Towards classification of Fracton phases: the multipole algebra

TL;DR

The paper introduces the multipole algebra as an extended space-time symmetry framework to encode fracton phenomenology, deriving invariant field theories and a partial gauging that recovers symmetric tensor gauge theories and generalized gauge theories. By incorporating polynomial shift symmetries, it explains how multipole moments are conserved and how gauge structures emerge, and shows concrete realizations in 2D and 3D, including the U(1) Haah code and fractal operators via charge condensation. It also relates the continuum EFT approach to polynomials over finite fields, connects to crystalline extensions, and demonstrates how condensation and finite-field formulations yield fracton SPTs and Haah-like codes. The framework provides a systematic method to classify and construct fracton phases, elastic/gauge duals, and potential topological orders, while highlighting open questions about stability, boundary effects, and full gauging procedures. Overall, the work offers a unifying perspective that ties symmetry, gauge structure, and fracton mobility to a versatile algebraic foundation with broad physical implications.

Abstract

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 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. 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 SPTs emerge. Finally, the relation between the present approach and the formalism based on polynomials over finite fields is discussed.

Figures (8)

• Figure 1: (a) The charge configuration corresponding to $q^{ij}\propto\sigma^1$. (b) The charge configuration corresponding to $q^{ij}\propto\sigma^3$. (c) A more convenient basis of charge configurations is obtained by applying (a) and inverse of (a) at plaquettes labeled by star and $-1$ star correspondingly.
• Figure 2: (a)-(c) The elementary charge configurations, corresponding to $D_\alpha$, for the degenerate theory, characterized by \ref{['eq:MAdipole2D1']}-\ref{['eq:MAdipole2D2']}, with invariant derivatives given by \ref{['eq:invDdipole']}.(d) Application of the charge configuration corresponding to $D_2$ results in hopping of the $(n,m)$ dipole in $x$-direction. (e) Application of the charge configuration corresponding to $D_3$ results in hopping of the $(n,m)$ dipole in $y$-direction.
• Figure 3: (a) The elementary charge configurations, corresponding to $D_\alpha$, for the effective theory for the $U(1)$ Haah code \ref{['eq:HaahDerivatives']}. charge configurations. (b) A different basis of elementary charge configurations. The first two configurations are precisely the ones studied in [bulmash2018generalized], while the last charge configuration is allowed by symmetries and is linearly independent from others.
• Figure 4: Elementary charge configurations in the $\mathbb Z_2$ version of the theory. Since the even charges have disappeared into the vacuum the first configuration turned into a hopping operator, in the $(1,1)$ direction.
• Figure 5: Hopping operator, in the $(1,1)$ direction. This operator corresponds to the $D_1$ invariant derivative after the charge-2 condensation. Thus a single charge is a dimension-$1$ particle moving in the $(1,1)$ direction.