Abstract
We revisit a model for gapped fractonic order in (2+1) dimensions (a symmetric-traceless tensor gauge theory with conservation of dipole and trace-quadrupole moments described in \cite{Prem:2017kxc}) and compute its ground-state entropy on $\mathbb R^2$. Along the way, we quantize the theory on open subsets of $\mathbb R^2$ which gives rise to gapless edge excitations that are Lifshitz-type scalar theories. We additionally explore varieties of gauge-invariant extended operators and rephrase the fractonic physics in terms of the local deformability of these operators. We explore similarities of this model to the effective field theories describing quantum Hall fluids: in particular, quantization of dipole moments through a novel compact symmetry leads us to interpret the vacuum of this theory as a dipole condensate atop of which dipoles with fractionalized moments appear as quasi-particle excitations with Abelian anyonic statistics. This interpretation is reflected in the subleading topological entanglement correction to the entropy. We extend this result to a series of models with conserved multipole moments.
Highlights
There are standard string-like operators, but there are novel strip-like operators whose restricted deformability encodes the underlying fractonic physics
We have shown that this model possess many similarities to conventional topological order: gapped bulk spectrum, gapped edge spectrum, and “anyon" statistics encoded by the string operators wrapping dipolar defects
An appealing physical picture of this topological order as a dipolar condensate emerges when allowing dipole moments to be quantized by an invariance under a novel set of large gauge transformations
Summary
The history of using quantum field theory as a tool for describing low energy phases of matter is one with many successes. The authors noted that the subsequent gapped theory displays both fractonic excitations (associated to the conservation of dipole moment discussed above) as well characteristics familiar to the Abelian Chern-Simons descriptions of quantum Hall fluids, leading the authors to coin it a “dipolar quantum Hall fluid." More recently, the connection between tensor gauge theories and quantum Hall physics has been noticed in [28]. We perform a calculation of the entanglement entropy of the ground-state of this dipolar condensate and show that it takes the form of two separate Abelian topological orders (associated to the two independent dipole orientations) We regard this as the major result of this paper: to our knowledge this is both the first calculation of the entanglement entropy of a tensor gauge theory as well as the first entanglement entropy calculation of a fractonic model using continuum quantum field theory techniques. Answer for the entanglement entropy (appendix A), and details on large gauge transformations (appendix B)
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