Emergent gravity of fractons: Mach’s principle revisited

Abstract

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.

Fracton phases of matter

“…Various other aspects of the dynamics were studied in [13], such as an anomalous exponent in the "tail" of the ballistically spreading peak. It was also shown that fractons could attract each other under random unitary evolution, consistent with the gravitational analysis of Reference [14].…”

“…At this level, it can already schematically be seen that fractons exert an effective attraction on each other, since a pair of fractons will slow down as they begin to move apart. This logic can be placed on firmer footing by studying the semi-classical equations of motion of the system, which indicate that a fracton moves along geodesic-like curves of an effective geometry dictated by the positions of all other fractons in the system [14]. In this sense, the interaction between fractons is purely geometric, just as in general relativity.…”

Finite-temperature screening of U (1) fractons

“…As discussed in earlier work 16 , a fracton will acquire finite mobility in the presence of a finite-density distribution of other fractons. In the presence of fracton density n, the typical velocity of a fracton will behave as v ∼ n. Once again, we expect that thermalization will only occur after fractons have moved a distance on the order of the interparticle spacing, n −1/3 .…”

“…We expect the same qualitative physics should hold for any U (1) model with these basic characteristics. The model we will work with has been studied in some detail in previous literature [1][2][3]16 . We will review the most important features of this phase here, though we refer the reader to these previous works for more details.…”

“…Such an immobile excitation is called a fracton, or a 0-dimensional particle. Theoretical evidence for these immobile particles has been seen for more than a decade [6][7][8][9][10][11] , but interest in the topic has particularly surged just in the last few years [2][3][4][5][12][13][14][15][16][17][18][19] , establishing the subject as a new subfield of condensed matter physics.…”