Subdimensional particle structure of higher rank $U\left(1\right)$ spin liquids

Spin liquids are conventionally described by gauge theories with a vector gauge field. However, there exists a wider class of spin liquids with higher rank tensors as the gauge variable. In this work, we focus on (3+1)-dimensional spin liquids described by $U\left(1\right)$ symmetric tensor gauge theories, which have recently been shown to be stable gapless spin liquids. We investigate the particle structure of these tensor gauge theories and find that they have deep connections with the “fracton” models recently discovered by Vijay, Haah, and Fu. Tensor gauge theories have more conservation laws than the simple charge conservation law of rank 1 theories. These conservation laws place severe restrictions on the motion of particles. Particles in some models are fully immobile (fractons), while other models have particles restricted to motion along lower-dimensional subspaces.

Figure 1

Increasing an off-diagonal component of $E_{ij}$ by 1 creates excitations at the four corners of a plaquette, in a quadrupolar configuration.

Figure 2

Increasing a diagonal component of $E_{ij}$ by 1 creates three excitations in a line, again with vanishing charge and dipole moment.

Figure 3

Increasing a diagonal component of $E_{ij}$ by 1 creates two oppositely directed vector charges on adjacent collinear links.

Figure 4

Increasing an off-diagonal component of $E_{ij}$ by 1 creates a “loop” of vector charge around a plaquette, in such a way that both linear and angular momentum are conserved.

Figure 5

We can regard the spins of this system as living on the sites of a square lattice, in a scalar representation.

Figure 6

We could regard the same system of spins as living on the links of a tilted square lattice, in a vector representation.