Twisted Lattice Gauge Theory: Membrane Operators, Three-loop Braiding and Topological Charge
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| Format: | Preprint |
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2024
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| _version_ | 1866913682141741056 |
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| author | Huxford, Joe Nguyen, Dung Xuan Kim, Yong Baek |
| author_facet | Huxford, Joe Nguyen, Dung Xuan Kim, Yong Baek |
| contents | 3+1 dimensional topological phases can support loop-like excitations in addition to point-like ones, allowing for non-trivial 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+1d 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 dimensional topological phases. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_13042 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Twisted Lattice Gauge Theory: Membrane Operators, Three-loop Braiding and Topological Charge Huxford, Joe Nguyen, Dung Xuan Kim, Yong Baek Strongly Correlated Electrons High Energy Physics - Theory 3+1 dimensional topological phases can support loop-like excitations in addition to point-like ones, allowing for non-trivial 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+1d 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 dimensional topological phases. |
| title | Twisted Lattice Gauge Theory: Membrane Operators, Three-loop Braiding and Topological Charge |
| topic | Strongly Correlated Electrons High Energy Physics - Theory |
| url | https://arxiv.org/abs/2401.13042 |