -- and describe a mechanism based on anomalies that applies in all dimensions.
Most topological properties of quantum systems, such as the braiding of anyons or protected edge modes, are easiest to define in systems with an energy gap. While some topological phenomena persist in the presence of gapless modes, this paper explores such properties that are only possible without a gap. These systems are here dubbed intrinsically gapless topological phases. The authors study a simple model -- a hole-doped one-dimensional antiferromagnet -- and describe a mechanism based on anomalies that applies in all dimensions.
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.
We consider a one-dimensional, time-reversal-invariant system with attractive interactions and spin-orbit coupling. Such a system is gapless due to the strong quantum fluctuations of the superconducting order parameter. However, we show that a sharply defined topological phase with protected, exponentially localized edge states exists. If one of the spin components is conserved, the protection of the edge modes can be understood as a consequence of the presence of a spin gap. In the more general case, the localization of the edge states arises from a gap to single-particle excitations in the bulk. We consider specific microscopic models and demonstrate both analytically and numerically (using density matrix renormalization group calculations) that they can support the topologically nontrivial phase.
We show that topology can protect exponentially localized, zero energy edge modes at critical points between one-dimensional symmetry-protected topological phases. This is possible even without gapped degrees of freedom in the bulk-in contrast to recent work on edge modes in gapless chains. We present an intuitive picture for the existence of these edge modes in the case of noninteracting spinless fermions with time-reversal symmetry (BDI class of the tenfold way). The stability of this phenomenon relies on a topological invariant defined in terms of a complex function, counting its zeros and poles inside the unit circle. This invariant can prevent two models described by the same conformal field theory (CFT) from being smoothly connected. A full classification of critical phases in the noninteracting BDI class is obtained: Each phase is labeled by the central charge of the CFT, c∈1/2N, and the topological invariant, ω∈Z. Moreover, c is determined by the difference in the number of edge modes between the phases neighboring the transition. Numerical simulations show that the topological edge modes of critical chains can be stable in the presence of interactions and disorder.
Topology is a fascinating phenomena in condensed-matter physics typically associated with a bulk gap. However, recent research shifts focus to quantum critical points or phases that exhibit nontrivial topological properties. Here we explore a cluster-Ising chain with long-range antiferromagnetic interactions that decay as a power law with distance. Using large-scale density matrix renormalization group simulations, we demonstrate that the nontrivial topology at the critical point remains stable against long-range interactions, resulting in a topologically nontrivial critical line. Moreover, even within the gapped region, the interplay between topology and long-range interaction can give rise to a topological phase featuring algebraically decaying correlations and edge modes, similar to gapless topological phases. We refer to this phase as the algebraic topological phase, which exhibits nontrivial gapless topological behaviors and arises solely from long-range interactions without short-range counterparts. The findings pave the way for more studies on topological states in long-range systems. Long-range interactions play a crucial role in condensed-matter systems and can lead to range of emerging and exotic physical phenomenon. Here using simulations, the authors investigate the non-trivial topological properties that can arise in a cluster Ising chain with long-range antiferromagnetic interactions, finding evidence for what they classify as an algebraic symmetry-protected topological phase.
We introduce topological invariants for gapless systems and study the associated boundary phenomena.More generally, the symmetry properties of the low-energy conformal field theory (CFT) provide discrete invariants establishing the notion of symmetry-enriched quantum criticality.The charges of nonlocal scaling operators, or more generally, of symmetry defects, are topological and imply the presence of localized edge modes.We primarily focus on the 1 þ 1d case where the edge has a topological degeneracy, whose finite-size splitting can be exponential or algebraic in system size depending on the involvement of additional gapped sectors.An example of the exponential case is given by tuning the spin-1 Heisenberg chain to a symmetry-breaking Ising phase.An example of the algebraic case arises between the gapped Ising and cluster phases: This symmetry-enriched Ising CFT has an edge mode with finite-size splitting scaling as 1=L 14 .In addition to such new cases, our formalism unifies various examples previously studied in the literature.Similar to gapped symmetry-protected topological phases, a given CFT can split into several distinct symmetry-enriched CFTs.This raises the question of classification, to which we give a partial answer-including a complete characterization of symmetry-enriched 1 þ 1d Ising CFTs.Nontrivial topological invariants can also be constructed in higher dimensions, which we illustrate for a symmetry-enriched 2 þ 1d CFT without gapped sectors.
Deconfined criticality and gapless topological states have recently attracted growing attention, as both phenomena go beyond the traditional Landau paradigm. However, the deep connection between these two critical states, particularly in lattice realization, remains insufficiently explored. In this Letter, we reveal that certain deconfined criticality can be regarded as an intrinsically gapless topological state without gapped counterparts in a one dimensional lattice model. Using a combination of field-theoretic arguments and large-scale numerical simulations, we establish the global phase diagram of the model, which features deconfined critical lines separating two distinct spontaneous symmetry breaking ordered phases. More importantly, we unambiguously demonstrate that the mixed anomaly inherent to deconfined criticality enforces topologically robust edge modes near the boundary, providing a general mechanism by which deconfined criticality manifests as a gapless topological state. Our findings not only offer a new perspective on deconfined criticality but also deepen our understanding of gapless topological phases of matter.