Three-dimensional (3D) gapped topological phases with fractional excitations are divided into two subclasses: one has topological order with point-like and loop-like excitations fully mobile in the 3D space, and the other has fracton order with point-like excitations constrained in lower-dimensional subspaces. These exotic phases are often studied by exactly solvable Hamiltonians made of commuting projectors, which, however, are not capable of describing those exhibiting surface states with gapless chiral dispersion. Here we introduce a systematic way, based on cellular construction recently proposed for 3D topological phases, to construct another type of exactly solvable models in terms of coupled quantum wires with given inputs of cellular structure, two-dimensional Abelian topological order, and their gapped interfaces. We show that our models can describe both 3D topological and fracton orders (and even their hybrid) and study their universal properties such as quasiparticle statistics and topological ground-state degeneracy. We also apply this construction to two-dimensional coupled-wire models with ordinary topological orders and translation-symmetry-enriched topological orders. Our results pave the way for effective quantum field theory descriptions or microscopic model realizations of fracton orders with chiral gapless surface states.Three-dimensional (3D) gapped topological phases with fractional excitations are divided into two subclasses: one has topological order with point-like and loop-like excitations fully mobile in the 3D space, and the other has fracton order with point-like excitations constrained in lower-dimensional subspaces. These exotic phases are often studied by exactly solvable Hamiltonians made of commuting projectors, which, however, are not capable of describing those exhibiting surface states with gapless chiral dispersion. Here we introduce a systematic way, based on cellular construction recently proposed for 3D topological phases, to construct another type of exactly solvable models in terms of coupled quantum wires with given inputs of cellular structure, two-dimensional Abelian topological order, and their gapped interfaces. We show that our models can describe both 3D topological and fracton orders (and even their hybrid) and study their universal properties such as quasiparticle statistics and topological ground-state degeneracy. We also apply this construction to two-dimensional coupled-wire models with ordinary topological orders and translation-symmetry-enriched topological orders. Our results pave the way for effective quantum field theory descriptions or microscopic model realizations of fracton orders with chiral gapless surface states.

arXiv:2112.07926Physical Review ResearchNov 3, 20233 citations

Topological Orders with Spins and Fermions

Laura Ortiz Martín

Springer thesesJan 1, 20190 citations

Topological spin liquids

Antônio Sérgio Teixeira Pires

Institute of Physics eBooksApr 1, 20210 citations

Fragmented Topological Excitations in Generalized Hypergraph Product Codes

Meng-Yuan Li, Yue Wu

Product code construction is a powerful tool for constructing quantum stabilizer codes, which serve as a promising paradigm for realizing fault-tolerant quantum computation. Furthermore, the natural mapping between stabilizer codes and the ground states of exactly solvable spin models also motivates the exploration of many-body orders in the stabilizer codes. In this work, we investigate the fracton topological orders in a family of codes obtained by a recently proposed general construction. More specifically, this code family can be regarded as a class of generalized hypergraph product (HGP) codes. We term the corresponding exactly solvable spin models \textit{orthoplex models}, based on the geometry of the stabilizers. In the 3D orthoplex model, we identify a series of intriguing properties within this model family, including non-monotonic ground state degeneracy (GSD) as a function of system size and non-Abelian lattice defects. Most remarkably, in 4D we discover \textit{fragmented topological excitations}: while such excitations manifest as discrete, isolated points in real space, their projections onto lower-dimensional subsystems form connected objects such as loops, revealing the intrinsic topological nature of these excitations. Therefore, fragmented excitations constitute an intriguing intermediate class between point-like and spatially extended topological excitations. In addition, these rich features establish the generalized HGP codes as a versatile and analytically tractable platform for studying the physics of fracton orders.Product code construction is a powerful tool for constructing quantum stabilizer codes, which serve as a promising paradigm for realizing fault-tolerant quantum computation. Furthermore, the natural mapping between stabilizer codes and the ground states of exactly solvable spin models also motivates the exploration of many-body orders in the stabilizer codes. In this work, we investigate the fracton topological orders in a family of codes obtained by a recently proposed general construction. More specifically, this code family can be regarded as a class of generalized hypergraph product (HGP) codes. We term the corresponding exactly solvable spin models \textit{orthoplex models}, based on the geometry of the stabilizers. In the 3D orthoplex model, we identify a series of intriguing properties within this model family, including non-monotonic ground state degeneracy (GSD) as a function of system size and non-Abelian lattice defects. Most remarkably, in 4D we discover \textit{fragmented topological excitations}: while such excitations manifest as discrete, isolated points in real space, their projections onto lower-dimensional subsystems form connected objects such as loops, revealing the intrinsic topological nature of these excitations. Therefore, fragmented excitations constitute an intriguing intermediate class between point-like and spatially extended topological excitations. In addition, these rich features establish the generalized HGP codes as a versatile and analytically tractable platform for studying the physics of fracton orders.

quant-pharXiv:2601.09850Physical review. B./Physical review. BJan 16, 20260 citations

Lattice model for fermionic toric code

Zheng‐Cheng Gu, Zhenghan Wang, Xiao-Gang Wen

The ${Z}_{2}$ topological order in ${Z}_{2}$ spin liquid and in the exactly solvable Kitaev toric code model is the simplest topological order for 2 + 1D bosonic systems. More general 2 + 1D bosonic topologically ordered states can be constructed via exactly solvable string-net models. However, the most important topologically ordered phases of matter are arguably the fermionic fractional quantum Hall states. Topological phases of matter for fermion systems are strictly richer than their bosonic counterparts because locality has different meanings for the two kinds of systems. In this paper, we describe a simple fermionic version of the toric code model to illustrate many salient features of fermionic exactly solvable models and fermionic topologically ordered states.The ${Z}_{2}$ topological order in ${Z}_{2}$ spin liquid and in the exactly solvable Kitaev toric code model is the simplest topological order for 2 + 1D bosonic systems. More general 2 + 1D bosonic topologically ordered states can be constructed via exactly solvable string-net models. However, the most important topologically ordered phases of matter are arguably the fermionic fractional quantum Hall states. Topological phases of matter for fermion systems are strictly richer than their bosonic counterparts because locality has different meanings for the two kinds of systems. In this paper, we describe a simple fermionic version of the toric code model to illustrate many salient features of fermionic exactly solvable models and fermionic topologically ordered states.

arXiv:1309.7032Physical Review BAug 26, 201456 citations

The

{Z}_{2}

topological order in

{Z}_{2}

spin liquid and in the exactly solvable Kitaev toric code model is the simplest topological order for 2 + 1D bosonic systems. More general 2 + 1D bosonic topologically ordered states can be constructed via exactly solvable string-net models. However, the most important topologically ordered phases of matter are arguably the fermionic fractional quantum Hall states. Topological phases of matter for fermion systems are strictly richer than their bosonic counterparts because locality has different meanings for the two kinds of systems. In this paper, we describe a simple fermionic version of the toric code model to illustrate many salient features of fermionic exactly solvable models and fermionic topologically ordered states.