In this thesis, we explore the aspects of symmetry, topology and anomalies in quantum matter with entanglement from both condensed matter and high energy theory viewpoints. The focus of our research is on the gapped many-body quantum systems including symmetry-protected topological states and topologically ordered states. Chapter 1. Introduction. Chapter 2. Geometric phase, wavefunction overlap, spacetime path integral and topological invariants. Chapter 3. Aspects of Symmetry. Chapter 4. Aspects of Topology. Chapter 5. Aspects of Anomalies. Chapter 6. Quantum Statistics and Spacetime Surgery. Chapter 7. Conclusion: Finale and A New View of Emergence-Reductionism. (Thesis supervisor: Prof. Xiao-Gang Wen)
In this thesis, we explore the aspects of symmetry, topology and anomalies in quantum matter with entanglement from both condensed matter and high energy theory viewpoints. The focus of our research is on the gapped many-body quantum systems including symmetry-protected topological states and topologically ordered states. Chapter 1. Introduction. Chapter 2. Geometric phase, wavefunction overlap, spacetime path integral and topological invariants. Chapter 3. Aspects of Symmetry. Chapter 4. Aspects of Topology. Chapter 5. Aspects of Anomalies. Chapter 6. Quantum Statistics and Spacetime Surgery. Chapter 7. Conclusion: Finale and A New View of Emergence-Reductionism. (Thesis supervisor: Prof. Xiao-Gang Wen)
Symmetry is a fundamental organising principle in theoretical physics, underpinning the conservation of physical quantities and the characterisation of phases of matter. It is closely related to topology, leading to generalisations such as higher-form and non-invertible symmetries, with applications in electromagnetism, the standard model of particle physics, condensed matter, and quantum computation. This thesis harnesses generalised symmetries to derive universal results in gapped and gapless phases of matter, focusing particularly on topologically ordered systems and conformal field theories (CFTs). Topological order is characterised by long-range entangled gapped phases, namely ground states that cannot be disentangled by finite-depth quantum circuits. At low energies it is described via topological quantum field theories (TQFTs). One part of this thesis focusses on extracting patterns of long-range entanglement from the low-energy TQFT for a class of topological orders described by abelian BF theory. By considering algebras of topological operators restricted to subregions, a new measure of entanglement, termed essential topological entanglement, is introduced to refine topological entanglement entropy. This is complemented by a detailed analysis of the edge-mode spectrum of abelian BF theory and its contribution to entanglement entropy across different topologies. A key mathematical result, enabling the above analysis, is the generalisation of Kac–Moody current algebras and their representation theory to higher dimensions. On the gapless front, the thesis considers CFTs with continuous higher-form symmetries, invertible or non-invertible. It is shown that they also possess infinite-dimensional current algebras akin to those described above. Their representation theory allows for a complete characterisation of the quantum states of these theories across arbitrary spatial topologies. In four dimensions, CFTs with continuous one-form symmetries can be described in terms of free photons. Combining this with the previous analysis leads to a one-to-one correspondence between states on particular topologies and line operators, i.e. wires carrying distributions of electric and magnetic charge.
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.
T HE CLASSIFICATION OF DISTINCT PHASES OF MATTER is a central theme in condensed matter physics. A phase of matter can be viewed as an equivalence class of physical states that share a certain set of properties. In the case of topological quantum matter, these properties are associated with topological invariants that take specific values for each equivalence class. Perturbing a gapped state that corresponds to a given topological phase (i.e., smoothly deforming the Hamiltonian of the system) does not affect the topological invariant, as long as the energy gap remains finite. The value of this invariant changes only if the system undergoes a quantum phase transition, which is signaled by the collapse of the gap. Full topological protection can only emerge in interacting systems and requires the presence of long-range entanglement, which is at the origin of intrinsic topological order. However, topological properties protected by symmetries emerge even in noninteracting systems and, perhaps surprisingly, can be described within the framework of good old band theory of solids. Systems belonging to this category are generically called topological insulators, after the class of materials that is largely responsible for the recent surge in research activity on topological matter. In addition, topological band theory considerations can also be applied to superconductors (and superfluids), as described at the mean-field level using Bogoliubov-de Gennes theory. This chapter summaries the key ideas behind the topological classification of noninteracting systems and discusses the basic properties of various classes of topological insulators and superconductors. For in-depth discussions and related developments the reader is referred to a number of books [48, 402, 153] and review papers [211, 212, 357, 22] specifically dedicated to this subject.
The classification of distinct phases of matter is a central theme in condensed matter physics. A phase of matter can be viewed as an equivalence class of physical states that share a certain set of properties. In the case of topological quantum matter, these properties are associated with topological invariants that take specific values for each equivalence class. Perturbing a gapped state that corresponds to a given topological phase (i.e., smoothly deforming the Hamiltonian of the system) does not affect the topological invariant, as long as the bulk energy gap remains finite. The value of this invariant changes only if the system undergoes a quantum phase transition, which is signaled by the collapse of the gap. Full topological protection can only emerge in interacting systems and requires the presence of long-range entanglement, which is at the origin of intrinsic topological order. However, topological properties protected by symmetries emerge even in noninteracting systems and, perhaps surprisingly, can be described within the framework of good old band theory of solids. Systems belonging to this category are generically called topological insulators, after the class of materials that is largely responsible for the recent surge in research on topological matter. In addition, topological band theory considerations can also be applied to superconductors (and superfluids), as described at the mean-field level using Bogoliubov–de Gennes theory. This chapter summarizes the key ideas behind the topological classification of noninteracting systems and discusses the basic properties of various classes of topological insulators and superconductors. For in-depth discussions and related developments the reader is referred to a number of books [ 54 , 175 , 469 ] and review papers [ 23 , 239 , 240 , 421 ] specifically dedicated to this subject.
We present the correspondence between symmetry-protected topological (SPT) phases and their anomalous boundary states, based on examples in various spacetime dimensions. Through the study of the effect of interactions on these SPT phases, we discovered a new formalism of quantum anomalies, associated with discrete spacetime (such as time-reversal and spatial reflection) symmetries in particular, to classify distinct interacting topological phases. An example is the Z2 classification of the (2+1)d topological insulator protected by charge U(1) and time-reversal (or CP) symmetries, which can be deduced by the form of the global U(1) gauge anomalies on its edge theories defined on closed unorientable manifolds. In this case, the nontrivial phase (in free systems) is robust against electron interactions. Another example is the (3+1)d topological superconductor protected by only time-reversal or reflection symmetry. For this system, we identified the bulk phase by studying the global gravitational anomalies of the surface theories formulated on unorientable spacetime manifolds, and also discussed its connection to the collapse of the non-interacting classification by an integer Z to Z16, in the presence of interactions. We also revisit the problem of gauging a discrete internal symmetry in theories of chiral (Weyl) fermions in 3+1 dimensions – which have not been fully understood so far – from the perspective of fermionic SPT phases in 4+1 dimensions. Comparing with the previous results, we give a complete answer for the anomalies constraints on the discrete symmetry, as our approach is based on purely geometrical considerations, namely, our assumption is more fundamental and general. Furthermore, our result also provides an understanding of gapped states of fermions with anomalous discrete symmetries, and we present a model, based on weak coupling, for constructing these anomalous gapped states.
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.