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.
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.
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.
Fractonic phases are new phases of matter that host excitations with restricted mobility. We show that a certain class of gapless fractonic phases are realized as a result of spontaneous breaking of continuous higher-form symmetries whose conserved charges do not commute with spatial translations. We refer to such symmetries as nonuniform higher-form symmetries. These symmetries fall within the standard definition of higher-form symmetries in quantum field theory, and the corresponding symmetry generators are topological. Worldlines of particles are regarded as the charged objects of 1-form symmetries, and mobility restrictions can be implemented by introducing additional 1-form symmetries whose generators do not commute with spatial translations. These features are realized by effective field theories associated with spontaneously broken nonuniform 1-form symmetries. At low energies, the theories reduce to known higher-rank gauge theories such as scalar/vector charge gauge theories, and the gapless excitations in these theories are interpreted as Nambu-Goldstone modes for higher-form symmetries. Due to the nonuniformity of the symmetry, some of the modes acquire a gap, which is the higher-form analogue of the inverse Higgs mechanism of spacetime symmetries. The gauge theories have emergent nonuniform magnetic symmetries, and some of the magnetic monopoles become fractonic. We identify the ’t Hooft anomalies of the nonuniform higher-form symmetries and the corresponding bulk symmetry-protected topological phases. By this method, the mobility restrictions are fully determined by the choice of the commutation relations of charges with translations. This approach allows us to view existing (gapless) fracton models such as the scalar/vector charge gauge theories and their variants from a unified perspective and enables us to engineer theories with desired mobility restrictions.
FINDING: Fractonic Chern-Simons theory extends topological gauge theory to higher-rank symmetric tensor gauge fields, enabling fracton phases with restricted mobility and subdimensional topological order. | MATH: Higher-rank gauge fields: Aij (symmetric tensor) with gauge transformation δAij=∂i∂jα; fractonic Chern-Simons action: S=4πk∫ϵμνρAμ∂νAρ generalized to higher-rank; fracton charge conservation: ∂i∂jJij=0; dipole moment conservation: ∫xiρ constant. | CONNECTION: Crystallographic group symmetries (e.g., cubic, hexagonal) constrain allowed higher-rank tensor gauge fields; lattice structures (e.g., square, triangular) yield subdimensional particle mobility; no direct golden ratio or base-60 link found. | DEPTH: 8 — Fractonic Chern-Simons theory unifies higher-rank gauge theory with topological order, revealing new quantum phase Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
We present an effective field theory approach to the fracton phases. The approach is based on 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 the symmetries responsible for the conservation of multipole moments local, while keeping rotation and translations symmetries global. It is shown that upon such gauging one finds the symmetric tensor gauge theories, as well as the generalized gauge theories discussed recently in the literature. We refer to all such theories as multipole gauge theories. The outcome of the gauging procedure depends on the choice of the multipole algebra. In particular, we show how to construct an effective theory for the U(1) version of the Haah code based on the principles of symmetry and provide a two-dimensional example with operators supported on a Sierpinski triangle. We show that upon condensation of charged excitations, fracton phases of both types as well as various Symmetry-protected topological phases emerge. Finally, the relation between the present approach and the formalism based on polynomials over finite fields is discussed.
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.
Fractons emerge from many-body systems, featuring subdimensional particles with restricted mobility. These particles have attracted interest for their roles across disciplines, including topological quantum codes, quantum field theory, emergent gravity, and quantum information. They display unique nonequilibrium behaviors such as nonergodicity and glassy dynamics. This review offers a structured overview of fracton phenomena, especially those of gapless fracton liquids, which enable collective modes similar to gauge fluctuations in Maxwell's electromagnetic framework, yet their phenomena are distinguished by a unique conservation law that restricts the mobility of individual charges and monopoles. We delve into the theoretical basis of three-dimensional (3D) fracton liquids, exploring emergent symmetric tensor gauge theories and their properties. We also discuss the material realization of fracton liquids in Yb-based pyrochlore lattices and other synthetic quantum matter platforms.