Emergent gravity of fractons: Mach’s principle revisited

Recent work has established the existence of stable quantum phases of matter described by symmetric tensor gauge fields, which naturally couple to particles of restricted mobility, such as fractons. We focus on a minimal toy model of a rank 2 tensor gauge field, consisting of fractons coupled to an emergent graviton (massless spin-2 excitation). We show how to reconcile the immobility of fractons with the expected gravitational behavior of the model. First, we reformulate the fracton phenomenon in terms of an emergent center of mass quantum number, and we show how an effective attraction arises from the principles of locality and conservation of center of mass. This interaction between fractons is always attractive and can be recast in geometric language, with a geodesiclike formulation, thereby satisfying the expected properties of a gravitational force. This force will generically be short-ranged, but we discuss how the power-law behavior of Newtonian gravity can arise under certain conditions. We then show that, while an isolated fracton is immobile, fractons are endowed with finite inertia by the presence of a large-scale distribution of other fractons, in a concrete manifestation of Mach’s principle. Our formalism provides suggestive hints that matter plays a fundamental role, not only in perturbing, but in creating the background space in which it propagates.

Figure 1

A dipole (particle hopping) operator is disallowed by the dipole conservation law.

Figure 2

One acceptable creation operator is a square quadrupole configuration, amounting to a transverse dipole hop.

Figure 3

Another acceptable operator is the linear quadrupole configuration, amounting to a longitudinal dipole hop.

Figure 4

The left fracton can hop down by emitting a virtual dipole, which then propagates to the right fracton, where its absorption results in an upwards hop.

Figure 5

A fracton hops by emitting a dipole which begins to propagate. The emitted dipole can change its moment by emitting a second dipole. When the dipoles are gapless, such branching processes have non-negligible amplitude.

Figure 6

We here illustrate a planar cross section (in the $xy$ plane) of the cubic lattice on which our model is defined. The off diagonal $A_{xy}$ rotor lives on each of the plaquettes. The remaining off diagonals live in other planar cross sections, not pictured here. All diagonal elements ($A_{xx}$, $A_{yy}$, $A_{zz}$) live on each vertex. (We have spread out the three colored circles on each vertex simply for ease of illustration. All three rotors exist at the same location.)