Systematic construction of gapped nonliquid states

Abstract

Using Abelian and non-Abelian topological orders in two-dimensional (2D) space and the different ways to glue them together via their gapped boundaries, we propose a systematic way to construct three-dimensional (3D) gapped states (and in other dimensions). The resulting states are called cellular topological states, which include gapped nonliquid states, as well as gapped liquid states in some special cases. Some new fracton states with fractal excitations are constructed even using 2D Z 2 topological order. More general cellular topological states can be constructed by connecting 2D domain walls between different 3D topological orders. The constructed cellular topological states can be viewed as fixed-point states for a reverse renormalization of gapped nonliquid states.

Gapped boundaries and string-like excitations in (3+1)d gauge models of topological phases

“…It would certainly be interesting to consider how this generalizes in higher dimensions. Once this more general scenario is well-understood, we could then apply our results to so-called fracton models, which were recently suggested in [74][75][76] to have an interpretation in terms of defect TQFTs. A related question would be to study invertible domain walls such as duality defects and derive the underlying mathematical structure in category theoretical terms.…”

A degeneracy bound for homogeneous topological order

“…At least, all the cubic codes [13], the X-cube model, and the checkerboard model [15] have homogeneous topological order; see Section 3.1 below. Rigorous verification is anticipated for other models [6,7] in flat space or general manifolds [4,16].…”

“…Beyond this aspect, there is not much that is purely topological in fracton phases: there exist analogs of Wilson loop operators but they only give a many-to-one map into homology groups; continuum field theories have been studied [3][4][5] but complete data of operators on the ground state subspace still depend on geometric details. Recently [6][7][8], it is proposed that fracton phases are obtained by stitching together blocks of conventional topological order (anomalous or not), providing a machinery to write a vast number of examples. This construction requires so many algebraic quantities and parameters, including length scales of constituent blocks, that we are motivated to pause and ask what it means for a many-body state to represent a quantum phase of homogeneous matter.…”

Lattice Clifford fractons and their Chern-Simons-like theory

“…, whereas the coefficient k must be an odd integer, in accordance with (43). In order to compare with [20], we just need to rename the fields and the derivative operators according to A 1 → −A 2 , A 2 → A 1 , D 1 → −D 2 , and D 2 → D 1 (see equation ( 95) of [20]), which leave both the action and the commutation relation in (48) unchanged. In this form, we can immediately compare with the results of [20] with the following identification between the parameters k = s/2.…”

“…Nevertheless, gapped fracton models can be obtained from higher-rank gauge theories via the Higgs mechanism [40,41]. Gapped 3D fractons can also be obtained by either stacking [34,[42][43][44][45][46][47] or glueing [48][49][50] known (2 + 1)-dimensional topological orders.…”