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.
The Chern-Simons theory is a powerful tool in the study of 2+1D topological phases, especially quantum Hall systems. Here, the authors extend the framework by allowing an infinite number of gauge fields and use it to study the 3+1D fracton order. Fracton order has been studied mostly using lattice models and higher-rank gauge theories. The introduction of the Chern-Simons formulation allows the user to take advantage of theoretical tools developed over the past decades for quantum Hall systems. In particular, the authors are able to study foliated fracton order more systematically and identify a new type of fracton order, not possible in previous formulations.
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
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.
The present understanding of nonperturbative ground states in the fractional quantum Hall effect is based on effective theories of the Jain ‘composite fermion’ excitations. We review the approach based on matrix variables, i.e. D0 branes, originally introduced by Susskind and Polychronakos. We show that the Maxwell–Chern–Simons matrix gauge theory provides a matrix generalization of the quantum Hall effect, where the composite-fermion construction naturally follows from gauge invariance. The matrix ground states obtained by suitable projections of higher Landau levels are found to be in one-to-one correspondence with the Laughlin and Jain hierarchical states. The matrix theory possesses a physical limit for commuting matrices that could be reachable while staying in the same phase.
We derive an effective field theory for a type-II fracton starting from the Haah code on the lattice. The effective topological theory is not given exclusively in terms of an action; it must be supplemented with a condition that selects physical states. Without the constraint, the action only describes a type-I fracton. The constraint emerges from a condition that cube operators multiply to the identity, and it cannot be consistently implemented in the continuum theory at the operator level, but only in a weaker form, in terms of matrix elements of physical states. Informed by these studies and starting from the opposite end, i.e., the continuum, we discuss a Chern-Simons-like theory that does not need a constraint or projector, and yet has no mobile excitations. Whether this continuum theory admits a lattice counterpart remains unanswered.