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arXiv:2508.12444 · National Science Review

Non-Abelian statistics for bosonic symmetry-protected topological phases

Hao Wang, Bo Wang, Jian-Song Hong, Xiong-Jun Liu

Published Jul 18, 2026• 0 citations
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Abstract

Symmetry-protected non-Abelian (SPNA) statistics opens new frontiers in quantum statistics and enriches the schemes for topological quantum computing. In this work, we propose a new paradigm of SPNA statistics in one-dimensional correlated bosonic symmetry-protected topological (SPT) phases and uncover exotic universal features from a systematic investigation. In particular, we show that for generic bosonic SPT phases described by real Hamiltonians, the SPNA statistics of topological zero modes fall into two distinct classes. The first class exhibits conventional braiding of hard-core bosonic zero modes. Furthermore, we discover a second class of unconventional braiding statistics characterized by a nonlinear transformation, featuring a fractionalization of the first class and reminiscent of the non-Abelian statistics of symmetry-protected Majorana pairs. The two distinct classes of statistics have topological origin in classifying non-Abelian Berry phases for braiding processes of real-Hamiltonian systems, distinguished by whether the holonomy involves a reflection operation. To illustrate, we focus on a specific bosonic SPT phase with particle-hole symmetry, and demonstrate that both classes of braiding statistics can be feasibly realized in tri-junctions with and without the aid of a controlled defect, respectively. Analytic and numerical results are given. We demonstrate how to encode logical qubits and implement both single- and two-qubit gates using the two classes of SPNA statistics. Finally, we propose feasible experimental schemes to observe these predictions and identify the parameter regimes for the high-fidelity braiding, paving the way for the experimental observation of our results in the near future.

DOI
10.1093/nsr/nwag444

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Symmetry-protected non-Abelian (SPNA) statistics opens new frontiers in quantum statistics and enriches the schemes for topological quantum computing. In this work, we propose a new paradigm of SPNA statistics in one-dimensional correlated bosonic symmetry-protected topological (SPT) phases and uncover exotic universal features from a systematic investigation. In particular, we show that for generic bosonic SPT phases described by real Hamiltonians, the SPNA statistics of topological zero modes fall into two distinct classes. The first class exhibits conventional braiding of hard-core bosonic zero modes. Furthermore, we discover a second class of unconventional braiding statistics characterized by a nonlinear transformation, featuring a fractionalization of the first class and reminiscent of the non-Abelian statistics of symmetry-protected Majorana pairs. The two distinct classes of statistics have topological origin in classifying non-Abelian Berry phases for braiding processes of real-Hamiltonian systems, distinguished by whether the holonomy involves a reflection operation. To illustrate, we focus on a specific bosonic SPT phase with particle-hole symmetry, and demonstrate that both classes of braiding statistics can be feasibly realized in tri-junctions with and without the aid of a controlled defect, respectively. Analytic and numerical results are given. We demonstrate how to encode logical qubits and implement both single- and two-qubit gates using the two classes of SPNA statistics. Finally, we propose feasible experimental schemes to observe these predictions and identify the parameter regimes for the high-fidelity braiding, paving the way for the experimental observation of our results in the near future.
Non-Abelian anyons have garnered extensive attention for obeying exotic non-Abelian statistics and having potential applications to fault-tolerant quantum computing. While the prior research has predominantly focused on non-Abelian statistics without the necessity of symmetry protection, recent progresses have shown that symmetries can play essential roles and bring a notion of the symmetry-protected non-Abelian (SPNA) statistics. In this work, we extend the concept of SPNA statistics to strongly-correlated systems which host parafermion zero modes (PZMs). This study involves a few fundamental results proved here. First, we unveil a generic unitary symmetry mechanism that protects PZMs from local couplings. Then, with this symmetry protection, the PZMs can be categorized into two nontrivial sectors, each maintaining its own parity conservation, even though the whole system cannot be dismantled into separate subsystems due to nonlinear interactions. Finally, by leveraging the parity conservation of each sector and the general properties of the effective braiding Hamiltonian, we prove rigorously that the PZMs intrinsically obey SPNA statistics. To further confirm the results, we derive the braiding matrix at a tri-junction. We also propose a correlated quantum nanowire model that accommodates a pair of PZMs protected by mirror symmetry and satisfying the generic theory. This work shows a broad spectrum of strongly-correlated systems capable of hosting fractional SPNA quasiparticles and enriches our comprehension of fundamental quantum statistics linked to the symmetries that govern the exchange dynamics.
Abstract Fractional statistics is one of the most intriguing features of topological phases in 2D. In particular, the so-called non-Abelian statistics plays a crucial role towards realizing topological quantum computation. Recently, the study of topological phases has been extended to 3D and it has been proposed that loop-like extensive objects can also carry fractional statistics. In this work, we systematically study the so-called three-loop braiding statistics for 3D interacting fermion systems. Most surprisingly, we discover new types of non-Abelian three-loop braiding statistics that can only be realized in fermionic systems (or equivalently bosonic systems with emergent fermionic particles). On the other hand, due to the correspondence between gauge theories with fermionic particles and classifying fermionic symmetry-protected topological (FSPT) phases with unitary symmetries, our study also gives rise to an alternative way to classify FSPT phases. We further compare the classification results for FSPT phases with arbitrary Abelian unitary total symmetry G f and find systematical agreement with previous studies.
We develop a complete theory for symmetry protected non-Abelian statistics of Majorana Kramers' pairs (MKPs) in time-reversal (TR) invariant topological superconductors, with fundamental results being uncovered. By introducing an effective Hamiltonian approach to describe the braiding of MKPs, we show that the non-Abelian braiding is protected when the effective Hamiltonian exhibits a new TR-like antiunitary symmetry, which is satisfied if the system is free of dynamical noise. Importantly, even the dynamical noise cannot cause error in braiding, unless the noise correlation function breaks a dynamical TR symmetry. This is a profound result and generalizes the TR symmetry protection of MKPs to the dynamical regime. Moreover, the resulted error by noise is shown to be a higher-order effect, compared with the decoherence of Majorana qubits without TR symmetry protection. This study completes the theory of symmetry-protected non-Abelian statistics of MKPs, and shows that the non-Abelian braiding of MKPs is well observable and may have versatile applications to future quantum computation technologies.
It has been shown that the symmetry-protected topological (SPT) phases with finite Abelian symmetries can be described by Chern-Simons field theory. We propose a topological response theory to uniquely identify the SPT orders, which allows us to obtain a systematic scheme to classify bosonic SPT phases with any finite Abelian symmetry group. We point out that even for finite Abelian symmetry, there exist bosonic SPT phases beyond the current Chern-Simons theory framework. We also apply the theory to fermionic SPT phases with Zm symmetry and find the classification of SPT phases depends on the parity of m: for even m there are 2m classes, m out of which is intrinsically fermionic SPT phases and cannot be realized in any bosonic system. Finally we propose a classification scheme of fermionic SPT phases for any finite, Abelian symmetry.
Floquet topological matter has emerged as one exciting platform to explore rich physics and game-changing applications of topological phases. As one remarkable and recently discovered feature of Floquet symmetry protected topological (SPT) phases, in principle, a simple periodically driven system can host an arbitrary number of topological protected zero edge modes and $\ensuremath{\pi}$ edge modes, with Majorana zero modes and Majorana $\ensuremath{\pi}$ modes as examples protected by the particle-hole symmetry. This work advocates a new route to holonomic quantum computation by exploiting the co-existence of many Floquet SPT edge modes, all of which have trivial dynamical phases during a computation protocol. As compelling evidence supporting this ambitious goal, three pairs of Majorana edge modes, hosted by a periodically driven one-dimensional (1D) superconducting superlattice, are shown to suffice to encode two logical qubits, realize quantum gate operations, and execute two simple quantum algorithms through adiabatic lattice deformation. When compared with early studies on quantum computation based on Majorana zero modes of topological quantum wires, significant resource saving is now made possible by use of Floquet SPT phases.
Topological materials are of great interest for applications in quantum computing, providing intrinsic robustness against environmental noises. A popular direction is to look for Majorana modes in integrated systems interfaced with superconducting materials. However, is superconductivity necessary for materials to exhibits non-abelian statistics? Here we predict with strong theoretical and numerical evidences that there exist topologically phases in a class of one-dimensional single crystals, which contain large bandgaps and are within experimental reach. Specifically, the nontrivial Zak phases are associated with gapless boundary states, which provides the non-Abelian statistics required for constructing topologically-protected quantum gates, even without superconductivity and magnetic field. Another anomalous feature of these materials is that the electric polarization is quantized and a transverse field induces a longitudinal polarization. This novel material property provides an experimentally-accessible mechanism for braiding the non-Abelian anyons.