The discovery of generalized symmetries has led to powerful new insights into quantum matter. They have been used to classify new families of quantum phases, place constraints on phases realizable in a given physical system, and conceptually unify seemingly disparate phenomena. In many ways, they prove just as powerful as traditional symmetries at organizing and constraining the theories that describe quantum matter. In this thesis, we attempt a unification of such constraints by developing a holographic correspondence between (generalized) symmetries and topological orders, called the Sym/TO correspondence. For any (finite internal) symmetry of a quantum system in d (spatial) dimensions, we associate with it a unique topological order in d + 1 dimensions, called its Symmetry Topological Order (SymTO). We devise an operator algebraic recipe to compute the SymTO data for any lattice spin model, demonstrating it in a number of examples. We then use the SymTO to classify possible quantum phases allowed by the symmetry—we call this a generalized Landau paradigm. Besides classifying phases, we also identify constraints on the phase transitions between them using a SymTO-resolved modular bootstrap. We test this framework in a quantum spin chain with non-invertible symmetries. We discover a new Kramers-Wannier-like duality and a rich phase diagram including a noninvertible symmetry-enriched incommensurate phase. The translation symmetry of the spin chain has a nontrivial interplay with the lattice Kramers-Wannier duality, which matches the anomaly of the corresponding non-invertible symmetry in the low-energy effective field theory. Finally, we explore such unusual anomaly-matching mechanisms in more detail in the context of the chiral anomaly of a single massless Dirac fermion, demonstrating a novel lattice realization of chiral symmetries and their anomaly.
Abstract These lecture notes are mainly concerned with how the concept of topological insulators is generalized when interactions are included. More generally, the interplay of symmetry and topology is discussed. Traditionally, phases of matter were distinguished on the basis of symmetry alone. On the other hand, fractional quantum Hall phases are examples of topological states whose essential character does not require a discussion of symmetry. However, topological insulators are an example of a new phase of matter that combines both symmetry and topology.
For most of the last century, condensed matter physics has been dominated by band theory and Landau's symmetry breaking theory. In the last twenty years, however, there has been an emergence of a new paradigm associated with fractionalization, emergent gauge bosons and fermions, topological order, string-net condensation, and long range entanglements. These new physical concepts are so fundamental that they may even influence our understanding of the origin of light and electrons in the universe. This book is a pedagogical and systematic introduction to the new concepts and quantum field theoretical methods in condensed matter physics. It discusses many basic notions in theoretical physics which underlie physical phenomena in nature, including a notion that unifies light and electrons. Topics covered include dissipative quantum systems, boson condensation, symmetry breaking and gapless excitations, phase transitions, Fermi liquids, spin density wave states, Fermi and fractional statistics, quantum Hall effects, topological/quantum order, and spin liquid and string-net condensation. Methods discussed include the path integral, Green's functions, mean-field theory, effective theory, renormalization group, bosonization in one- and higher dimensions, non-linear sigma-model, quantum gauge theory, dualities, projective construction, and exactly soluble models beyond one-dimension.
Recently, a new class of quantum phases of matter---symmetry protected topological states, such as topological insulators---has attracted much attention. In presence of interactions, group cohomology provides a classification of these [Chen et al., Phys. Rev. B 87, 155114 (2013)]. These phases have short-ranged entanglement and no topological order in the bulk. However, when long-range entangled topological order is present, it is much less understood how to classify quantum phases of matter in the presence of global symmetries. Here we present a classification of bosonic gapped quantum phases with or without long-range entanglement in the presence or absence of on-site global symmetries. In
2+1 dimensions, the quantum phases in the presence of a global symmetry group
SG, and with topological order described by a finite gauge group
GG, are classified by the cohomology group ${H}^{3}(SG\ifmmode\times\else\texttimes\fi{}GG,U(1))$. Generally, in
d+1 dimensions, such quantum phases are classified by ${H}^{d+1}(SG\ifmmode\times\else\texttimes\fi{}GG,U(1))$. Although we only partially understand to what extent our classification is complete, we present an exactly solvable local bosonic model, in which the topological order is emergent, for each given class in our classification. When the global symmetry is absent, the topological order in our models is described by the general Dijkgraaf-Witten discrete gauge theories. When the topological order is absent, our models become the exactly solvable models for symmetry protected topological phases [Chen et al., Phys. Rev. B 87, 155114 (2013)]. When both the global symmetry and the topological order are present, our models describe symmetry enriched topological phases. Our classification includes, but goes beyond, the previously discussed projective symmetry group classification. Measurable signatures of these symmetry enriched topological phases and generalizations of our classification are discussed.
Abstract Classifying and understanding phases of matter is an important task in condensed matter physics. The class of ‘conventional’ symmetry-broken phases is well understood in terms of Landau’s theory. A paradigmatic example is the Z2-symmetric Ising model with symmetric (paramagnetic) and symmetry-broken (ferromagnet) phases, which can be distinguished by measuring the magnetization as a local order parameter. Topological phases of matter, however, are less understood and no complete classification is known. These lecture notes discuss schemes that provide an understanding and characterization of certain topological phases of matter in models with local Hamiltonians. The main focus is on symmetry-protected topological (SPT) phases in one-dimensional bosonic systems. On the basis of the entanglement properties of such systems, a motivation is given for the matrix product state (MPS) representation of ground states. Using the MPS framework, it is demonstrated how SPT phases can be classified using projective representations of the symmetries.