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部分次元粒子：格子模型、場の理論、背景の数学的構造

Taiichi Nakanishi

Published Mar 23, 2026• 0 citations
Publisher

Abstract

Topologically ordered phases and their continuum descriptions provide a rich arena in which quantum many-body systems, algebraic topology, and quantum field theory meet.Recently, foliated fracton phases with subdimensional particles, which are excitations constrained to move only along lower-dimensional submanifolds, have emerged as a new class of quantum matter with potential applications to robust quantum information storage.These phases are naturally tied to foliation structures of the spatial manifold and exhibit unusual ground state degeneracies and subsystem conservation laws that go beyond conventional topological order.In this thesis we investigate subdimensional particles from three complementary viewpoints: lattice stabilizer codes, mathematical structures on manifolds, and continuum gauge theories.As a warm-up, we revisit the toric code and show how its stabilizer structure, logical operators, and ground state degeneracy are encoded in cellular (co)homology.This "triangle of ideas" relating lattice models, BF-type topological quantum field theories, and topological invariants serves as a guiding paradigm.We then extend this perspective to foliated fracton phases, focusing on the plaquette Ising model and the X-cube model as paradigmatic examples.We clarify how their subextensive ground state degeneracies and rigid line and membrane operators are organized by foliation-sensitive (co)homological data of the underlying manifold.Motivated by these observations, we construct a Godbillon-Vey U(1) BF-like gauge theory coupled to a codimension-one foliation.By promoting the redundancies of the foliation data to gauge symmetries, the resulting continuum theory depends only on the foliation class and naturally realizes Wilson and higher-form operators whose mobility is restricted to the leaves.In (2+1)-and (3+1)-dimensions we show that this theory reproduces, respectively, the ground state degeneracy and operator algebra of the plaquette Ising and X-cube models, providing a foliated analogue of the BF-toric code correspondence.Finally, we introduce matter field theories coupled to the foliation background and to the BF-like gauge fields.For complex scalars, fermions, and compact scalars we construct conserved currents that obey leafwise conservation laws enforcing subdimensional kinematics.We close by outlining extensions to higher-codimension foliations, more general G-structures, and symmetry topological field theories, and by emphasizing a generalized triangle relating microscopic models, geometric invariants, and continuum field theories for subdimensional particle systems.

DOI
10.14989/doctor.k26503

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Topologically ordered phases and their continuum descriptions provide a rich arena in which quantum many-body systems, algebraic topology, and quantum field theory meet.Recently, foliated fracton phases with subdimensional particles, which are excitations constrained to move only along lower-dimensional submanifolds, have emerged as a new class of quantum matter with potential applications to robust quantum information storage.These phases are naturally tied to foliation structures of the spatial manifold and exhibit unusual ground state degeneracies and subsystem conservation laws that go beyond conventional topological order.In this thesis we investigate subdimensional particles from three complementary viewpoints: lattice stabilizer codes, mathematical structures on manifolds, and continuum gauge theories.As a warm-up, we revisit the toric code and show how its stabilizer structure, logical operators, and ground state degeneracy are encoded in cellular (co)homology.This "triangle of ideas" relating lattice models, BF-type topological quantum field theories, and topological invariants serves as a guiding paradigm.We then extend this perspective to foliated fracton phases, focusing on the plaquette Ising model and the X-cube model as paradigmatic examples.We clarify how their subextensive ground state degeneracies and rigid line and membrane operators are organized by foliation-sensitive (co)homological data of the underlying manifold.Motivated by these observations, we construct a Godbillon-Vey U(1) BF-like gauge theory coupled to a codimension-one foliation.By promoting the redundancies of the foliation data to gauge symmetries, the resulting continuum theory depends only on the foliation class and naturally realizes Wilson and higher-form operators whose mobility is restricted to the leaves.In (2+1)-and (3+1)-dimensions we show that this theory reproduces, respectively, the ground state degeneracy and operator algebra of the plaquette Ising and X-cube models, providing a foliated analogue of the BF-toric code correspondence.Finally, we introduce matter field theories coupled to the foliation background and to the BF-like gauge fields.For complex scalars, fermions, and compact scalars we construct conserved currents that obey leafwise conservation laws enforcing subdimensional kinematics.We close by outlining extensions to higher-codimension foliations, more general G-structures, and symmetry topological field theories, and by emphasizing a generalized triangle relating microscopic models, geometric invariants, and continuum field theories for subdimensional particle systems.
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Fracton order is an intriguing new type of order which shares many common features with topological order, such as topology-dependent ground-state degeneracies, and excitations with mutual statistics.However, it also has several distinctive geometrical aspects, such as excitations with restricted mobility, which naturally lead to effective descriptions in terms of higher-rank gauge fields.In this paper, we investigate possible effective field theories for three-dimensional fracton order, by presenting a general philosophy whereby topological-like actions for such higher-rank gauge fields can be constructed.Our approach draws inspiration from Chern-Simons and BF theories in 2+1 dimensions, and imposes constraints binding higher-rank gauge charge to higher-rank gauge flux.We show that the resulting fractonic Chern-Simons and BF theories reproduce many of the interesting features of their familiar two-dimensional cousins.We analyze one example of the resulting fractonic Chern-Simons theory in detail, and show that upon quantization it realizes a gapped fracton order with quasiparticle excitations that are mobile only along a subset of one-dimensional lines, and display a form of fractional self-statistics.The ground-state degeneracy of this theory is both topology and geometry dependent, scaling exponentially with the linear system size when the model is placed on a three-dimensional torus.By studying the resulting quantum theory on the lattice, we show that it describes a Z s generalization of the Chamon code.
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In this work, we develop a coupled layer construction of fracton topological orders in d=3d=3d=3 spatial dimensions. These topological phases have subextensive topological ground-state degeneracy and possess excitations whose movement is restricted in interesting ways. Our coupled layer approach is used to construct several different fracton topological phases, both from stacked layers of simple d=2d=2d=2 topological phases and from stacks of d=3d=3d=3 fracton topological phases. This perspective allows us to shed light on the physics of the X-cube model recently introduced by Vijay, Haah, and Fu, which we demonstrate can be obtained as the strong-coupling limit of a coupled three-dimensional stack of toric codes. We also construct two new models of fracton topological order: a semionic generalization of the X-cube model, and a model obtained by coupling together four interpenetrating X-cube models, which we dub the `four color cube model''. The couplings considered lead to fracton topological orders via mechanisms we dub ``p-string condensation'' and ``p-membrane condensation'', in which strings or membranes built from particle excitations are driven to condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the phases we construct to be easily studied in terms of more familiar degrees of freedom. Our work raises the possibility of studying fracton topological phases from within the framework of topological quantum field theory, which may be useful for obtaining a more complete understanding of such phases.
Fracton phases are new types of phases of matter characterized by subsystem global symmetry, which is a generalized global symmetry whose symmetry operator is partially topological. Their continuum low-energy effective descriptions admit two different formulations: an exotic quantum field theory (QFT) using exotic tensor gauge fields, and a foliated QFT constructed from a foliation structure and foliated gauge fields. For certain fracton QFTs, these two descriptions are equivalent, which is called the foliated-exotic duality. In this dissertation, we extend the foliated-exotic duality by combining it with the anomaly inflow mechanism for 't Hooft anomalies of subsystem symmetries. This dissertation has two main results. First, we discuss the exotic and foliated BFBFBF theories in 2+1 dimensions, which exhibit the mixed 't Hooft anomaly of ZN×ZN\mathbb{Z}_N \times \mathbb{Z}_NZN​×ZN​ subsystem symmetry. This anomaly is captured by a subsystem symmetry-protected topological (SSPT) phase for ZN×ZN\mathbb{Z}_N \times \mathbb{Z}_NZN​×ZN​ subsystem symmetry in one dimension higher. By extending the foliated-exotic duality in the fractonic BFBFBF theory to the SSPT phase, we establish the field correspondences in the SSPT phase and construct the foliated description of the SSPT phase. Second, we discuss the exotic ϕϕϕ-theory in 2+1 dimensions -- a fractonic gapless scalar field theory, which has the 't Hooft anomaly of U(1)×U(1)U(1) \times U(1)U(1)×U(1) subsystem symmetry. The anomaly is captured by an SSPT phase for U(1)×U(1)U(1) \times U(1)U(1)×U(1) subsystem symmetry in 3+1 dimensions via the anomaly inflow mechanism. Extending the foliated-exotic duality to the ϕϕϕ-theory, we establish field correspondences in the ϕϕϕ-theory and construct the foliated ϕϕϕ-theory that is equivalent to the exotic ϕϕϕ-theory. This provides the first example of the foliated-exotic duality in gapless theories.
We broaden the scope of quantum field theory by introducing a general class of discrete gauge theories that realize either topological order or fracton behavior across dimensions. We start from translation-invariant systems endowed with unconventional charge-conservation laws, which we term \textit{exponential polynomial symmetries}. Gauging these symmetries yields ZN\mathbb{Z}_NZN​ gauge theories in 2D that exhibit topological order whose quasiparticles have constrained mobility and whose ground-state degeneracy shows ultraviolet (UV) dependence. These features are reminiscent of spatial symmetry-enriched topological order, wherein quasiparticle excitations transform nontrivially under lattice translations. We further propose a Chern-Simons variant that produces non-CSS stabilizer codes and outline a framework for exponentially symmetric subsystem SPT phases. Finally, we extend this gauging procedure to 3D, obtaining new variants of fracton topological order.