Global dipole symmetry, compact Lifshitz theory, tensor gauge theory, and fractons

Abstract

The 2+1d continuum Lifshitz theory of a free compact scalar field plays a prominent role in a variety of quantum systems in condensed matter physics and high energy physics. It is known that in compact space, it has an infinite ground state degeneracy. In order to understand this theory better, we consider two candidate lattice regularizations of it using the modified Villain formalism. We show that these two lattice theories have significantly different global symmetries (including a dipole global symmetry), anomalies, ground state degeneracies, and dualities. In particular, one of them is self-dual. Given these theories and their global symmetries, we can couple them to corresponding gauge theories. These are two different U (1) tensor gauge theories. The resulting models have excitations with restricted mobility, i.e., fractons. Finally, we give an exact lattice realization of the fracton/lineon-elasticity dualities for the Lifshitz theory, scalar and vector charge gauge theories.

Anomalous hydrodynamics with triangular point group in 2 + 1 dimensions

“…The somewhat unexpected connection between this theory and foliated quantum field theory raises the question of what other (non-fractonic) phenomena could be captured within the latter framework. Alternatively, the length scale could be tied to the size of the system, which would lead to a more subtle manifestation of the UV-IR mixing that arises in theories with exotic symmetry [30][31][32][33][34][35][36][37][38]. Understanding the length scale a is also interesting, because anomaly coefficients (in this case proportional to a −1 ) are RG-invariant, which is in tension with the naive hydrodynamic scaling dimensions of operators in our classical field theory.…”

“…In a complementary thread of research, a series of papers over the past few years [30][31][32][33][34][35][36][37][38] has posed a simple question: what happens when a quantum field theory has an unusual global symmetry? For example, suppose that there is a U(1) symmetry on each plane of a three-dimensional cubic lattice.…”

Foliated-Exotic Duality in Fractonic BF Theories

“…Along the way, we find that there are gauge-invariant operators that cannot be remotely detected by other spatially placed operators, but represents a time-like symmetry [35]. This makes a contrast to the case of ordinary topological order or topological field theory, where every operator is remotely detectable.…”

“…when C 1 surrounds C 0 1 [36][37][38][39]. Without the defect operator F q inside C 1 , the b operator T q becomes trivial, which corresponds to a time-like symmetry [35]. 6 For the later purpose, it will be convenient to consider the case when…”

“…This quadrupole operator is a product of the gauge-invariant operators Ṽ q localized at the corners of the rectangle, which is a time-like symmetry [35]. The operator T p can detect the fracton operator F q :…”