SK moinuddin, Pradip Mukherjee
A new representation of the scalar electrodynamics is discovered which gives a more redundant description of electromagnetic theory and suitable to construct an appropriate matter action which contains two global symmetries . The symmetries of the model when localized by gauge principle , provides a co-variant fracton gauge theory.
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Covariant Fracton Electrodynamics in Six Dimensions
本文提出六维协变Maxwell型分数电动力学，通过标量规范对称性构建应力能张量，揭示其在六维中成为总微分的结构，实现电荷和偶极矩的守恒。
Fracton and Non-Lorentzian Particle Duality: Gauge Field Couplings and Geometric Implications
本研究系统探究电磁场中分形子与非洛伦兹粒子（卡罗尔、伽利略粒子）的对偶性，通过构建正则作用量、对称性分析及规范代数，建立耦合框架并推导几何约束，统一分形子动力学与物极端极限，为相关领域提供新见解。
Unifying fractons, gravitons and photons from a gauge theoretical approach
本文基于规范理论方法，采用带有规范群指标的矢量场（Einstein-Cartan变量）重新构造相对论性无隙fracton理论，证明该模型统一了线性引力、fractons与Maxwell方程，揭示了自旋液体等系统的带电相，并恢复了Blasi-Maggiore规范模型。
Elastic description of planar quasicrystals can be formulated as an interplay between two Goldstone fields corresponding to phonon and phason degrees of freedom. We reformulate this description as a gauge theory with one gauge field that is symmetric under exchange of indices and one that is not. We also show which topological defects in quasicrystals can be succinctly incorporated in the dual description and interpret them as fractonic excitations. Finally we calculate the static interaction potential between defects in a quasicrystal with fivefold symmetry. This is done in the limit of a sma…
Gapless fracton phases are characterized by the conservation of certain charges and their higher moments. These charges generically couple to higher rank gauge fields. In this paper we study systems conserving charge and dipole moment, and construct the corresponding gauge fields propagating in arbitrary curved backgrounds. The relation between the symmetries of these class of systems and spacetime transformations is discussed. In fact, we argue that higher rank symmetric gauge theories are closer to gravitational fields than to a standard gauge theory.
Gauging fractons and linearized gravity
针对 fractons 协变规范场论，本文提出张量型替代规范固定，克服标量条件冻结 Landau 规范且无法给出线性化引力极限的缺陷。
Fractonic self-duality and covariant magnetic fractons
本文通过双势张量规范场框架推广电磁对偶性，提出协变磁分形子并实现分形子自对偶性，同时证明该体系中不存在类Witten效应，排除了分形子双子出现的可能性。
Fracton gauge fields from higher-dimensional gravity
本文通过Poincaré代数的高维Aristotelian收缩，从相对论性高维理论经维度约化导出fracton电动力学，并给出了弯曲空间下的规范不变推广。
A powerful mechanism for constructing gauge theories is to start from a theory with a global symmetry, then apply the "gauge principle," which demands that this symmetry hold locally. For example, the global phase rotation of a system of conserved charges can be promoted to a local phase rotation by coupling to an ordinary U(1) vector gauge field. More recently, a class of particles has been studied featuring not only charge conservation, but also conservation of higher moments, such as dipole moment, which leads to severe restrictions on the mobility of charges. These particles, called fracto…
We consider a vector gauge theory in 2 + 1 dimensions of the type recently proposed by Radzihovsky and Hermele [1] to describe fracton phases of matter. The theory has U(1)XU(1) vector gauge fields coupled to an additional vector field with a non conventional gauge symmetry. We added to the theory scalar matter in order to break the gauge symmetry. We analyze non trivial configurations by reducing the field equations to first order self dual (BPS) equations which we solved numerically. We have found vortex solutions for the gauge fields which in turn generate for the extra vector field non-tri…
We present a realization of fracton-elasticity duality purely formulated in terms of ordinary gauge fields, encompassing standard elasticity and incommensurate crystals as those describing twisted bilayer graphene, quasicrystals or more general moiré lattices. Our construction comprises a description of all types of two-dimensional defects: disclinations, dislocations, discompressions and point-like defects, and takes into account body forces and impurities. The original form of the duality in terms of tensor gauge fields is recovered after partial gauge fixing. We identify the coupling of eac…
Fracton Matter
本文回顾了fracton领域，展示了对称张量与耦合矢量规范理论如何从二维量子晶体弹性中涌现，并通过fracton-弹性对偶性与广义Higgs机制描述量子熔化相变及fracton相。
Fracton Gauge Theories in Curved Spacetimes
本文通过将偶极对称性重新表述为传统矢量规范场，实现了分形子规范理论在弯曲时空中的协变规范不变耦合，并识别出保持全局偶极对称性的弯曲几何，推导出相应的能量-动量守恒 Ward 恒等式。
Covariant fracton gauge theory with boundary
本文研究4维线性化引力与协变分形子理论组合模型在平直边界上的效应，导出类Maxwell-Chern-Simons的3维无迹张量理论，并发现广义U(1) Kaç-Moody代数与偶极矩守恒。
In this work we explore the interplay between global symmetry and the mobility of quasiparticle excitations. We show that fractonic matter naturally appears in a three dimensional U(1) gauge theory, enriched by global U(1) and translational symmetries, via the mechanism of anyonic spin-orbital coupling. We develop a systematic understanding of such symmetry-enforced mobility restrictions in terms of the classification of U(1) gauge theories enriched by U(1) and translational symmetries. We provide a unified construction of these phases by gauging layered symmetry-protected topological phases.
We introduce a new kind of foliated quantum field theory (FQFT) of gapped fracton orders in the continuum. FQFT is defined on a manifold with a layered structure given by one or more foliations, which each decompose spacetime into a stack of layers. FQFT involves a new kind of gauge field, a foliated gauge field, which behaves similar to a collection of independent gauge fields on this stack of layers. Gauge invariant operators (and their analogous particle mobilities) are constrained to the intersection of one or more layers from different foliations. The level coefficients are quantized and …
We analyze the "higher rank" gauge theories, that capture some of the phenomenology of the Fracton order. It is shown that these theories lose gauge invariance when arbitrarily weak and smooth curvature is introduced. We propose a resolution to this problem by introducing a theory invariant under area-preserving diffeomorphisms, which reduce to the "higher rank" gauge transformations upon linearization around a flat background. The proposed theory is geometric in nature and is interpreted as a theory of chiral topological elasticity. This theory exhibits some of the Fracton phenomenology. We e…
We argue that quiver gauge theories with $SU(N)$ gauge groups give rise to lattice gauge theories with matter possessing fractonic properties, where the lattice is the quiver itself. This idea extends a recent proposal by Razamat. This class of theories exhibit a $\mathbb{Z}_N$ 1-form global symmetry that can be used to classify their phases. The order parameter of this transition is the expectation value of Wilson loops, which correspond to mesonic operators in the underlying quiver gauge theory. We discuss how this perspective naturally fits with the deconstruction of a higher dimensional th…
Covariant Fractons and Weitzenböck Torsion
本文证明协变分形子规范理论的解构成 Moller-Hayashi-Shirafuji 引力理论解空间的一个子扇区，从而揭示协变分形子与引力的联系及其作为新型规范理论的意义。
We study complex scalar theories with dipole symmetry and uncover a no-go theorem that governs the structure of such theories and which, in particular, reveals that a Gaussian theory with linearly realised dipole symmetry must be Carrollian. The gauging of the dipole symmetry via the Noether procedure gives rise to a scalar gauge field and a spatial symmetric tensor gauge field. We construct a worldline theory of mobile objects that couple gauge invariantly to these gauge fields. We systematically develop the canonical theory of a dynamical symmetric tensor gauge field and arrive at scalar cha…
Quasi-topological fractons: a 3D dipolar gauge theory
本文提出了一种三维偶极规范理论，研究了秩-2张量场的准拓扑性质，发现其在壳时能动张量消失并具有三个自由度，揭示了广义平面电磁学和受限移动矢量电荷的分形子行为。
We present a dual formulation of the Cosserat theory of elasticity. In this theory a local element of an elastic body is described in terms of local displacement and local orientation. Upon the duality transformation these degrees of freedom map onto a coupled theory of a vector-valued one-form gauge field and an ordinary $U(1)$ gauge field. We discuss the degrees of freedom in the corresponding gauge theories, the defect matter and coupling to the curved space.
We consider the theory of a symmetric tensor field in 4D, invariant under a subclass of infinitesimal diffeomorphism transformations, where the vector diff parameter is the 4-divergence of a scalar parameter. The resulting gauge symmetry characterizes the "fracton" quasiparticles and identifies a theory which depends on a dimensionless parameter, which cannot be reabsorbed by a redefinition of the tensor field, despite the fact that the theory is free of interactions. This kind of "electromagnetic gauge symmetry" is weaker that the original diffeomorphism invariance, in the sense that the most…
2+1d Compact Lifshitz Theory, Tensor Gauge Theory, and Fractons
本文利用改进的Villain形式构造2+1维紧致Lifshitz理论的两种晶格正则化，揭示其对称性、反常、简并度与对偶性差异，并耦合U(1)张量规范理论得到分形子模型及精确对偶实现。
Emergent fracton strings from covariant bi-form gauge field theory
本文提出了一种协变场论框架，用于描述分数弦状物体的四阶张量规范场理论，通过 symmetry 原理自然涌现出分数弦激发，揭示了高阶规范场与引力结构的深层联系。
We study field theories with global dipole symmetries and gauge dipole symmetries. The famous Lifshitz theory is an example of a theory with a global dipole symmetry. We study in detail its 1+1d version with a compact field. When this global symmetry is promoted to a $U(1)$ dipole gauge symmetry, the corresponding gauge field is a tensor gauge field. This theory is known to lead to fractons. In order to resolve various subtleties in the precise meaning of these global or gauge symmetries, we place these 1+1d theories on a lattice and then take the continuum limit. Interestingly, the continuum …
We review what is known about fracton phases of quantum matter. Fracton phases are characterized by excitations that exhibit restricted mobility, being either immobile under local Hamiltonian dynamics, or mobile only in certain directions. They constitute a new class of quantum state of matter, which does not wholly fit into any of the existing paradigms, but which connects to areas including glassy quantum dynamics, topological order, spin liquids, elasticity theory, quantum information theory, and gravity. We begin by discussing gapped fracton phases, which may be described using exactly sol…
Recent work has established the existence of stable quantum phases of matter described by symmetric tensor gauge fields, which naturally couple to particles of restricted mobility, such as fractons. We focus on a minimal toy model of a rank 2 tensor gauge field, consisting of fractons coupled to an emergent graviton (massless spin-2 excitation). We show how to reconcile the immobility of fractons with the expected gravitational behavior of the model. First, we reformulate the fracton phenomenon in terms of an emergent center of mass quantum number, and we show how an effective attraction arise…
We study models with fracton-like order based on $\mathbb{Z}_2$ lattice gauge theories with subsystem symmetries in $d=2$ and $d=3$ spatial dimensions. The $3d$ model reduces to the $3$-dimensional Toric Code when subsystem symmetry is broken, giving an example of a subsystem symmetry enriched topological phase (SSET). Although not topologically protected, its ground state degeneracy has as leading contribution a term which grows exponentially with the square of the linear size of the system. Also, there are completely mobile gauge charges living along with immobile fractons. Our method shows …
Fracton theories possess exponentially degenerate ground states, excitations with restricted mobility, and nontopological higher-form symmetries. This paper shows that such theories can be defined on arbitrary spatial lattices in three dimensions. The key element of this construction is a generalization of higher-form gauge theories to so-called $\mathfrak{F}_p$ gauge theories, in which gauge transformations of rank-$k$ fields are specified by rank-$(k - p)$ gauge parameters. The $\mathbb{Z}_2$ rank-two theory of type $\mathfrak{F}_2$, placed on a cubic lattice and coupled to scalar matter, is…
The asymptotic structure of gauge theories describing fracton interactions is analyzed. Two sets of asymptotic conditions are proposed. Both encompass all known solutions, lead to finite charges and resolve the problem of the divergent energy coming from the monopole contribution. While the first set leads to the expected fracton symmetry algebra, including a dipole charge, the second set provides a soft infinite-dimensional extension of it. These soft charges provide evidence of a rich infrared structure for fracton-like theories and provide one corner of a possible fracton infrared triangle.
We consider XY-spin degrees of freedom on an FCC lattice, such that the system respects some subsystem global symmetry. We then gauge this global symmetry and study the corresponding $U(1)$ gauge theory on the FCC lattice. Surprisingly, this $U(1)$ gauge theory is dual to the original spin system. We also analyze a similar $\mathbb{Z}_N$ gauge theory on that lattice. All these systems are fractonic. The $U(1)$ theories are gapless and the $\mathbb{Z}_N$ theories are gapped. We analyze the continuum limits of all these systems and present free continuum Lagrangians for their low-energy physics.…
We offer a fractonic perspective on a familiar observation -- a flat sheet of paper can be folded only along a straight line if one wants to avoid the creation of additional creases or tears. Our core underlying technical result is the establishment of a duality between the theory of elastic plates and a fractonic gauge theory with a second rank symmetric electric field tensor, a scalar magnetic field, a vector charge, and a symmetric tensor current. Bending moment and momentum of the plate are dual to the electric and magnetic fields, respectively. While the flexural waves correspond to the q…
We study novel three-dimensional gapped quantum phases of matter which support quasiparticles with restricted mobility, including immobile "fracton" excitations. So far, most existing fracton models may be instructively viewed as generalized Abelian lattice gauge theories. Here, by analogy with Dijkgraaf-Witten topological gauge theories, we discover a natural generalization of fracton models, obtained by twisting the gauge symmetries. Introducing generalized gauge transformation operators carrying an extra phase factor depending on local configurations, we construct a plethora of exactly solv…
A symmetry principle for gauge theories with fractons
本文提出一类无能隙分形子相源于守恒荷不与空间平移对易的非均匀高形式对称性的自发破缺，揭示了其与高秩规范理论的联系，为设计和统一理解分形子模型提供了新视角。
Worldline Formulations of Covariant Fracton Theories
本文提出三种世界线模型，用于研究协变碎片理论的协变框架，通过BRST量子化方法验证了理论的正确性，并探讨了规范固定问题。
Fracton-elasticity duality on curved manifolds
本文在弯曲流形上构建晶体弹性的张量规范对偶理论，将度规自由度映射为三指标张量、弹性自由度映射为对称规范场，并揭示拓扑缺陷作为分形子激发，从而导出同时约束位移与引力缺陷的广义滑移约束。
Covariant field theory of 3D massive fractons
本文构建了三维大质量分形子的协变规范理论，通过类陈-西蒙斯项同时产生拓扑质量和内禀分形物质，传播一个有质量和一个无质量自由度，并统一描述内禀与外在分形子系统。
We describe topologically ordered and fracton ordered states on novel geometries which do not have an underlying manifold structure. Using tree graphs such as the $k$-coordinated Bethe lattice ${\cal B}(k)$ and a hypertree called the $(k,n)$-hyper-Bethe lattice ${\cal HB}(k,n)$ consisting of $k$-coordinated hyperlinks (defined by $n$ sites), we construct multidimensional arboreal arenas such as ${\cal B}(k_1) \square {\cal B}(k_2)$ by the notion of a graph Cartesian product $\square$. We study various quantum systems such as the ${\mathbb Z}_2$ gauge theory, generalized quantum Ising models (G…
We study the role of "$θ$ terms" in the action for three-dimensional $U(1)$ symmetric tensor gauge theories, describing quantum phases of matter hosting gapless higher-spin gauge modes and gapped subdimensional particle excitations, such as fractons. In Maxwell theory, the $θ$ term is a total derivative which has no effect on the gapless photon, but has two important, closely related consequences: attaching electric charge to magnetic monopoles (the Witten effect) and leading to a Chern-Simons theory on the boundary. We will find that a similar story holds in the higher-spin $U(1)$ gauge theor…
On the stability of fracton gravity
研究fracton引力（限制为纵向微分同胚的线性化引力变体）的稳定性，发现自旋-1部分存在动力学不稳定性，虽可通过调整耦合消除，但其哈密顿量始终无下界，且辅助场或规范固定条件无法解决该问题。
Anomalies in Covariant Fracton Theories
本文利用BRST上同调方法系统计算了协变分形子物质在对称二阶规范张量耦合下的量子反常，给出二维、四维及任意维度的结果，并指出高阶导数标量场理论可作为协变分形子物质的实例。
By gauging a higher-moment polynomial degree global symmetry and a discrete charge conjugation (i.e., particle-hole) symmetry coupled to matter fields (two symmetries mutually non-commutative), we derive a new class of higher-rank tensor non-abelian gauge field theory with dynamically gauged fractonic matter fields: Non-abelian gauged fractons interact with a hybrid class of higher-rank (symmetric or generic non-symmetric) tensor gauge theory and anti-symmetric tensor topological field theory, generalizing [arXiv:1909.13879, 1911.01804]. We also apply a quantum phase transition similar to that…
A relativistic non-perturbative local model of fractons and its non-local perturbative hidden sector
该研究从第一性原理构建了标量分形子物质与对称张量规范场耦合的协变局域模型，解决了直接微扰失效问题，推导出色散关系并发现六种相及分形子束缚态证据。
We use Dirac matrix representations of the Clifford algebra to build fracton models on the lattice and their effective Chern-Simons-like theory. As an example we build lattice fractons in odd $D$ spatial dimensions and their $(D+1)$ effective theory. The model possesses an anti-symmetric $K$ matrix resembling that of hierarchical quantum Hall states. The gauge charges are conserved in sub-dimensional manifolds which ensures the fractonic behavior. The construction extends to any lattice fracton model built from commuting projectors and with tensor products of spin-$1/2$ degrees of freedom at t…
We extend recent work on hydrodynamics with global multipolar symmetries -- known as "fracton hydrodynamics" -- to systems in which the multipolar symmetries are gauged. We refer to the latter as "fracton magnetohydrodynamics", in analogy to conventional magnetohydrodynamics (MHD), which governs systems with gauged charge conservation. We show that fracton MHD arises naturally from higher-rank Maxwell's equations and in systems with one-form symmetries obeying certain constraints; while we focus on "minimal" higher-rank generalizations of MHD that realize diffusion, our methods may also be use…
We introduce two exotic lattice models on a general spatial graph. The first one is a matter theory of a compact Lifshitz scalar field, while the second one is a certain rank-2 $U(1)$ gauge theory of fractons. Both lattice models are defined via the discrete Laplacian operator on a general graph. We unveil an intriguing correspondence between the physical observables of these lattice models and graph theory quantities. For instance, the ground state degeneracy of the matter theory equals the number of spanning trees of the spatial graph, which is a common measure of complexity in graph theory …
Lattice Quantum Villain Hamiltonians: Compact scalars, $U(1)$ gauge theories, fracton models and Quantum Ising model dualities
本文为紧致标量和阿贝尔规范理论构造了Villain哈密顿量，通过提升Villain整数消除缺陷，使格点模型保持与连续统模型一致的对称性和反常结构，并揭示了模型间的对偶性，应用于分形子物理模型。
We construct a purely fermionic system with spontaneously broken supersymmetry that shares the common feature with a fracton phase of matter. Our model is gapless due to the Nambu-Goldstone mechanism. It shows a ground-state degeneracy with the "Area-law" entropy due to fermionic subsystem symmetries. In the strongly coupled limit, it becomes a variant of the Nicolai model, and the ground-state degeneracy shows the "Volume-law" entropy. Gauging the fermionic subsystem symmetry has an t'Hooft anomaly by itself, but the would-be gauged theory may possess a fermionic defect that is immobile in ce…
We present an effective field theory approach to the Fracton phases. The approach is based the notion of a multipole algebra. It is an extension of space(-time) symmetries of a charge-conserving matter that includes global symmetries responsible for the conservation of various components of the multipole moments of the charge density. We explain how to construct field theories invariant under the action of the algebra. These field theories generally break rotational invariance and exhibit anisotropic scaling. We further explain how to partially gauge the multipole algebra. Such gauging makes t…
We extend our exploration of nonstandard continuum quantum field theories in 2+1 dimensions to 3+1 dimensions. These theories exhibit exotic global symmetries, a peculiar spectrum of charged states, unusual gauge symmetries, and surprising dualities. Many of the systems we study have a known lattice construction. In particular, one of them is a known gapless fracton model. The novelty here is in their continuum field theory description. In this paper, we focus on models with a global $U(1)$ symmetry and in a followup paper we will study models with a global $\mathbb{Z}_N$ symmetry.
Zero- and Finite-Temperature Casimir Effect in a Fracton Gauge Field
该研究分析分形规范场在平行边界间的零温和有限温卡西米尔效应，发现快支速度为光速时压强为电磁值的2.21倍，高温极限比值为2.5，揭示多支结构的谱特征。
We consider two-dimensional QED with several fermion flavors on a finite spatial circle. A modified version of the model with {\em flavor-dependent} boundary conditions $ψ_p(L) = e^{2πip/ N} ψ_p(0)$, $p = 1, \ldots , N$ is discussed ($N $ is the number of flavors). In this case a non-contactable contour in the space of the gauge fields is {\em not} determined by large gauge transformations. The Euclidean path integral acquires the contribution from the gauge field configurations with fractional topological charge. The configuration with $ν= 1/N$ is responsible for the formation of the fermion …
We formulate a new class of tensor gauge field theories in any dimension that is a hybrid class between symmetric higher-rank tensor gauge theory (i.e., higher-spin gauge theory) and anti-symmetric tensor topological field theory. Our theory describes a mixed unitary phase interplaying between gapless and gapped topological order phases (which can live with or without Euclidean, Poincaré or anisotropic symmetry, at least in ultraviolet high or intermediate energy field theory, but not yet to a lattice cutoff scale). The "gauge structure" can be compact, continuous, abelian or non-abelian. Our …
Fractons are a new type of quasiparticle which are immobile in isolation, but can often move by forming bound states. Fractons are found in a variety of physical settings, such as spin liquids and elasticity theory, and exhibit unusual phenomenology, such as gravitational physics and localization. The past several years have seen a surge of interest in these exotic particles, which have come to the forefront of modern condensed matter theory. In this review, we provide a broad treatment of fractons, ranging from pedagogical introductory material to discussions of recent advances in the field. …
Gapped Interfaces in Fracton Models and Foliated Fields
本文利用分层规范场论和晶格模型系统研究3+1维分形子相的有能隙界面，在X-cube模型及其与toric code模型之间发现多种新型有能隙边界和界面，包括线性子转换、Kramers-Wannier对偶型界面和电磁对偶界面。
Fractons are particles with restricted mobility. We give a symmetry-based derivation of effective field theories of gapless phases with fractonic topological defects, such as solids and supersolids, using a coset construction. The resulting theory is identified as the Cosserat elasticity theory, which reproduces the conventional symmetric elasticity theory at low energies. The construction can be viewed as a dynamical realization of the inverse Higgs mechanism. We incorporate topological defects such as dislocations and disclinations, which are nontrivially related by the Bianchi identities of…
We reformulate known exotic theories (including theories of fractons) on a Euclidean spacetime lattice. We write them using the Villain approach and then we modify them to a convenient range of parameters. The new lattice models are closer to the continuum limit than the original lattice versions. In particular, they exhibit many of the recently found properties of the continuum theories including emergent global symmetries and surprising dualities. Also, these new models provide a clear and rigorous formulation to the continuum models and their singularities. In appendices, we use this approa…
Electric-Magnetic Duality for Symmetric Tensor Gauge Theories and Immobile $p$-branes
本文研究对称张量规范理论中分形子的电磁对偶，证明其对偶仅在四维时空成立，并利用双形式微积分解释高阶理论的层级结构，同时讨论p-膜的受限不可动性及BPS自对偶条件。
The Use of Torsion in Supergravity Uplifts and Covariant Fractons
本论文研究了超引力理论中扭率在提升和协变裂变理论中的作用，探讨了超引力理论的对偶性及异常计算方法。
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