Fractional excitations in foliated fracton phases

Abstract

Fractional excitations in fracton models exhibit novel features not present in conventional topological phases: their mobility is constrained, there are an infinitude of types, and they bear an exotic sense of 'braiding'. Hence, they require a new framework for proper characterization. Based on our definition of foliated fracton phases in which equivalence between models includes the possibility of adding layers of gapped 2D states, we propose to characterize fractional excitations in these phases up to the addition of quasiparticles with 2D mobility. That is, two quasiparticles differing by a set of quasiparticles that move along 2D planes are considered to be equivalent; likewise, 'braiding' statistics are measured in a way that is insensitive to the attachment of 2D quasiparticles. The fractional excitation types and statistics defined in this way provide a universal characterization of the underlying foliated fracton order which can subsequently be used to establish phase relations. We demonstrate as an example the equivalence between the X-cube model and the semionic X-cube model both in terms of fractional excitations and through an exact mapping. Contents

Fracton phases of matter

“…It was shown that by changing the number of sets of 2D layers and the type of string-net or the type of membranes, it is possible to construct models which are equivalent to a variety of known fracton models, including the X-cube model and its Z N generalization, stacks of Toric Code, and the lineon model discussed in Ref. 36. A particularly nice feature of the string-membrane-net construction is that it readily gives rise to a field theory description of the constructed models.…”

Quantum robustness of fracton phases

“…The state |0 X-Cube corresponds to an infinite equal-amplitude superposition of spin product states in the thermodynamic limit resulting from the action of an arbitrary combination of cube operators on the fully polarized state |⇑ . This state can therefore be visualized as a generalized loop soup of flipped spins on rectangular prisms [71].…”

“…The easiest way to manipulate them is by acting with σ x on suitable links [68]. Note that the different orientations κ are important as only b s. This two-particle excitation is called lineon, as it can only move on a one-dimensional line [68,71]. The action of σ x (ν,n) on the ground state creates two lineons as there are exactly fourB (κ) s that do not commute with σ x ν,n .…”

Anisotropic layer construction of anisotropic fracton models

“…The resulting fracton phases have quasiparticles with spatially anisotropic mobility, in contrast to the "isotropic" layer constructions [15,24,56,60] which yield the X-cube model [7] or their relatives with the same mobility of quasiparticles in all three directions, and turn out to be described by exactly solvable models proposed by Shirley, Slagle, and Chen in Ref. [40]. As the models obtained by our construction take simpler forms than the previous models, they may help us to seek experimental realizations of fracton topological orders, e.g., in 2d spin-liquid candidate materials with "bad" two-dimensionality.…”