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. Here we 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.

Figure 1

When the dipoles are densely packed, the notion of separate species of dipoles breaks down. As seen in the top row, this configuration of charges could be regarded as closely packed $y$-oriented dipoles. But, as seen in the bottom row, the same configuration of charge could be regarded as closely packed $x$-oriented dipoles. If this charge configuration were set in motion, there would be a fundamental ambiguity in defining a “dipole current.” This ambiguity is connected with the fact that the current tensor of this theory must be a symmetric tensor.

Figure 2

The microscopic model takes the form of a lattice rotor model, where each independent component of the tensor corresponds to a separate rotor. Diagonal components of the tensor live on each site of a cubic lattice (a two-dimensional cross section of which is pictured above). Off-diagonal components live on the appropriate plaquettes of the lattice, with $A_{xy}$ on plaquettes in the $xy$ plane, for example.