Dualities are central to our understanding of physics, in particular in the study of quantum phases of matter. In this dissertation, we explore exact lattice dualities that relate various topological phases of matter, which elucidates the interrelations between them. Moreover, searching for new dualities can even lead to the theoretical realization of new phases of matter. In the first half, we study local dualities, and how they can be used as quantum circuits to prepare Symmetry-Protected Topological (SPT) phases. We construct explicit quantum circuits that prepare fermionic SPT phases, which gives insight into the strongly interacting nature of these phases in contrast to previous free-fermion realizations. Moreover, we discuss a family of dualities, which not only generates the entanglement necessary to prepare SPT phases, but also generates anomalous symmetries which arise at transitions between such topological phases, giving new insights into topological quantum criticality. In the second half, we explore non-local dualities, exemplified by the famous Kramers-Wannier or Jordan-Wigner dualities in one dimension. we give a systematic method to generalize such non-local dualities to systems with different symmetries and to higher dimensions, which leads to explicit lattice models realizing new phases of matter. In particular, we uncover new families of fracton phases such as (i) Phases where immobile fracton excitations are emergent fermions (ii) Fracton phases that are enriched by spatial and/or time-reversal symmetries (iii) phases that "hybridizes" between conventional topological orders and fracton orders, some of which hosts non-abelian fracton excitations. Finally, we show that the two types of dualities can be related by an extra ingredient: measurement. This gives a practical method to realize non-local dualities in near-term quantum devices, paving a way to efficiently prepare a large class of topological phases of matter in experiments.
Dualities are central to our understanding of physics, in particular in the study of quantum phases of matter. In this dissertation, we explore exact lattice dualities that relate various topological phases of matter, which elucidates the interrelations between them. Moreover, searching for new dualities can even lead to the theoretical realization of new phases of matter. In the first half, we study local dualities, and how they can be used as quantum circuits to prepare Symmetry-Protected Topological (SPT) phases. We construct explicit quantum circuits that prepare fermionic SPT phases, which gives insight into the strongly interacting nature of these phases in contrast to previous free-fermion realizations. Moreover, we discuss a family of dualities, which not only generates the entanglement necessary to prepare SPT phases, but also generates anomalous symmetries which arise at transitions between such topological phases, giving new insights into topological quantum criticality. In the second half, we explore non-local dualities, exemplified by the famous Kramers-Wannier or Jordan-Wigner dualities in one dimension. we give a systematic method to generalize such non-local dualities to systems with different symmetries and to higher dimensions, which leads to explicit lattice models realizing new phases of matter. In particular, we uncover new families of fracton phases such as (i) Phases where immobile fracton excitations are emergent fermions (ii) Fracton phases that are enriched by spatial and/or time-reversal symmetries (iii) phases that "hybridizes" between conventional topological orders and fracton orders, some of which hosts non-abelian fracton excitations. Finally, we show that the two types of dualities can be related by an extra ingredient: measurement. This gives a practical method to realize non-local dualities in near-term quantum devices, paving a way to efficiently prepare a large class of topological phases of matter in experiments.
In this thesis, we study gapped topological phases of matter in systems with strong inter-particle interaction. They are challenging to analyze theoretically, because interaction not only gives rise to a plethora of phases that are otherwise absent, but also renders methods used to analyze non-interacting systems inadequate. By now, people have had a relatively systematic understanding of topological orders in two spatial dimensions. However, less is known about the higher dimensional cases. In Chapter 2, we will explore three dimensional long-range entangled topological orders in the framework of Walker-Wang models, which are a class of exactly solvable models for three-dimensional topological phases that are not known previously to be able to capture these phases. We find that they can represent a class of twisted discrete gauge theories, which were discovered using a different formalism. Meanwhile, a systematic theory of bosonic symmetry protected topological (SPT) phases in all spatial dimensions have been developed based on group cohomology. A generalization of the theory to group supercohomology has been proposed to classify and characterize fermionic SPT phases in all dimensions. However, it can only handle cases where the symmetry group of the system is a product of discrete unitary symmetries. Furthermore, the classification is known to be incomplete for certain symmetries. In Chapter 3, we will construct an exactly solvable model for the two-dimensional time-reversal-invariant topological superconductors, which could be valuable as a first attempt to a systematic understanding of strongly interacting fermionic SPT phases with anti-unitary symmetries in terms of exactly solvable models. In Chapter 4, we will propose an alternative classification of fermionic SPT phases using the spin cobordism theory, which hopefully can capture all the phases missing in the supercohomology classification. We test this proposal in the case of fermionic SPT phases with Z2 symmetry, where Z2 is either time-reversal or an internal symmetry. We find that cobordism classification correctly describes all known fermionic SPT phases in space dimensions less than or equal to 3.
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.
We introduce a generalization of conventional lattice gauge theory to describe fracton topological phases, which are characterized by immobile, pointlike topological excitations, and subextensive topological degeneracy. We demonstrate a duality between fracton topological order and interacting spin systems with symmetries along extensive, lower-dimensional subsystems, which may be used to systematically search for and characterize fracton topological phases. Commutative algebra and elementary algebraic geometry provide an effective mathematical tool set for our results. Our work paves the way for identifying possible material realizations of fracton topological phases.
This dissertation discusses some applications of conformal field theories (CFTs) in topological phases of matter. The first part is devoted to a discussion of coupled wire constructions of some novel quantum Hall systems. Through a theoretical coupled wire model, we construct strongly correlated electronic integer quantum Hall states with filling factor 16. The edge state is a bosonic chiral (E8)1 CFT, which is closely related to topological paramagnets in (3+1)d. As a distinguishing feature, these states support electric and thermal Hall transport violating the Wiedemann-Franz law. We further construct two descendant non-Abelian quantum Hall states at filling 8, each carrying bosonic chiral (G2)1 or (F4)1 edge theories, and hosting Fibonacci anyonic excitations in the bulk. Finally, we discover a new notion of particle-hole conjugation based on the E8 state that relates the G2 and F4 Fibonacci states, which is reminiscent of similar physics in half-filled Landau level. The second part is focused on the surface topological orders of 3D bulk topological systems. Symmetry protected and symmetry-enriched topological (SPT/SET) phases in three dimensions are quantum systems that support non-trivial two-dimensional surface states. These surface states develop finite excitation energy gaps when the relevant symmetries are broken. On the other hand, one-dimensional gapless modes can populate along interfaces that separate adjacent gapped surface domains with distinct symmetry-breaking orders. A surface strip pattern in general reduces the low-energy SPT/SET surface degrees of freedom onto a 2D array of gapless 1D channels. These channels can be coupled to one another by quasiparticle tunneling, and these inter-wire interactions collectively provide an effective description of the surface state. In this part, we study a general class of symmetry-preserving or breaking SPT/SET surface states that admit finite excitation energy gaps and Abelian topological orders via the coupled wire construction. In particular, we focus on the prototype Abelian surface topological orders that fall under the ADE classification of simply-laced Lie algebras. We also elaborate on the emergent symmetry and duality properties of the coupled wire models. The third part is to discuss the relation between the conformal boundary state and (2+1)d SPT phases. We propose a diagnostic tool for detecting nontrivial symmetry-protected topological (SPT) phases protected by a symmetry group G in 2 + 1 dimensions. Our method is based on directly studying the 1 + 1-dimensional anomalous edge conformal field theory (CFT) of SPT phases. We claim that if the CFT is the edge theory of an SPT phase, then there must be an obstruction to cutting it open. This obstruction manifests as the non-existence of boundary states in the CFT that preserves both the conformal symmetry and the global symmetry G. We discuss the relation between edgeability and gappability in the presence of G. We study several examples including time-reversal symmetric topological insulators, ZN symmetric bosonic SPT phases, and Z2 × Z2 symmetric topological superconductors.
Recent work has constructed higher-dimensional analogs of non-invertible symmetries similar to 1+1d Kramers-Wannier duality. Although their continuum descriptions often treat purely gravitational topological terms as inessential counterterms, these terms can have an essential lattice manifestation: they distinguish states prepared by finite-depth quantum circuits (FDQCs) from those entangled by nontrivial quantum cellular automata (QCAs). Motivated by this mismatch, we show that QCAs associated with gravitational topological responses arise in several related settings: (1) lattice realizations of projective SL(2,ZN) relations generated by topological operations on symmetries; (2) squares of dualities that generalize the relation between fermionization and Kramers-Wannier duality; (3) lattice implementations of QCAs through higher-form gauging; and (4) invertible phases protected by generalized time-reversal symmetries. We derive new projective SL(2,ZN) relations whose projective phases are gravitational topological responses constructed from Stiefel-Whitney classes. We furthermore give a general protocol for preparing the associated QCA-entangled states using finite-depth unitary circuits, measurements, and error correction. These results unify the study of gravitational topological responses in field theories, higher dimensional dualities, and quantum cellular automata.
Topological quantum phases of matter are often characterized by the presence of fractionalized quasiparticles, which exhibit non-trivial braiding statistics or carry fractional quantum numbers, or of protected gapless surface states. In this thesis, we study topological phases in two and three spatial dimensions, from the perspective of searching for new exotic quantum phases and of characterizing their experimental signatures.\nWe first study topological defects in fermionic paired superfluids and discover that in the presence of a multiply quantized vortex, such a state hosts unpaired fermions in the BCS regime. We predict that these unpaired fermions will result in an experimentally measurable deviation of the system's angular momentum from its value in the BEC regime. Focusing on two-dimensions, we then study superconductors coupled to dynamically fluctuating electromagnetism, and establish a universal framework for studying the low-energy physics of such systems. We derive topological field theories describing all spin-singlet superconductors which naturally capture the interplay of symmetry and topology in these gapped states.\nThe remainder of this thesis is focused on a novel class of long-range entangled states of matter, known as “fracton” phases, in both two- and three-dimensions. These phases exhibit an intriguing phenomenology, most surprising of which is the presence of excitations which are immobile in isolation. Studying the non-equilibrium dynamics of gapped fracton phases, we discover that they naturally exhibit glassy quantum dynamics in the absence of quenched disorder, and hence may have potential technological applications as robust quantum memories. Finally, we conclude with a description of fracton phases as higher-rank tensor gauge theories and discuss the emergent phases of matter in which a finite density of fractons or their bound states may exist.