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 geodesic-like 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.Comment: 12+4 page

Generalized Electromagnetism of Subdimensional Particles: A Spin Liquid Story

It has recently been shown that there exists a class of stable gapless spin liquids in 3+1 dimensions described by higher rank tensor U(1) gauge fields, giving rise to an emergent tensor electromagnetism. The tensor gauge field of these theories couples naturally to subdimensional particles (such as fractons), which are restricted by gauge invariance to move only along lower-dimensional subspaces of the system. We here work out some of the basic generalized electromagnetic properties of subdimensional particles coupled to tensor electromagnetism, such as generalized electrostatic fields, potential formulations, Lorentz forces, Maxwell equations, and Biot-Savart laws. Some concepts from conventional electromagnetism will carry over directly, while others require significant modification.Comment: 14+4 page

Finite-Temperature Screening of U(1) Fractons

We investigate the finite-temperature screening behavior of three-dimensional U(1) spin liquid phases with fracton excitations. Several features are shared with the conventional U(1) spin liquid. The system can exhibit spin liquid physics over macroscopic length scales at low temperatures, but screening effects eventually lead to a smooth finite-temperature crossover to a trivial phase at sufficiently large distances. However, unlike more conventional U(1) spin liquids, we find that complete low-temperature screening of fractons requires not only very large distances, but also very long timescales. At the longest timescales, a charged disturbance (fracton) will acquire a screening cloud of other fractons, resulting in only short-range correlations in the system. At intermediate timescales, on the other hand, a fracton can only be partially screened by a cloud of mobile excitations, leaving weak power-law correlations in the system. Such residual power-law correlations may be a useful diagnostic in an experimental search for U(1) fracton phases.Comment: 8+2 page

Entanglement Entropy of U(1) Quantum Spin Liquids

We here investigate the entanglement structure of the ground state of a (3+1)-dimensional U(1) quantum spin liquid, which is described by the deconfined phase of a compact U(1) gauge theory. A gapless photon is the only low-energy excitation, with matter existing as deconfined but gapped excitations of the system. It is found that, for a given bipartition of the system, the elements of the entanglement spectrum can be grouped according to the electric flux between the two regions, leading to a useful interpretation of the entanglement spectrum in terms of electric charges living on the boundary. The entanglement spectrum is also given additional structure due to the presence of the gapless photon. Making use of the Bisognano-Wichmann theorem and a local thermal approximation, these two contributions to the entanglement (particle and photon) are recast in terms of boundary and bulk contributions, respectively. Both pieces of the entanglement structure give rise to universal subleading terms (relative to the area law) in the entanglement entropy, which are logarithmic in the system size (log L), as opposed to the subleading constant term in gapped topologically ordered systems. The photon subleading logarithm arises from the low-energy conformal field theory and is essentially local in character. The particle subleading logarithm arises due to the constraint of closed electric loops in the wavefunction and is shown to be the natural generalization of topological entanglement entropy to the U(1) spin liquid. This contribution to the entanglement entropy can be isolated by means of the Grover-Turner-Vishwanath construction (which generalizes the Kitaev-Preskill scheme to three dimensions).Comment: 15+6 page