Systematic Constructions of Fracton Theories

TL;DR

This work introduces the $\,\mathfrak F_2\,$ class of higher-rank gauge theories to realize fracton physics on arbitrary three-dimensional lattices, extending beyond cubic lattices and standard tensor gauge theories. By defining a generalized boundary operator $\partial_2$ and a duality framework, the authors construct a $\,\mathbb Z_2\,$ rank-two theory on a cubic lattice that is exactly dual to the X-cube model, and they formulate pure, Coulomb-regime, and matter-coupled $\,\mathfrak F_2\,$ theories on general lattices, including nonabelian generalizations. The continuum descriptions reveal a perturbed gapless fracton regime and a hollow rank-two tensor structure with constrained excitations and extensive two-form symmetries, while lattice analyses on BCC lattices show how global and subdimensional symmetries can vary with lattice geometry. Collectively, the results broaden the landscape of fracton models, provide constructive tools for building new fracton phases on arbitrary lattices, and open questions about dynamics, phase structure, and mathematical underpinnings of $\partial_2$-based gauge theories. The work thus paves the way for a systematic exploration of fracton physics through generalized higher-form gauge theories across diverse lattice geometries and gauge groups.

Abstract

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 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 $\mathrm{U}(1)$ rank-two theory of type $\mathfrak{F}_2$ 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 $\mathfrak{F}_2$ theory also show gapless fracton behavior; on a cubic lattice they have a conserved $\mathrm{U}(1)$ charge and dipole moment, but these particular charges are not necessarily conserved on more general lattices. The construction straightforwardly generalizes to $\mathfrak{F}_2$ theories of nonabelian rank-two gauge fields, giving first examples of pure nonabelian higher-rank theories.

Figures (8)

• Figure 1: Left, A small part of a cubic lattice, with three Gauss operators \ref{['Xcube G']} on three different sites $v_1$, $v_2$, $v_3$ indicated in red, green, and blue. Each Gauss operator is a product of electric operators $X_\ell$ on links of the same color. Links used to construct each $G_v^i$ emanate from $v$ and are perpendicular to direction $i$. Right, the configuration of electric flux lines in an example gauge-invariant state: flux lines either go straight or split into two lines perpendicular to each other and to the original line. A "glueball" of electric flux lives on the cube in the lower left corner.
• Figure 2: Two examples of two-form electric symmetry operators in a single plane $p$ of the cubic lattice. The operators $U_{l^\vee_1}$ and $U_{l^\vee_2}$ from \ref{['def Ul']} are defined in terms of closed lines $l^\vee_1$ and $l^\vee_2$ on the square lattice dual to the slice $p$: the operators are given by products of all $X_\ell$'s whose links pierce the lines $l^\vee$. The line $l^\vee_1$ wraps around the torus and the associated generator is nontrivial: the line can be smoothly deformed within $p$ but cannot be shrunk to a point. On the other hand, $U_{l_2^\vee}$ is always equal to the identity because it can be represented as a product of Gauss operators $G^z$.
• Figure 3: The shaded area is a two-chain $b^\vee$ on the dual lattice, called a belt. The red links it corresponds to on the original lattice host the operators that make up electric two-form symmetry generators $U_{b^\vee}$.
• Figure 4: Left, the local operators $W_c$ and $X_\ell$ of the X-cube model. Right, their dual operators, as per eq. \ref{['dual global']}. The cube operator $W_c$, shown in blue, maps to a matter momentum operator $X_c^\vee$ that lives on the site dual to cube $c$. The electric field operator $X_\ell$, shown in red, maps to the product of five operators: a gauge field position operator $Z_\ell^\vee$ on the plaquette dual to link $\ell$, and four matter position operators $Z_{c_i}^\vee$ living on sites dual to the cubes $c_i$ that share the link $\ell$.
• Figure 5: The action of $\partial_2$ on chains of all possible ranks in 3d. Two-chains (surfaces) are reduced to their corners, and three-chains (cubes) are reduced to their edges. One- and zero-chains are annihilated. If chains live in a vector space with integer coefficients, like $C_2(\mathbb M, \mathbb Z)$, the resulting one- and zero-chains will typically have coefficients $\pm 1$ depending on the choice of orientation when defining $\partial_2$.