Topological Orders with Spins and Fermions: Quantum Phases and Computation

Topological orders and associated topological protected excitations satisfying Abelian or non-Abelian statistics have been widely explored in various platforms. In recent years, the Majorana zero modes with these features have been realized in experiments based on semiconductor and superconducting hybrid structures, opening an exciting avenue for realization of fault-tolerant topological quantum computations. The $\mathbb{Z}_3$ parafermions are regarded as the most natural generation of the Majorana fermions for realizing of more intriguing topological orders. In this work, we investigate the topological phase and emergent fermions in an extended parafermion chain using exact numerical methods. This model exhibits rich variety of phases, including threefold degenerate topological ferromagnetic phase, trivial paramagnetic phase, XX phase, dimer phase, chiral phase and commensurate phase. We present a smooth connection between this model and the extended spin- ${1\over 2}$ XX model, by which we can generalize their measurements, such as the order parameters and long-range spin-spin correlations {\it etc}, to fully characterize the properties of these phases and map out their corresponding phase diagram. Strikingly, we find that all the phase boundaries finally merge to a single supercritical point, determined by the extended three-state Potts model. This approach opens a wide range of intriguing applications in investigating the phases and emergent phenomena in other $\mathbb{Z}_k$ parafermion models, and may be also illuminating for their experimental realizations.Topological orders and associated topological protected excitations satisfying Abelian or non-Abelian statistics have been widely explored in various platforms. In recent years, the Majorana zero modes with these features have been realized in experiments based on semiconductor and superconducting hybrid structures, opening an exciting avenue for realization of fault-tolerant topological quantum computations. The $\mathbb{Z}_3$ parafermions are regarded as the most natural generation of the Majorana fermions for realizing of more intriguing topological orders. In this work, we investigate the topological phase and emergent fermions in an extended parafermion chain using exact numerical methods. This model exhibits rich variety of phases, including threefold degenerate topological ferromagnetic phase, trivial paramagnetic phase, XX phase, dimer phase, chiral phase and commensurate phase. We present a smooth connection between this model and the extended spin-${1\over 2}$ XX model, by which we can generalize their measurements, such as the order parameters and long-range spin-spin correlations {\it etc}, to fully characterize the properties of these phases and map out their corresponding phase diagram. Strikingly, we find that all the phase boundaries finally merge to a single supercritical point, determined by the extended three-state Potts model. This approach opens a wide range of intriguing applications in investigating the phases and emergent phenomena in other $\mathbb{Z}_k$ parafermion models, and may be also illuminating for their experimental realizations.

Jan 10, 20180 citations

Sixteen-Fold Way for Fermionic Topological Orders

Ryohei Kobayashi, Abhinav Prem, Matthew Yu

Fermionic topological orders can host 't Hooft anomalies with no bosonic counterpart. We identify a new sixteen-fold family of (2+1)D fermionic topological orders, forming a fermionic analogue of Kitaev's sixteen-fold way. This family is distinguished by the mod 16 't Hooft anomaly of a $\mathbb{Z}_2$ one-form symmetry, generated in each theory by a single nontrivial $\mathbb{Z}_2$ anyon. This intrinsically fermionic anomaly permits anyon spins that are forbidden in bosonic phases; the simplest new example is an Abelian fermionic topological order containing a single $\mathbb{Z}_2$ Abelian anyon of spin 1/8. Each theory can be realized as the gapped boundary of a (3+1)D fermionic symmetry-protected topological (SPT) phase protected by the $\mathbb{Z}_2$ one-form symmetry, which acquires a $\mathbb{Z}_{16}$ classification once the spacetime spin structure is twisted by the one-form symmetry. We realize these phases microscopically via lattice models built from Walker-Wang models coupled to local fermions.Fermionic topological orders can host 't Hooft anomalies with no bosonic counterpart. We identify a new sixteen-fold family of (2+1)D fermionic topological orders, forming a fermionic analogue of Kitaev's sixteen-fold way. This family is distinguished by the mod 16 't Hooft anomaly of a $\mathbb{Z}_2$ one-form symmetry, generated in each theory by a single nontrivial $\mathbb{Z}_2$ anyon. This intrinsically fermionic anomaly permits anyon spins that are forbidden in bosonic phases; the simplest new example is an Abelian fermionic topological order containing a single $\mathbb{Z}_2$ Abelian anyon of spin 1/8. Each theory can be realized as the gapped boundary of a (3+1)D fermionic symmetry-protected topological (SPT) phase protected by the $\mathbb{Z}_2$ one-form symmetry, which acquires a $\mathbb{Z}_{16}$ classification once the spacetime spin structure is twisted by the one-form symmetry. We realize these phases microscopically via lattice models built from Walker-Wang models coupled to local fermions.

cond-mat.str-elCross-listedarXiv:2606.28682Jun 30, 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

Topological Phases of Matter and Quantum Computation

Paul Bruillard, Carlos Ortiz Marrero, Julia Plavnik

Contemporary mathematics - American Mathematical SocietyMar 27, 20206 citations

Dynamical classification of topological quantum phases

Lin Zhang, Long Zhang, Sen Niu +1 more

arXiv:1802.10061中国科学通报：英文版Sep 29, 2018150 citations

\mathbb{Z}_3

parafermions are regarded as the most natural generation of the Majorana fermions for realizing of more intriguing topological orders. In this work, we investigate the topological phase and emergent fermions in an extended parafermion chain using exact numerical methods. This model exhibits rich variety of phases, including threefold degenerate topological ferromagnetic phase, trivial paramagnetic phase, XX phase, dimer phase, chiral phase and commensurate phase. We present a smooth connection between this model and the extended spin-

{1\over 2}

XX model, by which we can generalize their measurements, such as the order parameters and long-range spin-spin correlations {\it etc}, to fully characterize the properties of these phases and map out their corresponding phase diagram. Strikingly, we find that all the phase boundaries finally merge to a single supercritical point, determined by the extended three-state Potts model. This approach opens a wide range of intriguing applications in investigating the phases and emergent phenomena in other

\mathbb{Z}_k

parafermion models, and may be also illuminating for their experimental realizations.

\mathbb{Z}_2

one-form symmetry, generated in each theory by a single nontrivial

\mathbb{Z}_2

anyon. This intrinsically fermionic anomaly permits anyon spins that are forbidden in bosonic phases; the simplest new example is an Abelian fermionic topological order containing a single

\mathbb{Z}_2

Abelian anyon of spin 1/8. Each theory can be realized as the gapped boundary of a (3+1)D fermionic symmetry-protected topological (SPT) phase protected by the

\mathbb{Z}_2

one-form symmetry, which acquires a

\mathbb{Z}_{16}

classification once the spacetime spin structure is twisted by the one-form symmetry. We realize these phases microscopically via lattice models built from Walker-Wang models coupled to local fermions.

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.