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arXiv:1403.7437 · Physical Review Letters

Braiding Statistics of Loop Excitations in Three Dimensions

Chenjie Wang, Michael Levin

Published Aug 19, 2014• 193 citations
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Abstract

While it is well known that three dimensional quantum many-body systems can support nontrivial braiding statistics between particlelike and looplike excitations, or between two looplike excitations, we argue that a more fundamental quantity is the statistical phase associated with braiding one loop α around another loop β, while both are linked to a third loop γ. We study this three-loop braiding in the context of (Z(N))(K) gauge theories which are obtained by gauging a gapped, short-range entangled lattice boson model with (Z(N))(K) symmetry. We find that different short-range entangled bosonic states with the same (Z(N))(K) symmetry (i.e., different symmetry-protected topological phases) can be distinguished by their three-loop braiding statistics.

DOI
10.1103/physrevlett.113.080403

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While it is well known that three dimensional quantum many-body systems can support nontrivial braiding statistics between particlelike and looplike excitations, or between two looplike excitations, we argue that a more fundamental quantity is the statistical phase associated with braiding one loop α around another loop β, while both are linked to a third loop γ. We study this three-loop braiding in the context of (Z(N))(K) gauge theories which are obtained by gauging a gapped, short-range entangled lattice boson model with (Z(N))(K) symmetry. We find that different short-range entangled bosonic states with the same (Z(N))(K) symmetry (i.e., different symmetry-protected topological phases) can be distinguished by their three-loop braiding statistics.
Braiding phenomena, from the charge-flux Aharonov-Bohm effect to anyonic statistics in fractional quantum Hall systems, are paradigmatic manifestations of topology in quantum physics. Ordinary mutual braiding between ppp- and qqq-dimensional excitations occurs in d=p+q+2d=p+q+2d=p+q+2 spatial dimensions. In this work, we introduce a universal construction of mutual statistics in the adjacent dimension d=p+q+1d=p+q+1d=p+q+1, applicable to excitations obeying ZN\mathbb Z_NZN​ fusion for arbitrary NNN and all excitation dimensions ppp and qqq. The corresponding invariant is the Berry phase accumulated in a simple 4N4N4N-step microscopic unitary process built from local excitation operators on lattices. This process measures the linking of one excitation with the NNN-fold fusion junction of the other, encompassing particle-particle statistics in one dimension, particle-loop statistics in two dimensions, and loop-loop or particle-membrane statistics in three dimensions. We establish the quantization and bilinearity of the invariant and show that its field-theory response is governed by the Bockstein homomorphism, motivating the name Bockstein braiding statistics. Interpreting the excitation operators as open symmetry operators turns the same invariant into a direct microscopic diagnostic of mixed anomalies between symmetries. We demonstrate this diagnostic in a (1+1)D spin chain, where the nontrivial Bockstein braiding phase proves the mixed anomaly between the spin-flip symmetry ∏X\prod X∏X and the nearest-neighbor controlled-ZZZ symmetry ∏CZ\prod CZ∏CZ. We construct explicit (2+1)D and (3+1)D lattice analogs, yielding new anomalous symmetry pairs, and apply the framework to strongly coupled (3+1)D continuum gauge theories. Nontrivial Bockstein braiding rules out a fully symmetric gapped phase, obstructs simultaneous condensation of the two excitations, and implies fractionalization of higher-form symmetries.
(3+1)(3+1)(3+1)-dimensional topological phases can support loop-like excitations in addition to point-like ones, allowing for nontrivial loop-loop and point-loop braiding statistics not permitted to point-like excitations alone. Furthermore, these loop-like excitations can be linked together, changing their properties. In particular, this can lead to distinct three-loop braiding, involving two loops undergoing an exchange process while linked to a third loop. In this work, we investigate the loop-like excitations in a (3+1)(3+1)(3+1)-dimensional Hamiltonian realization of Dijkgraaf-Witten theory through direct construction of their membrane operators, for a general finite Abelian group and 4-cocycle twist. Using these membrane operators, we find the braiding relations and fusion rules for the loop-like excitations, including those linked to another loop-like excitation. Furthermore, we use these membrane operators to construct projection operators that measure the topological charge and show that the number of distinct topological charges measured by the 2-torus matches the ground-state degeneracy of the model on the 3-torus, explicitly confirming a general expectation for topological phases. This direct construction of the membrane operators sheds significant light on the key properties of the loop-like excitations in (3+1)(3+1)(3+1)-dimensional topological phases.
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.
Topological phases of matter are characterised by long-range entanglement between their constituent degrees of freedom, which allows them to host excitations with non-trivial exchange statistics. While topological phases in 2+1d are relatively well studied, less is known about their 3+1d counterparts. In this thesis, we examine features of these higher-dimensional phases in a tractable example model, based on a generalisation of lattice gauge theory called higher lattice gauge theory. As well as hosting point-like excitations, the model supports loop-like excitations with non-trivial loop-loop and point-loop braiding statistics. We explicitly construct operators to produce and move these excitations, and use these to find the loop-loop and point-loop braiding relations. These creation operators also reveal that some of the excitations are confined, costing energy to separate, while others are condensed and can be produced locally. This is discussed in the context of condensation-confinement transitions between different cases of this model. We also explain a method for measuring topological charge and use explicit measurement operators to re-derive a relationship between the number of charges measured by a 2-torus and the ground-state degeneracy of the higher lattice gauge theory model on the 3-torus. From these measurement operators, we can see that the ground state degeneracy on the 3-torus is related to the number of types of linked loop-like excitations.
Topological quantum field theory (TQFT) is a very powerful theoretical tool to study topological phases and phase transitions. In 2+1d2+1d2+1d, it is well known that the Chern-Simons theory captures all the universal topological data of topological phases, e.g., quasiparticle braiding statistics, chiral central charge, and even provides us a deep insight for the nature of topological phase transitions. Recently, topological phases of quantum matter are also intensively studied in 3+1d3+1d3+1d and it has been shown that looplike excitation obeys the so-called three-loop-braiding statistics. In this paper, we will try to establish a TQFT framework to understand the quantum statistics of particle and looplike excitation in 3+1d3+1d3+1d. We will focus on Abelian topological phases for simplicity, however, the general framework developed here is not limited to Abelian topological phases.
We study $(d\ensuremath{-}1)$-dimensional excitations in the ddd-dimensional color code that are created by transversal application of the Rd{R}_{d}Rd​ phase operators on connected subregions of qubits. We find that such excitations are the superpositions of electric charges and can be characterized by the fixed-point wave functions of $(d\ensuremath{-}1)$-dimensional bosonic symmetry-protected topological (SPT) phases with ${({\mathbb{Z}}_{2})}^{\ensuremath{\bigotimes}d}$ symmetry. While these SPT excitations are localized on $(d\ensuremath{-}1)$-dimensional boundaries, their creation requires operations acting on all qubits inside the boundaries, reflecting the nontriviality of emerging SPT wave functions. Moreover, these SPT excitations can be physically realized as transparent gapped domain walls which exchange excitations in the color code. Namely, in the three-dimensional color code, the domain wall, associated with the transversal R3{R}_{3}R3​ operator, exchanges a magnetic flux and a composite of a magnetic flux and the looplike SPT excitation, revealing rich possibilities of boundaries in higher-dimensional TQFTs. We also find that magnetic fluxes and the looplike SPT excitations exhibit nontrivial three-loop braiding statistics in three dimensions as a result of the fact that the R3{R}_{3}R3​ phase operator belongs to the third level of the Clifford hierarchy. We believe that the connection between SPT excitations, fault-tolerant logical gates and gapped domain walls, established in this paper, can be generalized to a large class of topological quantum codes and TQFTs.