Fracton topological order via coupled layers

Abstract

In this work, we develop a coupled layer construction of fracton topological orders in d = 3 spatial dimensions. These topological phases have subextensive topological ground-state degeneracy and possess excitations whose movement is restricted in interesting ways. Our coupled layer approach is used to construct several different fracton topological phases, both from stacked layers of simple d = 2 topological phases and from stacks of d = 3 fracton topological phases. This perspective allows us to shed light on the physics of the X-cube model recently introduced by Vijay, Haah, and Fu, which we demonstrate can be obtained as the strong-coupling limit of a coupled three-dimensional stack of toric codes. We also construct two new models of fracton topological order: a semionic generalization of the X-cube model, and a model obtained by coupling together four interpenetrating X-cube models, which we dub the 'four color cube model". The couplings considered lead to fracton topological orders via mechanisms we dub "p-string condensation" and "p-membrane condensation", in which strings or membranes built from particle excitations are driven to condense. This allows the fusion properties, braiding statistics, and ground-state degeneracy of the phases we construct to be easily studied in terms of more familiar degrees of freedom. Our work raises the possibility of studying fracton topological phases from within the framework of topological quantum field theory, which may be useful for obtaining a more complete understanding of such phases.

Emergent gravity of fractons: Mach’s principle revisited

“…As a limiting case, some subdimensional particles are restricted to a zero-dimensional subspace and are totally immobile. These unconventional particles were given the name "fractons" [19,21], a topic which has been particularly active recently [22][23][24][25][26]. Fractons find a natural setting in the language of higher spin gauge fields, with the "generalized lattice gauge theory" introduced in Ref.…”

Finite-temperature screening of U (1) fractons

“…Such an immobile excitation is called a fracton, or a 0-dimensional particle. Theoretical evidence for these immobile particles has been seen for more than a decade [6][7][8][9][10][11] , but interest in the topic has particularly surged just in the last few years [2][3][4][5][12][13][14][15][16][17][18][19] , establishing the subject as a new subfield of condensed matter physics.…”

Fracton Topological Phases from Strongly Coupled Spin Chains

“…(14) is analogous to the coupled-layer constructions introduced in Refs. [17] and [18] of the main text. Indeed, the fracton topological order is obtained in our second step by strongly coupling orthogonal stacks of two-dimensional topologically ordered layers.…”

“…While these approaches can be used to understand the generic properties of fracton phases, the concrete spin models they provide are far from realistic as they involve interactions between many spins at the same time. From a more practical perspective, fracton phases can be constructed by coupling orthogonal stacks of two-dimensional topologically ordered layers [17,18]. This approach can lead to more realistic spin models involving only two-spin interactions [19], although it is not immediately clear what kind of fracton phase is obtained from a generic construction.…”

“…In a conceptual sense, the coupled-layer models of fracton phases introduced in Refs. [17] and [18] are thus an intermediate step between our coupled-spin-chain models and the commuting-projector models in Ref. [14].…”