Fracton fusion and statistics

Abstract

We introduce and develop a theory of fusion and statistical processes of gapped excitations in Abelian fracton phases. The key idea is to incorporate lattice translation symmetry via its action on superselection sectors, which results in a fusion theory endowed with information about the non-trivial mobility of fractons and sub-dimensional excitations. This results in a description of statistical processes in terms of local moves determined by the fusion theory. Our results can be understood as providing a characterization of translation-invariant fracton phases. We obtain simple descriptions of the fusion theory in the X-cube and checkerboard fracton models, as well as for gapped electric and magnetic excitations of some gapless U(1) tensor gauge theories. An alternate route to the X-cube model fusion theory is provided by starting with a system of decoupled twodimensional toric code layers, and giving a description of the p-string condensation mechanism within our approach. We discuss examples of statistical processes of fractons and sub-dimensional excitations in the X-cube and checkerboard models. As an application of the ideas developed, we prove that the X-cube and semionic X-cube models realize distinct translation-invariant fracton phases, even when the translation symmetry is broken corresponding to an arbitrary but finite enlargement of the crystalline unit cell. CONTENTS

Fracton phases of matter

“…For example, we can stipulate that t a (s 1 + s 2 ) = t a s 1 + t a s 2 , reflecting the fact that it does not matter whether we fuse two particles then translate them, or translate them first and then fuse them. Furthermore, there is a natural action of It can readily be checked that this formalism correctly captures the immobility of fractons and two-dimensional nature of dipoles in the X-cube model [40]. Furthermore, this logic can even be extended to construct fusion theories of gapless fracton models, such as the U (1) gauge theories.…”

“…In Reference [40], Pai and Hermele constructed a fusion theory capable of describing the quasiparticle content of fracton phases, along with various examples of nontrivial statistical processes. The key idea in this fusion theory is to consider the action of translation on the superselection sectors of the theory, which encodes the mobility of quasiparticles.…”

Gauging permutation symmetries as a route to non-Abelian fractons

“…A new chapter in the book of three-dimensional (3D) quantum phases of matter was opened with the discovery of fracton models [1][2][3][4][5][6][7][8], characterized by the presence of topological excitations with restricted mobility. These peculiar particles have attracted significant recent interest, thereby revealing intriguing connections to quantum information processing [9][10][11][12], topological order [13][14][15][16][17][18][19][20][21][22], sub-system symmetries [23][24][25][26][27][28][29][30], and slow quantum dynamics [1,3,[31][32][33]. Much of the phenomenology of fractons can also be realized in tensor gauge theories [34][35][36][37][38][39][40][41][42][43][44] with higher moment conservation laws, unveiling further connections of fractons with elasticity …”

“…[63], for instance, coupled layers of 2D Abelian topological orders to a 3D Abelian topological phase to reproduce the X-Cube model [8] and generalizations thereof-we anticipate that suitable non-Abelian generalizations of the "string-membrane-net" can produce fracton orders distinct from those found in the cage-net or twisted fracton models. Moreover, all of the aforementioned examples fall into the category of foliated type-I fracton phases, where the presence of excitations mobile along lines or planes allows generalized notions of braiding and statistics to be defined [19,21]. Unlike these examples, where notions of "non-Abelianness" 2 have been extended to subdimensional excitations [62,76], whether such notions extend to type-II models, lacking any mobile excitations, remains unclear.…”

Electric circuit realizations of fracton physics

“…This mobility restriction is naturally encoded in the higher moment conservation laws of such systems, such as conservation of dipole moment [23,25,26]. Fractons are notable both for their potential applications to quantum information storage [20,[27][28][29], as well as their prevalence across numerous domains of physics, including spin liquids [30][31][32][33][34][35][36][37][38][39][40][41][42][43][44], elasticity [45][46][47][48][49][50][51], localization [19,52,53], hole-doped antiferromagnets [54], gravity [55][56][57], Majorana systems [21,58,59], and deconfined quantum criticality [60].…”