Symmetries play a central role in organizing many-body quantum phases of matter and in revealing universal low-energy behavior. Systems with gapless excitations are especially rich, as symmetries can act in intricate ways on low-energy degrees of freedom, while enhanced correlations and entanglement open possibilities not available in gapped settings. This thesis studies two settings in which symmetry considerations give rise to qualitatively new phenomena in gapless quantum matter and correlated electronic systems. In the first part of the thesis, I study how the framework developed for gapped topological phases can be generalized to gapless systems, especially those arising at topological phase transitions. I show that symmetry can refine a fixed conformal field theory universality class into distinct symmetry-enriched critical phases that cannot be smoothly connected without breaking symmetry or leaving the critical theory. In Chapter 1, I show that transitions between distinct symmetry-protected topological phases can exhibit exotic boundary critical behavior not seen in gapped systems. In Chapter 2, I show that distinct gapless symmetry-enriched phases can meet at an anomalous multicritical point. In Chapter 3, I show that spatial interfaces provide an operational indicator of these phases, and that distinct symmetry-enriched criticalities are separated by non-invertible conformal defects. Together, these results show that subtle distinctions in symmetry-enriched criticality have robust and physically detectable consequences. In the second part of the thesis, I study how symmetry and strong interactions govern charge carriers in correlated electron systems. In Chapter 4, I develop a microscopic theory of the Josephson diode effect in Josephson junctions, showing that nonreciprocal supercurrents arise universally from finite Cooper-pair momentum when time-reversal and inversion symmetry are simultaneously broken. In Chapter 5, I study spin-polaron formation in a flat-band ferromagnetic Hubbard model and show that binding occurs in a regime where the conventional mechanisms based on kinetic frustration or nontrivial band topology do not apply, but instead exotic composite charge carriers arise from quantum geometry.
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 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.
We study several aspects of the realization of global symmetries in highly entangled phases of quantum matter. Examples include gapped topological ordered phases, gapless quantum spin liquids, and non-Fermi liquid phases. An insightful window into such phases is provided by recent developments in the theory of short-ranged entangled symmetry protected topological (SPT) phases. First, they generate useful no-go constraints on how global symmetry may be implemented in a highly entangled phase. Possible symmetry implementation in gapped topological phases and some proposed gapless spin/Bose liquids are examined in this light. We show that some previously proposed spin liquid states for 2D quantum magnets do not in fact have consistent symmetry implementation unless they occur as the surface of a 3D SPT phase. A second SPT-based insight into highly entangled states is the development of a view point of such states as SPT phases of one of the emergent excitations. We describe this in the specific context of time-reversal symmetric 3D U(1) quantum spin liquids with an emergent photon. Different such spin liquids are shown to be equivalent to different SPT insulating phases of the emergent monopole excitation of such phases. The highly entangled states also in turn enrich our understanding of SPT phases. We use the insights obtained from our results to provide an explicit construction of bosonic SPT phases in 3D in a system of coupled layers. This includes construction of a time-reversal symmetric SPT state that is not currently part of the cohomology classification of such states.
Quantum phenomena at macroscopic scales are fascinating, given that quantum mechanical effects usually manifest only at atomic scales due to their fragile nature. Enriched with symmetry, topology, and entanglement, quantum matters exhibit exotic emergent properties that defy our classical intuitions. This dissertation is devoted to the study of quantum matters, consisting of three distinct topics. The first part studies quantum phases of matter and their phase transitions, with an emphasis on the role of symmetry and their exotic natures. Employing both analytical and numerical methods, we study the deconfinement physics in quantum Hall systems and quantum magnets. The second part focuses on the physics of Moir\'e systems in two-dimensional materials that exhibit interesting single-particle as well as correlated many-body physics. Using a continuum model description, we illuminate several recent experimental findings in twisted double bilayer graphenes. The last part touches on a more unconventional topic, non-Hermitian systems. We provide mathematical foundations to classify their topological structure as well as the potential connection to anomalous or driven systems.
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.