Topological Defect Networks for Fractons of all Types

Abstract

Fracton phases exhibit striking behavior which appears to render them beyond the standard topological quantum field theory (TQFT) paradigm for classifying gapped quantum matter. Here, we explore fracton phases from the perspective of defect TQFTs and show that topological defect networks-networks of topological defects embedded in stratified 3+1D TQFTs-provide a unified framework for describing various types of gapped fracton phases. In this picture, the sub-dimensional excitations characteristic of fractonic matter are a consequence of mobility restrictions imposed by the defect network. We conjecture that all gapped phases, including fracton phases, admit a topological defect network description and support this claim by explicitly providing such a construction for many well-known fracton models, including the X-Cube and Haah's B code. To highlight the generality of our framework, we also provide a defect network construction of a novel fracton phase hosting non-Abelian fractons. As a byproduct of this construction, we obtain a generalized membrane-net description for fractonic ground states as well as an argument that our conjecture implies no type-II topological fracton phases exist in 2+1D gapped systems. Our work also sheds light on new techniques for constructing higher order gapped boundaries of 3+1D TQFTs. CONTENTS III. Haah's B Code from TopologicalDefects 9 A. Review of Haah's B code 9 B. Condensation on defects 9 C. Nets and relations 12 IV. Non-Abelian fracton model from topological defects in 3+1D D 4 gauge theory 12 A. Review of D 4 gauge theory 13 B. Condensation on defects 13 C. Mobility constraints 14 D. Nets and relations 15 E. Relation to other models 15 V. Classifying phases with topological defect networks 16 A. No type-II fractons in 2+1D 16 B. Phase preserving defect network equivalences 17 VI. Conclusion References A. Field Theory B. Fractal-like Lineon Model C. Z 3 type-I fracton model from coupled 2+1D layers D. X-Cube from defects lattice model E. Lattice model for B code from defects F. Lattice Hamiltonian for non-Abelian defect model 1. Brief review of 3+1D Lattice Gauge Theory 2. D 4 defect Hamiltonian 3. The 1-strata defect in the net-basis

Foliated Quantum Field Theory of Fracton Order

“…Additional terms can be added to the FQFT Lagrangian, which may realize more exotic fracton models [72][73][74][75][76][77][78]. It would also be interesting to reconsider the fractonic Higgs mechanism [79,80] now that we understand gapped fracton orders on curved foliations [46,47,50,81] and U (1) fracton models [35][36][37][38][39][40][41] on curved space [82]. Finally, FQFT could provide further insight on other works, such as the study of boundaries of fracton models [83] or models in higher dimensions [84].…”

“…Refs. [47][48][49][50] have shown that these fracton phases can be thought of as a topological quantum field theory (TQFT) that is embedded with stacks of interfaces (also called defects) upon which certain anyons are condensed. These interfaces are the so-called leaves (i.e.…”

Type-II fractons from coupled spin chains and layers

“…Fracton topological order is an exotic type of quantum order characterized by topological charges with restricted mobility due to superselection rules. To date, a wide range of 3D exactly solved models exhibiting fracton order have been discovered [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20] . The limited mobility of topological charges in these models has lead to topological quantum memories with unique advantages [21][22][23] .…”

“…This poses a significant barrier to any potential experimental realizations of fracton phases. To address this issue, several constructions of fracton models from coupled lower dimensional layers 19,20,[25][26][27][28][29][30][31][32] or spin chains 33,34 have been found, some with much lower degree interactions.…”

Fractonic topological phases from coupled wires

“…This observation has inspired more general constructions of fracton topological order [11][12][13][14][15][16][17] , as well as shedding light on the precise meaning of fracton phases [18][19][20] . More recently, systematic constructions of fracton models from networks of 2D TQFTs (possibly embedded in a 3D TQFT) have been proposed [21][22][23] , encompassing many existing models including type-II ex-amples. Ref.…”