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.
Highlights
Fracton order is a class of gapped phases of matter that exhibits a system-size-dependent ground-state degeneracy on a space of nontrivial topology
The physical properties of these models depend on the geometry of the underlying space; we find it to be more appropriate to refer to fracton systems as geometric order, as suggested in Ref. [26]
We have introduced the multipole algebra—an extension of space(time) symmetries that enforce conservation of certain multipole moments of the charge density
Summary
Fracton order is a class of gapped phases of matter that exhibits a system-size-dependent ground-state degeneracy on a space of nontrivial topology. The first substantial progress in a model-independent description of fractons was made in a series of papers [11,12,13], where it was explained that restricted mobility of type-I models can be incorporated into an effective field theory by enforcing a certain set of Gauss law constraints. We note in passing that the duality in the context of elasticity has been previously studied in a series of papers by Kleinert [22,23], where (Euclidean) symmetric tensor gauge theories (of vector charge type) were introduced; the glide constraint was omitted. The foliated spacetime has been used in Refs. [50,51,52,53] to describe the transport of energy and momentum, while the symmetric tensor d.o.f. have appeared in the context of nematic quantum Hall states and collective magnetoroton modes [54,55,56,57,58,59,60]
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