Abstract
Fracton order is an intriguing new type of order which shares many common features with topological order, such as topology-dependent ground state degeneracies, and excitations with mutual statistics. However, it also has several distinctive geometrical aspects, such as excitations with restricted mobility, which naturally lead to effective descriptions in terms of higher rank gauge fields. In this paper, we investigate possible effective field theories for 3D fracton order, by presenting a general philosophy whereby topological-like actions for such higher-rank gauge fields can be constructed. Our approach draws inspiration from Chern-Simons and BF theories in 2+1 dimensions, and imposes constraints binding higher-rank gauge charge to higher-rank gauge flux. We show that the resulting fractonic Chern-Simons and BF theories reproduce many of the interesting features of their familiar 2D cousins. We analyze one example of the resulting fractonic Chern-Simons theory in detail, and show that upon quantization it realizes a gapped fracton order with quasiparticle excitations that are mobile only along a sub-set of 1-dimensional lines, and display a form of fractional self-statistics. The ground state degeneracy of this theory is both topology- and geometry- dependent, scaling exponentially with the linear system size when the model is placed on a 3-dimensional torus. By studying the resulting quantum theory on the lattice, we show that it describes a $\mathbb{Z}_s$ generalization of the Chamon code.
Highlights
Topological quantum field theories (TQFTs) have been a powerful tool in developing our understanding of the possible strongly interacting, gapped phases of matter
Notice that our theory does not run into the subtle issues associated with discretizing and quantizing the regular 2D Chern-Simons theory. These subtle issues arise when, for example, canonically conjugate variables do not live on the same location, or when there are multiple natural choices to be made for the charge-vortex binding
Note that unlike the theories discussed in previous sections of this paper, coupling this theory to matter leads to dipolar excitations that are mobile in twodimensional planes
Summary
Topological quantum field theories (TQFTs) have been a powerful tool in developing our understanding of the possible strongly interacting, gapped phases of matter. The presence of such anomalous surfaces is surprising in light of the correspondence between our field theories and exactly solvable lattice models, which is not expected for systems with topologically protected gapless boundary modes This is one of several hints that the regularization may play a more fundamental role in quantizing our higher-rank ChernSimons theories than it does for TQFTs or critical theories. FRACTONIC CHERN-SIMONS AND BF THEORIES (in derivatives) gauge transformations lead to matter fields that are restricted to move on lines, and gauge-invariant “cage-net” operators similar to those previously discussed in the context of lattice fracton models [34,36,57]. We argue that though the resulting theory is gapless, it is interesting as the Chern-Simons term appears to overcome the theory’s expected confinement [56] in a manner very similar to the case of compact U(1) Maxwell-ChernSimons theory in 2 + 1 dimensions [58]
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