Many-body phenomena were always an integral part of physics comprising of collective behaviors through self-organization, in systems consisting of many components and degrees of freedom. We investigate the collective behaviors of strongly interacting particles confined in one dimension. We show that many-body orders with topological characteristics can be found at the Mott insulator limit for hardcore bosons, at different fillings, without considering the spin degree of freedom or long-range microscopic interactions. These orders have unique properties like weak or strong quantum correlations (entanglement), quantified by the entanglement entropy, edge excitations/modes and gapped energy spectrum with highly degenerate ground state, bearing resemblance to topologically ordered phases of matter.
Many-body phenomena were always an integral part of physics comprising of collective behaviors through self-organization, in systems consisting of many components and degrees of freedom. We investigate the collective behaviors of strongly interacting particles confined in one dimension. We show that many-body orders with topological characteristics can be found at the Mott insulator limit for hardcore bosons, at different fillings, without considering the spin degree of freedom or long-range microscopic interactions. These orders have unique properties like weak or strong quantum correlations (entanglement), quantified by the entanglement entropy, edge excitations/modes and gapped energy spectrum with highly degenerate ground state, bearing resemblance to topologically ordered phases of matter.
We investigate the self-organization of strongly interacting particles confined in 1D and 2D. We consider hardcore bosons in spinless Hubbard lattice models with short range interactions. We show that, many-body orders with topological characteristics emerge, at different energy bands separated by large gaps. These topological orders manifest in the way the particles organize in real space to form states with different energy. Each of these states contains topological defects/condensations whose Euler characteristic can be used as a topological number to categorize states belonging to the same energy band. We provide analytical formulas for this topological number and the full energy spectrum of the system for both sparcly and densely filled systems. Furthermore, we discuss the connection with the Gauss-Bonnet theorem of differential geometry, by using the curvature generated in real space by the particle structures. Our result is a demonstration of how topological orders can arise in many-body systems with simple underlying rules, without considering the spin or long-range microscopic interactions.
Topological phenomena in physical systems are determined by topological structures and are thus universal and protected against perturbations. We theoretically establish exotic topological phenomena and their consequences for experiments in crystalline systems such as topological insulators and topologically ordered phases. We show that protected one-dimensional fermionic modes may be associated with the line defects such as dislocations in three-dimensional topological insulators, and strong electron-electron repulsion may lead to topological Mott insulators via spontaneous spin-orbit correlations in three dimensions. We also predict anomalous Aharonov Bohm conductance oscillations maximized at half integer multiples of a flux quantum in a topological insulator nanowire with strong surface disorder, arising from surface curvature induced Berry phase. In addition, we classify three-dimensional inversion symmetric insulators and their quantized responses.Quantum entanglement provides a promising probe to the properties of many-body systems, especially topological phases not captured by local order parameters. We present a characterization of topological insulators using entanglement spectrum based only on bulk ground-state wave function. Further, by studying entanglement of trivial partitions, we establish topological order in candidate Gutzwiller projected wave functions for gapped spin liquids and Laughlin states; and with entanglement's dependence on the ground states for bipartition of a torus into two cylinders, we demonstrate a method to extract the modular matrices and statistics and braiding of quasiparticle excitations. Our method helps to determine the topological order with only the set of ground-state wave functions on a torus. Our variational Monte Carlo calculations of topological entanglement entropy agree well with theory. We also find a violation of the boundary law for a critical spin liquid of Gutzwiller projected Fermi sea on the triangular lattice, where the entanglement entropy's enhancement by a logarithmic factor reflects the presence of emergent fermions in a bosonic wave function.
Abstract We investigate the self-organization of point-particles with short-range interactions modeled via simple 1D and 2D Hubbard-like models. We show how various properties emerge such as, boson-like ordering leading to topological structures in real space, via the clustering of the particles at discrete energy levels, which can be analyzed using a network/graph mathematical language. We calculate analytically the number of clusters, the correlations between them and topological numbers like the Euler characteristic, deriving different organizational schemes and entanglement entropy scaling laws. All calculations are performed for an arbitrary number of particles N and energy of the system E . Our results demonstrate how orders with diverse topological/geometric and entanglement features, emerge in strongly interacting many-body systems, that follow minimal microscopic interaction rules.
We investigate topological phase transitions driven by interaction and identify a topological Mott insulator state in one-dimensional fermionic optical superlattices through a numerical density-matrix renormalization-group method. Remarkably, the low-energy edge excitations change from spin-1/2 fermionic single-particle modes to spin-1 bosonic collective modes across the phase transition. Due to spin-charge separation, the low-energy theory is governed by an effective spin superexchange model, whereas the charge degree of freedom is fully gapped out. Such topological Mott state can be characterized by a spin Chern number and gapless magnon modes protected by a finite spin gap. The proposed experimental setup is simple and may pave the way for the experimental observation of exotic topological Mott states.
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.
In this dissertation we study the properties of quantum many-body systems whose behavior is governed by strong correlation effects between large numbers of degrees of freedom. We begin with a review of various types of condensed matter settings and a general discussion of how interactions can lead to qualitatively new properties. In Chapter 2 we discuss a class of spin models which generate nontrivial realizations of a global symmetry, both in the presence and absence of topological order. In Chapter 3, we examine a model of a non-Fermi liquid. This model is built out of electron-like degrees of freedom and has a Fermi surface, yet has no stable quasiparticle excitations. In Chapter 4 we discuss some aspects of entanglement entropy in gapless multi-component systems, and suggest a way of measuring the topological properties of Kondo lattice wavefunctions.