Field Theories for type-II fractons

Abstract

We derive an effective field theory for a type-II fracton starting from the Haah code on the lattice. The effective topological theory is not given exclusively in terms of an action; it must be supplemented with a condition that selects physical states. Without the constraint, the action only describes a type-I fracton. The constraint emerges from a condition that cube operators multiply to the identity, and it cannot be consistently implemented in the continuum theory at the operator level, but only in a weaker form, in terms of matrix elements of physical states. Informed by these studies and starting from the opposite end, i.e., the continuum, we discuss a Chern-Simons-like theory that does not need a constraint or projector, and yet has no mobile excitations. Whether this continuum theory admits a lattice counterpart remains unanswered.

Boundary modes in the Chamon model

“…Equation ( 18) is equivalent to the Chern-Simons-like theory proposed in Refs. [9,20,21] with an additional copy. The two copies with l = 1, 2 are related by the lattice translation r → r + a s (1, 1, 1), which exchanges the sublattices AB ↔ CD.…”

“…They can be obtained through the extension of 2D topological order to three dimensions, either by directly constructing 3D spin models [1][2][3][4][5] or via stacking of 2D topological phases [10][11][12][13][14][15][26][27][28][29][30]. Fractonic phases can also be described by effective continuum theories, usually involving higher-rank gauge fields or exotic Chern-Simons theories [9,[17][18][19][20][21][31][32][33][34][35]. In the spirit of the latter, it is natural to ask whether a fractonic system can exhibit nontrivial boundary phenomena in analogy with 2D topological phases, where different gapped boundaries can be classified by the condensation of quasiparticles and are separated by quantum phase transitions [36][37][38][39][40][41][42].…”

Position-Dependent Excitations and UV/IR Mixing in the $\mathbb{Z}_{N}$ Rank-2 Toric Code and its Low-Energy Effective Field Theory

“…Thus, it would be interesting to studying the symmetry fractionalization patterns. Additionally, because excitations have directions which they can only hop by greater than one lattice site, it would be also be interesting to study the affect of extrinsic lattice defects, which would act an non-abelian excitations [28,[72][73][74][75][76].…”

Snowmass White Paper: Effective Field Theories for Condensed Matter Systems

“…In particular, it remains an outstanding open question to construct a continuum field theory description for one of the first fracton models, the Haah code [109]. (See, for example, [136,145,165] for developments in this direction.) Building a new framework of continuum field theory to incorporate these fracton phases of matter is important for understanding the universal properties of these models and for finding an organizing principle for classifying them.…”