This dissertation establishes and investigates new phenomena in diverse interacting many-body quantum systems guided by three distinct, but complementary, themes: (i) symmetry and topology, (ii) localization, and (iii) non-Fermi liquids. The first theme concerns how the interplay of symmetry and topology can offer robust protection for a many-body system. We investigate low-dimensional quantum fermionic models from a general structural perspective. These phases can exhibit fractionalized Majorana zero-energy modes on their boundary. We devise experimentally relevant nonlocal measurements that can be used to detect these topological phases. While our primary focus is on quantum systems, topologically protected behavior can arise in classical mechanical models as well. We extend a recent connection between the topological band theory of electrons and classical physics by proposing a mechanical analogue of a topological nodal semimetal.The second theme concerns that of many-body localization. We demonstrate that the combination of localization, symmetry, and topology can have radical consequences for quantum systems at high energies, such as the existence of protected gapless boundary modes. We show that, even at these high energies, quantum information can be preserved, and quantum coherence recovered. Quantum coherent dynamics in this regime is unexpected and of great interest for quantum computation.The third direction in our study of interacting many-body systems concerns non-Fermi liquids. We construct a non-Fermi liquid by bringing together a spin-orbit coupled Fermi surface and fluctuating magnetic order. Using newly developed analytic tools for strongly coupled systems, we demonstrate the stability of the non-Fermi liquid to ordering. This identifies an experimentally accessible candidate for exploring physics that lies beyond Fermi liquid theory.
This dissertation establishes and investigates new phenomena in diverse interacting many-body quantum systems guided by three distinct, but complementary, themes: (i) symmetry and topology, (ii) localization, and (iii) non-Fermi liquids. The first theme concerns how the interplay of symmetry and topology can offer robust protection for a many-body system. We investigate low-dimensional quantum fermionic models from a general structural perspective. These phases can exhibit fractionalized Majorana zero-energy modes on their boundary. We devise experimentally relevant nonlocal measurements that can be used to detect these topological phases. While our primary focus is on quantum systems, topologically protected behavior can arise in classical mechanical models as well. We extend a recent connection between the topological band theory of electrons and classical physics by proposing a mechanical analogue of a topological nodal semimetal.The second theme concerns that of many-body localization. We demonstrate that the combination of localization, symmetry, and topology can have radical consequences for quantum systems at high energies, such as the existence of protected gapless boundary modes. We show that, even at these high energies, quantum information can be preserved, and quantum coherence recovered. Quantum coherent dynamics in this regime is unexpected and of great interest for quantum computation.The third direction in our study of interacting many-body systems concerns non-Fermi liquids. We construct a non-Fermi liquid by bringing together a spin-orbit coupled Fermi surface and fluctuating magnetic order. Using newly developed analytic tools for strongly coupled systems, we demonstrate the stability of the non-Fermi liquid to ordering. This identifies an experimentally accessible candidate for exploring physics that lies beyond Fermi liquid theory.
This dissertation establishes and investigates new phenomena in diverse interacting many-body quantum systems guided by three distinct, but complementary, themes: (i) symmetry and topology, (ii) localization, and (iii) non-Fermi liquids. The first theme concerns how the interplay of symmetry and topology can offer robust protection for a many-body system. We investigate low-dimensional quantum fermionic models from a general structural perspective. These phases can exhibit fractionalized Majorana zero-energy modes on their boundary. We devise experimentally relevant nonlocal measurements that can be used to detect these topological phases. While our primary focus is on quantum systems, topologically protected behavior can arise in classical mechanical models as well. We extend a recent connection between the topological band theory of electrons and classical physics by proposing a mechanical analogue of a topological nodal semimetal.The second theme concerns that of many-body localization. We demonstrate that the combination of localization, symmetry, and topology can have radical consequences for quantum systems at high energies, such as the existence of protected gapless boundary modes. We show that, even at these high energies, quantum information can be preserved, and quantum coherence recovered. Quantum coherent dynamics in this regime is unexpected and of great interest for quantum computation.The third direction in our study of interacting many-body systems concerns non-Fermi liquids. We construct a non-Fermi liquid by bringing together a spin-orbit coupled Fermi surface and fluctuating magnetic order. Using newly developed analytic tools for strongly coupled systems, we demonstrate the stability of the non-Fermi liquid to ordering. This identifies an experimentally accessible candidate for exploring physics that lies beyond Fermi liquid theory.
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.
(English) In recent years, the evolution of quantum technologies has resulted in an unprecedented control over individual quantum particles and many-body systems. This remarkable progress has given rise to a new era, marked by the convergence of classical and quantum methodologies to investigate fundamental questions concerning the nature of quantum matter, improving our understanding of the role of entanglement in solid-state systems or the mechanisms behind high-energy physics. From analog quantum simulators to digital quantum computers, these advancements hold potential across diverse domains. This thesis explores the interplay between classical and quantum methods in understanding topological phases of matter. We concentrate on three distinct directions: non-conventional topological superconductors, interaction-induced topological phases in ultracold atom quantum simulators, and applications of variational quantum algorithms. Each trajectory relies on the combination of different techniques with the aim of understanding and characterizing topological phenomena in different settings. Exploring non-conventional topological superconductors involves extending the paradigmatic Kitaev chain model by incorporating additional terms in the Hamiltonian such as long-range interactions and quasi-periodic potentials. This investigation is relevant to better understand the impact of real-world conditions on the both the topological and localization properties of systems hosting non-local Majorana modes, which are promising candidates for topological quantum computation. In the realm of interacting systems, we explore the realization of interaction-induced topological phases in systems of ultracold atoms in optical lattices, both in one and two dimensions. The remarkable control and versatility of such platforms enable the simulation of both theoretical topological models and strongly correlated physics. Notably, the interplay between interactions and topology can give rise to intriguing phenomena, such as delocalized fractional charges and gapless topological phases,challenging existing intuition. We employ advanced numerical methods based on tensor networks to benchmark the experimental proposals that open the door to the realization and detection of novel many-body phases of matter, including topological quantum critical points and a higher-order topological Peierls insulator in Bose-Hubbard models with long-range interactions. Variational quantum algorithms, conversely, have the potential to efficiently tackle a wide range of problems, including ground state search, phase classification or accessing topological invariants. Despite current limitations in trainability and scalability, these hybrid classical-quantum algorithms provide practical insights into current quantum hardware capabilities and can inspire future architectures. We explore the application of variational quantum algorithms to shed light on topological phenomena, raising questions about their ability to discern topological phase transitions and compute topological invariants in situations where classical approaches fail. This thesis presents a comprehensive exploration of distinct approaches to topological quantum matter by leveraging quantum technologies and quantum-inspired classical algorithms. Our results not only advance our understanding of quantum systems but also pave the way for the realization and discovery of novel physics extending to quantum information processing, materials science, and beyond.
With the discovery of the integer quantum Hall effect by von Klitzing and collaborators in 1980, the mathematical field of topology entered the world of condensed matter physics. Almost three decades later, this eventually led to the theoretical prediction and the experimental realization of many intriguing topological materials and topology-based devices. In this Ph.D. thesis, we will study the interplay between topology and another key topic in condensed matter physics, namely the study of inter-particle interactions in many-body systems. This interplay is analyzed from two different perspectives. Firstly, we studied how the presence of electron-electron interactions affects single-electron injection into a couple of counter-propagating one-dimensional edge channels. The latter appear at the edges of topologically non-trivial systems in the quantum spin Hall regime and they can also be engineered by exploiting the integer quantum Hall effect. Because of inter-channel interactions, the injected electron splits up into a couple of counter-propagating fractional excitations. Here, we carefully study and discuss their properties by means of an analytical approach based on the Luttinger liquid theory and the bosonization method. Our results are quite relevant in the context of the so-called electron quantum optics, a fast developing field which deeply exploits the topological protection of one-dimensional edge states to study the coherent propagation of electrons in solid-state devices. As an aside, we also showed that similar analytical techniques can also be used to study the time-resolved dynamics of a Luttinger liquid subject to a sudden change of the interaction strength, a protocol known as quantum quench which is gaining more and more attention, especially within the cold-atoms community. Secondly, we study how inter-particle interactions can enhance the topological properties of strictly one-dimensional fermionic systems. More precisely, the starting point is the seminal Kitaev chain, a free-fermionic lattice model which hosts exotic Majorana zero-energy modes at its ends. The latter are extremely relevant in the context of topological quantum computation because of their non-Abelian anyonic exchange statistics. Here we show that, by properly adding electron-electron interactions to the Kitaev chain, it is possible to obtain lattice models which feature zero-energy parafermionic modes, an even more intriguing generalization of Majoranas. To this end, we develop at first an exact mapping between Z4 parafermions and ordinary fermions on a lattice. We subsequently exploit this mapping to analytically obtain an exactly solvable fermionic model hosting zero-energy parafermions. We study their properties and numerically investigate their signatures and robustness even when parameters are tuned away from the exactly solvable point.
Quantum phases of matter can be classified into two kinds: gapped or gapless, depending on whether the equilibrium ground state is separated from excited states by a finite energy window or not in the thermodynamic limit. Topology has played an important role in the understanding of both gapped and gapless phases of matter. This thesis uncovers several robust physical properties of both kinds of systems which are largely unexplored before. \\ For gapped systems, we study topological orders, which are characterized by the existence of anyonic quasiparticles with fractional braiding statistics. In the first part, we introduce a family of Abelian quantum Hall insulators, at filling factors ν=p/2q for bosonic systems and ν=p/(p+2q) for fermionic systems, with p and q being two coprime integers. These states are termed the \textit{nondiagonal} quantum Hall states, which are constructed in a coupled wire model that exhibits an intimate relation to the nondiagonal conformal field theory and has a constrained pattern of motion for anyons. We unveil a nontrivial interplay between charge symmetry and translation symmetry in this system, and establish the nondiagonal quantum Hall states as symmetry-enriched topological orders. In the second part, we introduce a related model of interacting electrons, but in the absence of a strong quantizing magnetic field, known as the \textit{toric code insulator}. It is constructed from a two-dimensional array of strongly coupled one-dimensional topological superconductors, which feature electron fractionalization and possesses anyonic excitations described by the Z2 topological order. The motion of anyons is again constrained, and different types of anyons can be related by the translation symmetry. Nondiagonal quantum Hall states and the toric code insulator are convincing examples that itinerant electronic systems can realize interesting symmetry-enriched topological orders.\\ For gapless quantum matters we focus on a particularly familiar class, the metals, but we uncover their topological aspects which were much less appreciated. Every metal is characterized by a Fermi surface of gapless excitations, and the Fermi surface surrounds a manifold in the momentum space, known as the Fermi sea. The ground state of a non-interacting metal, or a Fermi gas, consists of all single-electron states within the Fermi sea. The shape of the Fermi sea has a natural topological characterization in terms of the Euler characteristic χF, and a fundamental question that we address in this thesis is: \textbf{does χF imply any robust physical properties in metals, and how can they be measured?} In the third part of the thesis, χF is connected to physically measurable quantities, such as equal-time density-density correlation functions, and multipartite entanglement measures, such as the mutual information. Particularly, for three dimensions, we establish that the 4-partite mutual information is a robust probe of the Fermi sea topology even in the presence of Fermi-liquid interactions. In the last part, we focus on two dimensions, and propose an experimentally accessible platform to probe χF for any two-dimensional electron gas (2DEG) that respects time-reversal symmetry. The proposed experimental setup is a Josephson π-junction, and we predict that the electrical transport in the direction \textit{along} the junction exhibits a current-rectification effect that is controlled by the underlying Fermi sea topology. Due to the relation to Andreev state transport, we term this effect \textit{topological Andreev rectification}. The size of rectification is quantified by a \textit{nonlocal rectified conductance}, which is established to attain an integer quantized value that equals χF (in unit of e2/h). Extensive numerical simulations are performed to support our theoretical proposal, and material platforms such as InAs heterostructures and graphene are identified as promising candidates to demonstrate this effect.
Symmetry, topology and entanglement are three indispensable concepts for our understanding of quantum many-body systems. In this dissertation, we demonstrate the use and interplay of these concepts with four distinct topics. The first part focuses on two-dimensional (2D) symmetry protected topological (SPT) phases with lattice rotation symmetries. We start by discussing a prototypical example that exhibits interesting topological responses, and then investigate general aspects of rotation protected topology. In the second part, we report on a Hartree-Fock calculation of possible symmetry-broken phases in magic angle bilayer graphene at charge neutrality, assuming that the valley U(1) symmetry is preserved. Our solutions include both correlated insulators and nematic topological semimetals, whose relative stability can be tuned by weak strains. In the third part, inspired by the recent studies on gapless SPT states, we examine the fate of magnetic impurities at 2+1D quantum critical points. We introduced a series of large-N solvable impurity models which, as a consequence of topology, demonstrate intermediate coupling fixed points. In the last part, we develop a diagrammatic method to compute the entanglement properties of random mixed states which are obtained by partial tracing random pure states and are relevant for a variety of problems ranging from chaotic quantum dynamics to black hole physics.