States with gapless degrees of freedom are typically more complicated and less well- understood than systems possessing a gap. In this thesis, we study strongly-correlated systems described by gapless fermions. In most of the systems we consider, the fermionic excitations are emergent, i.e. they are not adiabatically connected to the electrons which constitute the fundamental building blocks of the system in question.\nWe start by proposing several novel states of matter which we claim are relevant to strongly-correlated systems. The rst state we present is a fractionalized Fermi liquid on the surface of a topological Kondo insulator. A Kondo insulator is a material which becomes insulating at low temperatures as a result of strong electron-spin interactions. When spin- orbit coupling is present, this insulator might be topological and, consequently, host robust gapless surface modes. Given the strong interactions and the decreased dimensionality of the surface, we propose that the spins and electrons there may decouple, resulting in the formation of a fractionalized Fermi liquid.\nWe next argue that quantum electrodynamics in 2+1 dimensions (QED3) with Nf = 4 fermion avours may describe a continuous, decon ned phase transition connecting the 120◦ coplanar Néel phase and the √12 × √12 valence bond solid phase of the triangular lattice antiferromagnet (AF). In addition to being a critical point, QED3 is also believed to describe a critical phase of matter called the Dirac spin liquid. Regardless of whether QED3 is manifest as a phase or a critical point, impurities and imperfections are always present in the real world, and it is therefore important to understand what e ects this may have. We show that when QED3 is perturbed by weak disorder, under certain circumstances, it ows to a new critical point in which both interactions and disorder are present.\nConclusively identifying a system described by QED3 has proven to be a di cult task, not only experimentally, but in numerical studies as well. Since QED3 is a critical theory, its spectrum on a torus is a universal quantity dependent only on low-energy degrees of freedom, such as the torus area. It follows that this data is accessible via exact diagonalization and can serve as an identifying signature of the theory. To this end, we calculate the QED3 torus spectrum using path integral methods.\nWe conclude this thesis with a comprehensive study of gapped Z2 spin liquids of the square lattice antiferrogmagnet, using a critical theory of Dirac fermions coupled to an SU(2) gauge eld as a starting point. There are many di erent Z2 spin liquid groundstates which preserve the same symmetry group. Even if one knows that a system has a Z2 spin liquid groundstate, it is by no means obvious which Z2 spin liquid is being realized. By starting from a gapless theory, the set of possibilities can be reduce considerably. Further, using certain recently proposed dualities, comparisons with theories formulated using bosons can be made.
This thesis explores the interaction of symmetries and quantum anomalies on the low-energy dynamics and phase transitions of strongly correlated quantum matter. Beyond the traditional Landau-Ginzburg-Wilson paradigm, we investigate how the embedding of microscopic lattice symmetries into emergent continuum symmetries governs the stability of quantum critical points in (2+1) dimensions. Through the lens of 't Hooft anomalies and anomaly matching, we establish a theoretical framework for understanding the fate of various exotic quantum phase transitions. A central application of this framework is understanding the stability of emergent conformal field theories (CFTs) that describe gapless quantum critical points. The exact nature of these phase transitions is chiefly governed by how the microscopic lattice symmetries impose stringent conditions on the relevant operators allowed in the low-energy effective theory. When physical symmetries suppress relevant perturbations, an interacting conformal fixed point, such as (2+1)-dimensional quantum electrodynamics (QED₃), can stably emerge. Chapters 2 and 3 explore the consequences of this symmetry embedding through the phenomenon of "unnecessary'' quantum critical points, which uniquely describe continuous transitions occurring entirely within a single phase of matter. We demonstrate that the Dirac spin liquid (DSL) in lattice magnets can surprisingly describe a stable continuous phase transition within a single, conventional, symmetry breaking phase. In chapter 4, we apply these tools to a class of quantum phase transitions between featureless bosonic atomic insulators. Despite lacking topological order or protected edge modes, these insulators represent distinct phases because they can be "obstructed,'' meaning their Wannier centers are not pinned to the physical atomic sites. Surprisingly, we find that the transition between these topologically trivial states can host a conformally invariant state described by QED₃, suggesting that even transitions between trivial phases can harbor rich, fractionalized physics. Lastly, chapter 5 analyzes transitions from a composite Fermi liquid (CFL) to a chiral superconductor, employing more traditional field-theoretic techniques and making concrete connections with recent experiments in moire materials.
The interplay among symmetry, topology and condensed matter systems has deepened our understandings of matter and lead to tremendous recent progresses in finding new topological phases of matter such as topological insulators, superconductors and semi-metals. Most examples of the aforementioned topological materials are free fermion systems, in this thesis, however, we focus on their strongly correlated counterparts where electron-electron interactions play a major role. With interactions, exotic topological phases and quantum critical points with fractionalized quantum degrees of freedom emerge. In the first part of this thesis, we study the problem of resonant tunneling through a quantum dot in a spinful Luttinger liquid. It provides the simplest example of a (0+1)d system with symmetry-protected phase transitions. We show that the problem is equivalent to a two channel SU(3) Kondo problem and can be mapped to a quantum Brownian motion model on a Kagome lattice. Utilizing boundary conformal field theory, we find the universal peak conductance and compute the scaling behavior of the resonance line-shape.\nFor the second part, we present a model of interacting Majorana fermions that describes a superconducting phase with a topological order characterized by the Fibonacci topological field theory. Our theory is based on a SO(7)1=SO(7)1/(G2)1 x (G2)1 coset construction and implemented by a solvable two-dimensional network model. In addition, we predict a closely related ''anti-Fibonacci'' phase, whose topological order is characterized by the tricritical Ising model. Finally, we propose an interferometer that generalizes the Z2 Majorana interferometer and directly probes the Fibonacci non-Abelian statistics.\nFor the third part, we argue that a correlated fluid of electrons and holes can exhibit fractional quantum Hall effects at zero magnetic field. We first show that a Chern insulator can be realized as a free fermion model with p-wave(m=1) excitonic pairing. Its ground state wavefunction is then worked out and generalized to m>1. We give several pieces of evidence that this conjectured wavefunction correctly describes a topological phase, dubbed ''fractional excitonic insulator'', within the same universality class as the corresponding Laughlin state at filling 1/m. We present physical arguments that gapless states with higher angular momentum pairing between energy bands are conducive to forming the fractional excitonic insulator in the presence of repulsive interactions. Without interactions, these gapless states appear at topological phase transitions which separate the trivial insulator from a Chern insulator with higher Chern number. Since the nonvanishing density of states at these higher angular momentum band inversion transitions can give rise to interesting many-body effects, we introduce a series of minimal lattice models realizations in two dimensions. We also study the effect of rotational symmetry broken electron-hole exciton condensation in our lattice models using mean field theory.
The quantum nature of electrons emerges in various exotic ways when they are put together with quantum many-body interactions. In Mott insulating systems with strong electron correlation, frustration and strong quantum fluctuations promote localized electron spins to form nontrivial quantum paramagnetic phases such as quantum spin liquid and valence bond solid preventing the development of conventional magnetic orders. In this thesis, we study such exotic quantum phases in various different frustrated quantum magnets. First, encouraged by the spin liquid phenomenology observed in Ba3XSb2O9 (X=Cu, Ni), we investigate Z2 spin liquids on a three dimensional triangular lattice by employing the Schwinger boson mean-field theory with projective symmetry group analysis. It is found that a nematic spin liquid is energetically favorable on this lattice structure rather than symmetric spin liquids. Next, we study various valence bond solid phases that can appear in kagome lattice antiferromagnets by generalizing the recently proposed phenomenological Z2 gauge theory for the topological Z2 spin liquid state found in the nearest-neighbor kagome lattice Heisenberg model. This work provides a bridge between the topological spin liquid and valence bond solid phases previously proposed for kagome lattice antiferromagnets via vison condensation in the gauge theory. Then, we consider optical conductivity in Mott insulators as an experimental probe to detect quantum paramagnetic phases like valence bond solid. Applying this idea to deformed kagome lattice antiferromagnet Rb2Cu3SnF12 with a 12-site valence bond solid state, we estimate the size of optical conductivity in this quantum magnet. After this, another quantum paramagnet BiCu2PO6 is carefully investigated to clarify its puzzling ground state nature and quasiparticle excitations. Utilizing recent high resolution neutron scattering data, we construct a microscopic spin model and also an effective theory for the low energy spin excitations in the compound. We find significant influences of magnetic anisotropies on the ground state and quasiparticle dynamics. Lastly, we investigate emergent quantum phases in the thin film geometries of the pyrochlore iridates, where a number of exotic quantum ground states are proposed to occur in bulk materials as a result of the interplay between electron correlation and strong spin-orbit coupling. The fate of these bulk phases, as well as the novel quantum states that may arise only in a thin film setting, are studied via a theoretical model that allows layer-dependent magnetic structures.
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.
The physics of strongly interacting particles is often extraordinarily rich, leading to the \textit{emergence} of fundamentally new phases and phenomena that cannot exist in the absence of correlations. This thesis is devoted to the study of different classes of quantum systems with such strong correlations. Programmable quantum simulators based on Rydberg atom arrays have recently emerged as versatile platforms to probe quantum many-body phases and their dynamics. In Chapter 2, we start with theories of quantum phase transitions (QPTs) in one-dimensional chains of Rydberg atoms. We demonstrate the existence of a novel QPT in the universality class of the
Z3 chiral clock model, calculate its critical exponents, and discuss their measurement with the quantum Kibble-Zurek mechanism. Proceeding to two dimensions, Chapter 3 describes the zero-temperature phase diagram of a square-lattice Rydberg atom array, revealing several experimentally realizable density-wave-ordered phases and QPTs. For a different geometry, namely, the kagome lattice studied in Chapter 4, we identify a regime that constitutes a likely candidate for hosting a phase with long-range quantum entanglement and topological order. Based on mappings to dimer models and gauge theories, we argue for the existence of a quantum spin liquid phase that has proved elusive in solid-state materials. In the next part of this thesis, we focus on similarly strongly correlated phases in the realm of quantum materials. Motivated by recent observations of an enhanced thermal Hall response in the high-temperature cuprate superconductors, in Chapter 5, we study the thermal Hall conductivity,
κxy, of chiral spin-liquid ans\"{a}tze on the square lattice, focusing on the contribution from neutral (bosonic or fermionic) spinons. Thereafter, going beyond mean-field theory, we consider the corrections due to emergent gauge fields, which yield a response of opposite sign to that of the fermionic matter. We additionally consider the contribution of phonons to
κxy and establish its relation to the spinon Hall viscosity. Thermal Hall measurements also leave intriguing fingerprints in Kitaev materials like
α-RuCl
3, and in this context, Chapter 6 investigates the unquantized
κxy obtained in a gapless spinon Fermi surface state.
Despite rapid progress in understanding gapped topological states, much less is known about gapless topological phases of matter, especially in strongly correlated electrons. In this work, we discuss a large class of robust gapless quantum spin liquids in frustrated magnets made of half-integer spins, which are described by gapless fermionic spinons coupled to dynamical
Z2 gauge fields. Requiring
U(1) spin conservation, time-reversal, and certain space-group symmetries, we show that certain spinon symmetry fractionalization class necessarily leads to a gapless spectrum. These gapless excitations are stable against any perturbations, as long as the required symmetries are preserved. Applying these gapless criteria to spin-
21 systems on square, triangular, and kagome lattices, we show that all gapped symmetric
Z2 spin liquids in Abrikosov-fermion representation can also be realized in Schwinger-boson representation. This leads to 64 gapped
Z2 spin liquids on square lattice, and 8 gapped states on both kagome and triangular lattices.