In this thesis, we present recent contributions to the study of Type-I non-abelian fracton models, which led us to propose the notion of generalized foliated fracton orders that captures the universal properties of both abelian and non-abelian Type-I fracton models.
\r\n\r\nFracton models are known for their exotic properties such as point-like excitations with restricted mobilities and robust topological ground state degeneracy that grows sub-extensively with the system size. A multitude of Type-I fracton models whose excitations obey either abelian or non-abelian fusion rules have recently been constructed. Among them, a large number of the abelian fracton models have been shown to possess foliation structures, where models of different system sizes can be related through the addition / removal of an entire piece of topologically ordered system on a sub-dimensional manifold via the action of a finite-depth local unitary circuit. In this thesis, this is referred to as the original foliation renormalization group (RG) scheme, which leads us to the notion of original foliation fracton orders. The Ising cage-net model and other similar non-abelian models are closely related to these abelian models in terms of their excitation structures and coupled layers construction etc. However, it was not known whether their fracton orders can also be understood within the original foliation framework. We address this problem in this thesis.
\r\n \r\nIn Chapter 2, we show that the Ising cage-net model does not fit into the original definition of foliated fracton orders, by calculating its ground state degeneracy. We realize that there exists naturally a more general way to define foliation -- the generalized foliation scheme (Chapter 3). The Ising cage-net and other similar non-abelian fracton models are foliated according to this generalized scheme. In the generalized foliation scheme, the RG transformation is defined by, from the excitation perspective, the condensation of planons / gauging subsystem symmetries. In terms of quantum circuits, this RG transformation is equivalent to a sequential linear-depth circuit that acts near a sub-dimensional manifold. With this definition, we can study phase relation of the Ising cage-net with other known fracton models. In Chapter 4, via gauging composite subsystem symmetries, we further show that the Ising cage-net belongs to the same generalized foliated fracton phases as the prototypical X-cube model. Furthermore, gauging composite subsystem symmetries opens up a new route to constructing non-abelian fracton models hosting exotic non-abelian fractons. An example is the tri-Ising-fracton model (Sec. 4.5).
", "doi": "10.7907/2x8m-j581", "publication_date": "2025", "thesis_type": "phd", "thesis_year": "2025" }, { "id": "thesis:15273", "collection": "thesis", "collection_id": "15273", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06022023-010804337", "type": "thesis", "title": "Fractonic Orders from Lattice Models and Field Theories", "author": [ { "family_name": "Ma", "given_name": "Xiuqi", "orcid": "000-0001-8294-2277", "clpid": "Ma-Xiuqi" } ], "thesis_advisor": [ { "family_name": "Chen", "given_name": "Xie", "orcid": "0000-0003-2215-2497", "clpid": "Chen-Xie" } ], "thesis_committee": [ { "family_name": "Alicea", "given_name": "Jason F.", "orcid": "0000-0001-9979-3423", "clpid": "Alicea-J" }, { "family_name": "Chen", "given_name": "Xie", "orcid": "0000-0003-2215-2497", "clpid": "Chen-Xie" }, { "family_name": "Hsieh", "given_name": "David", "orcid": "0000-0002-0812-955X", "clpid": "Hsieh-David" }, { "family_name": "Ni", "given_name": "Yi", "orcid": "0000-0002-5287-4258", "clpid": "Ni-Yi" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Fracton models are characterized by exotic features such as point-like excitations with restricted mobility, sub-extensive ground state degeneracy and UV/IR mixing. They have been studied previously using exactly solvable lattice models, higher rank gauge theories, etc. In an effort to classify fracton models into phases (i.e., fractonic orders), the so-called foliation structure has been introduced and shown to exist in many previously known models. A natural question then arises concerning the feasibility of the foliation paradigm in general. In this thesis, I study fracton models beyond the foliation paradigm and give simple diagnostics for the absence of a foliation structure. New notions of fractonic orders therefore need to be conceived, and I present such a conception which is a generalization of the foliation RG.
\r\n\r\nIn Chapters 2 - 4, I introduce new fracton models obtained from infinite-component Chern-Simons (CS\u221e) theories. By calculating observables such as ground state degeneracy and planon braiding statistics, I prove that most CS\u221e theories are not foliated. A CS\u221e theory can also be gapless with certain choices of parameters, and I show that such a theory is a stable gapless fracton model. Furthermore, I discuss topological features of a large subclass of gapless CS\u221e theories and present fully continuous effective field theories for this subclass.
\r\n\r\nIn Chapters 5 - 6, I discuss a new notion of fractonic orders by studying the example of the Ising cage-net model. I begin by calculating the ground state degeneracy of the model, which shows that the model is not foliated. The calculation uses an operator algebra approach which relies only on intrinsic physical properties of the model rather than microscopic details, and I establish the framework of this approach conceptually and via examples. I then argue why this intrinsic approach, despite being a tool for calculation initially, may be a useful characterization of a fractonic order. Finally, I present a generalized foliation RG scheme, apply it to the Ising cage-net model, and discuss its limitations.
", "doi": "10.7907/g80m-dy31", "publication_date": "2023", "thesis_type": "phd", "thesis_year": "2023" }, { "id": "thesis:14645", "collection": "thesis", "collection_id": "14645", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05272022-220331463", "primary_object_url": { "basename": "Wilbur_Shirley_Thesis.pdf", "content": "final", "filesize": 26187118, "license": "other", "mime_type": "application/pdf", "url": "/14645/1/Wilbur_Shirley_Thesis.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Structure of Entanglement in Fracton Phases of Matter", "author": [ { "family_name": "Shirley", "given_name": "Wilbur Eric", "orcid": "0000-0002-4970-1460", "clpid": "Shirley-Wilbur-Eric" } ], "thesis_advisor": [ { "family_name": "Chen", "given_name": "Xie", "orcid": "0000-0003-2215-2497", "clpid": "Chen-Xie" } ], "thesis_committee": [ { "family_name": "Preskill", "given_name": "John P.", "orcid": "0000-0002-2421-4762", "clpid": "Preskill-J" }, { "family_name": "Motrunich", "given_name": "Olexei I.", "orcid": "0000-0001-8031-0022", "clpid": "Motrunich-Olexei" }, { "family_name": "Hsieh", "given_name": "David", "orcid": "0000-0002-0812-955X", "clpid": "Hsieh-David" }, { "family_name": "Chen", "given_name": "Xie", "orcid": "0000-0003-2215-2497", "clpid": "Chen-Xie" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "This thesis discusses recent contributions to the theory of gapped fracton phases of matter, utilizing exactly solvable Hamiltonian models as the primary tool of study. A large component of the work revolves around the notion of a foliation structure, which is a defining feature of the long-range entanglement in certain gapped fracton states. We introduce this concept, identify its presence in a handful of prominent fracton models, and explore its consequences in terms of entanglement entropy and fractional excitations. A second major theme of the thesis is the characterization of gapped fracton states via emergent gauge theories based on discrete subsystem symmetries. We introduce a variety of novel fractonic gauge theories including twisted and fermionic variants, identify their emergence in a bevy of well-known models, and classify them with the use of novel topological invariants. We also establish a link between subsystem symmetry and entanglement renormalization group flow in fractal spin liquids.
", "doi": "10.7907/7b2c-2e06", "publication_date": "2022", "thesis_type": "phd", "thesis_year": "2022" }, { "id": "thesis:13644", "collection": "thesis", "collection_id": "13644", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:02262020-182938837", "type": "thesis", "title": "Information Scrambling in Quantum Many-Body Systems", "author": [ { "family_name": "Zhang", "given_name": "Yongliang", "orcid": "0000-0002-8246-3759", "clpid": "Zhang-Yongliang" } ], "thesis_advisor": [ { "family_name": "Chen", "given_name": "Xie", "orcid": "0000-0003-2215-2497", "clpid": "Chen-Xie" } ], "thesis_committee": [ { "family_name": "Brandao", "given_name": "Fernando", "orcid": "0000-0003-3866-9378", "clpid": "Brand\u00e3o-F-G-S-L" }, { "family_name": "Alicea", "given_name": "Jason F.", "orcid": "0000-0001-9979-3423", "clpid": "Alicea-J" }, { "family_name": "Chen", "given_name": "Xie", "orcid": "0000-0003-2215-2497", "clpid": "Chen-Xie" }, { "family_name": "Endres", "given_name": "Manuel A.", "orcid": "0000-0002-4461-224X", "clpid": "Endres-M" } ], "local_group": [ { "literal": "Institute for Quantum Information and Matter" }, { "literal": "Walter Burke Institute for Theoretical Physics" }, { "literal": "div_pma" } ], "abstract": "A closed quantum system never forgets its initial state, but the encoded information can get scrambled and become inaccessible without measuring a large fraction of all the system degrees of freedom. This scrambling can be diagnosed by studying the spatial spreading of initially local operators under the Heisenberg time evolution, and the decay of the out-of-time-ordered correlators (OTOC). What insights can OTOCs provide to understand the dynamics of quantum many-body systems? What are the characteristic behaviors of OTOCs during the time evolution? How is information scrambling affected by the dissipation in open quantum many-body systems?
\r\n\r\nWe first study slow scrambling in many-body localized systems via calculating various correlators, two-point retarded correlators and OTOCs. Comparing with retarded correlators, OTOCs provide more information about the dynamics. We find that disorder slows and partially halts the onset of information scrambling. Instead of ballistic spreading, propagation of information forms a logarithmic light cone.
\r\n\r\nNext, we study the finite-size scaling of OTOCs at late times in generic thermalizing quantum many-body systems. When energy is conserved, the late-time saturation value of the OTOC of generic traceless local operators scales as an inverse polynomial in the system size. This is in contrast to the inverse exponential scaling expected for chaotic dynamics without energy conservation.
\r\n\r\nWe also study information scrambling in open quantum many-body systems. We define a dissipative version of OTOC and study its behaviors in a prototypical chaotic quantum chain with dissipation. We find that dissipation leads to not only the overall decay of the scrambled information due to leaking, but also structural changes so that the information light cone can only reach a finite distance even when the effect of overall decay is removed.
\r\n\r\nFinally, we construct a family of local Hamiltonians for understanding the asymmetric information scrambling. Our models live on a one-dimensional lattice and exhibit asymmetric butterfly light cone between the left and right spatial directions.
", "doi": "10.7907/GDZ1-0G66", "publication_date": "2020", "thesis_type": "phd", "thesis_year": "2020" }, { "id": "thesis:11488", "collection": "thesis", "collection_id": "11488", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:04242019-205929726", "type": "thesis", "title": "Topological Phases of Matter: Exactly Solvable Models and Classification", "author": [ { "family_name": "Wang", "given_name": "Zitao", "orcid": "0000-0002-2326-2674", "clpid": "Wang-Zitao" } ], "thesis_advisor": [ { "family_name": "Chen", "given_name": "Xie", "orcid": "0000-0003-2215-2497", "clpid": "Chen-Xie" } ], "thesis_committee": [ { "family_name": "Kapustin", "given_name": "Anton N.", "orcid": "0000-0003-3903-5158", "clpid": "Kapustin-A" }, { "family_name": "Alicea", "given_name": "Jason F.", "orcid": "0000-0001-9979-3423", "clpid": "Alicea-J" }, { "family_name": "Endres", "given_name": "Manuel A.", "orcid": "0000-0002-4461-224X", "clpid": "Endres-M" }, { "family_name": "Chen", "given_name": "Xie", "orcid": "0000-0003-2215-2497", "clpid": "Chen-Xie" } ], "local_group": [ { "literal": "Institute for Quantum Information and Matter" }, { "literal": "Walter Burke Institute for Theoretical Physics" }, { "literal": "div_pma" } ], "abstract": "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.
", "doi": "10.7907/BXJR-1M62", "publication_date": "2019", "thesis_type": "phd", "thesis_year": "2019" } ]