The objective of the present work -- a literature review on both gapped and gapless fractonic theories -- is to pedagogically fill in the gaps between the research on fractons, and an undergraduate physics education (particularly quantum and statistical mechanics). Some familiarity with classical field theory is assumed. We will begin this work by reviewing the gauging of the Ising model to obtain the toric code. Then, following the chronological order of fracton research, we will establish the gauged spin model picture of gapped fracton theories. Next (after introducing lattice gauge theory) we will cover the developments on the tensor gauge theory front in describing gapless fracton theories. We then explain how conservation of dipole moment is key in the development of field-theoretic descriptions of fracton models, and conclude by providing future plans for work on gauging such field theories. Finally, we point out two more future plans: one being an established theoretical advance which in fact inspired this work, and the other being exciting experimental advances in ultracold atomic physics which could lead to the physical realization of fracton physics. Highlights of this work include: a derivation of the gauge field dynamics (field strength/curvature) term in the gauging of a spin system to obtain the toric code, a reduction of the 3 terms of the Kogut-Susskind lattice QED Hamiltonian to the 2 terms of the toric code, a proof of the correspondence between the invariance of a gauge field and a particular Gauss's law constraint in a tensor gauge theory context, a demonstration that conservation of dipole moment accounts for the same immobility/fractalization phenomenology that we also show from a gauged spin model perspective, and proof of the necessity for polynomial shift symmetries and higher-order spatial derivatives in a Lagrangian in order to have dipole moment conservation.
The objective of the present work -- a literature review on both gapped and gapless fractonic theories -- is to pedagogically fill in the gaps between the research on fractons, and an undergraduate physics education (particularly quantum and statistical mechanics). Some familiarity with classical field theory is assumed. We will begin this work by reviewing the gauging of the Ising model to obtain the toric code. Then, following the chronological order of fracton research, we will establish the gauged spin model picture of gapped fracton theories. Next (after introducing lattice gauge theory) we will cover the developments on the tensor gauge theory front in describing gapless fracton theories. We then explain how conservation of dipole moment is key in the development of field-theoretic descriptions of fracton models, and conclude by providing future plans for work on gauging such field theories. Finally, we point out two more future plans: one being an established theoretical advance which in fact inspired this work, and the other being exciting experimental advances in ultracold atomic physics which could lead to the physical realization of fracton physics. Highlights of this work include: a derivation of the gauge field dynamics (field strength/curvature) term in the gauging of a spin system to obtain the toric code, a reduction of the 3 terms of the Kogut-Susskind lattice QED Hamiltonian to the 2 terms of the toric code, a proof of the correspondence between the invariance of a gauge field and a particular Gauss's law constraint in a tensor gauge theory context, a demonstration that conservation of dipole moment accounts for the same immobility/fractalization phenomenology that we also show from a gauged spin model perspective, and proof of the necessity for polynomial shift symmetries and higher-order spatial derivatives in a Lagrangian in order to have dipole moment conservation.
Strongly correlated many-body systems provide a platform for novel phases of matter where constituent particles organize themselves in a variety of ways. At low temperature, these particles interact quantum mechanically and generate entanglement building up exotic quantum phases, such as topologcial order, where there can be emergent excitations which cannot be created locally. Such excitations, if gapped, are also called topological excitations.Fracton is one of such gapped point-like topological excitation in three dimensional system. Different from conventional topological excitation, it is immobile and was firstly discovered in exact solvable models exhibiting fracton topological order. This new order has sub-extensive topological ground-state degeneracy and generically also possesses other mobile excitations restricted to move in sub-dimensional spaces. Also, it has been noticed that the charges of symmetric-tensor U(1) gauge theories can be fractons. Its immobility is due to the existence of multiple conservation laws.In this thesis, I will present the relation among the gapped fracton topological order, gapless fracton phase described by U(1) symmetric tensor gauge theories, and ordinary topological ordered phases. Particularly, the fracton topological order exhibited in an exact solvable model called X- cube model can be constructed by coupling toric code layers. The mechanism leads to fracton topological orders is dubbed “p-string condensation” or “p-membrane condensation,” in which strings or membranes built up from particle excitations from layers of topological orders condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the resulting fracton order to be easily understood in terms of more familiar degrees of freedom. And the fracton topological order in the X-cube model can also be obtained from a particular rank-2 symmetric tensor gauge theory called scalar charge theory by a partial confinement transition followed by Higgs mechanism removing the gapless photon modes.
Fracton emerges from strongly interacting many-body systems whose excitations, referred to as sub-dimensional particles, have restricted mobility or kinetic motions. These entities have garnered significant interest due to their interdisciplinary implications spanning topological quantum codes, quantum field theory, emergent gravity, quantum information, and more, revealing unique nonequilibrium behaviors such as nonergodicity and glassy dynamics. This review presents a structured and educational overview of fracton phenomena, specifically focusing on gapless fracton liquids. Noteworthy for their compressibility and gapless excitations, fracton liquids facilitate collective modes reminiscent of those gauge fluctuations found in Maxwell's electromagnetic framework. However, they are distinct due to an additional higher-moment conservation law that restricts the mobility of individual charges and monopoles. Our exploration begins with the theoretical foundation of 3D fracton liquids, presenting a variety of emergent symmetric tensor gauge theories and reviewing their equilibrium and dynamic properties. Following this theoretical groundwork, we discuss the material realization of various fracton liquids in Yb-based breathing pyrochlore lattices, close-packed tiling structures and other synthetic quantum matter platforms. Additionally, we introduce a general protocol to foster and manipulate emergent fracton spin liquids in the realm of frustrated magnetism.
Fractons are gapped pointlike excitations in d=3 topological ordered phases whose motion is constrained. They have been discovered in several gapped models but a unifying physical mechanism for generating them is still missing. It has been noticed that in symmetric-tensor U(1) gauge theories, charges are fractons and cannot move freely due to, for example, the conservation of not only the charge but also the dipole moment. To connect these theories with fully gapped fracton models, we study Higgs and partial confinement mechanisms in rank-2 symmetric-tensor gauge theories, where charges or magnetic excitations, respectively, are condensed. Specifically, we describe two different routes from the rank-2 U(1) scalar charge theory to the X-cube fracton topological order, finding that a combination of Higgs and partial confinement mechanisms is necessary to obtain the fully gapped fracton model. On the other hand, the rank-2 Z2 scalar charge theory, which is obtained from the former theory upon condensing charge-2 matter, is equivalent to four copies of the d=3 toric code and does not support fracton excitations. We also explain how the checkerboard fracton model can be viewed as a rank-2 Z2 gauge theory with two different Gauss' law constraints on different lattice sites.
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 Fp gauge theories, in which gauge transformations of rank-k fields are specified by rank-(k−p) gauge parameters. The Z2 rank-two theory of type F2, placed on a cubic lattice and coupled to scalar matter, is shown to have a topological phase exactly dual to the well-known X-cube model. Generalizations of this example yield novel fracton theories. In the continuum, the U(1) rank-two theory of type F2 is shown to have a perturbatively gapless fracton regime that cannot be consistently interpreted as a tensor gauge theory of any kind. The compact scalar fields that naturally couple to this F2 theory also show gapless fracton behavior; on a cubic lattice they have a conserved U(1) charge and dipole moment, but these particular charges are not necessarily conserved on more general lattices. The construction straightforwardly generalizes to F2 theories of nonabelian rank-two gauge fields, giving first examples of pure nonabelian higher-rank theories.
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. We begin by demonstrating how the fracton phenomenon naturally arises as a consequence of higher moment conservation laws, often accompanied by the emergence of tensor gauge theories. We then provide a survey of fracton phases in spin models, along with the various tools used to characterize them, such as the foliation framework. We discuss in detail the manifestation of fracton physics in elasticity theory, as well as the connections of fractons with localization and gravitation. Finally, we provide an overview of some recently proposed platforms for fracton physics, such as Majorana islands and hole-doped antiferromagnets. We conclude with some open questions and an outlook on the field.
Abstract Recent theoretical research on tensor gauge theories led to the discovery of an exotic type of quasiparticles, dubbed fractons, that obey both charge and dipole conservation. Here we describe physical implementation of dipole conservation laws in realistic systems. We show that fractons find a natural realization in hole-doped antiferromagnets. There, individual holes are largely immobile, while dipolar hole pairs move with ease. First, we demonstrate a broad parametric regime of fracton behavior in hole-doped two-dimensional Ising antiferromagnets viable through five orders in perturbation theory. We then specialize to the case of holes confined to one dimension in an otherwise two-dimensional antiferromagnetic background, which can be realized via the application of external fields in experiments, and prove ideal fracton behavior. We explicitly map the model onto a fracton Hamiltonian featuring conservation of dipole moment. Manifestations of fractonicity in these systems include gravitational clustering of holes. We also discuss diagnostics of fracton behavior, which we argue is borne out in existing experimental results.