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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2403.01577 |
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| _version_ | 1866913252079828992 |
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| author | Chan, Ying Lan, Tian Wu, Linqian |
| author_facet | Chan, Ying Lan, Tian Wu, Linqian |
| contents | Given a modular tensor category $\mathscr{C}$, we construct an associative algebra $\mathrm{Tor({\mathscr{C}}})$, which we call the torus algebra. We prove that the torus algebra is semisimple by explicitly constructing all the simple modules. Suppose that a topological ordered phase described by $\mathscr{C}$ is put on a torus. Physically, each simple module over $\mathrm{Tor({\mathscr{C}}})$ consists of the low energy states on the torus with one anyon excitation, or equivalently, the ground states on a punctured torus where the anyon is enclosed by the puncture. Elements in $\mathrm{Tor({\mathscr{C}}})$ can be physically interpreted as anyon hopping processes on the torus. We give the precise formula how an arbitrary logical operator on the low energy states on a torus can be realized by moving anyons on the torus. Our work thus provides a theoretical proposal that the low energy states on a torus can serve as topological qudits and one can arbitrarily manipulate them by moving anyons around. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2403_01577 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Torus algebra and logical operators at low energy Chan, Ying Lan, Tian Wu, Linqian Strongly Correlated Electrons Mathematical Physics Given a modular tensor category $\mathscr{C}$, we construct an associative algebra $\mathrm{Tor({\mathscr{C}}})$, which we call the torus algebra. We prove that the torus algebra is semisimple by explicitly constructing all the simple modules. Suppose that a topological ordered phase described by $\mathscr{C}$ is put on a torus. Physically, each simple module over $\mathrm{Tor({\mathscr{C}}})$ consists of the low energy states on the torus with one anyon excitation, or equivalently, the ground states on a punctured torus where the anyon is enclosed by the puncture. Elements in $\mathrm{Tor({\mathscr{C}}})$ can be physically interpreted as anyon hopping processes on the torus. We give the precise formula how an arbitrary logical operator on the low energy states on a torus can be realized by moving anyons on the torus. Our work thus provides a theoretical proposal that the low energy states on a torus can serve as topological qudits and one can arbitrarily manipulate them by moving anyons around. |
| title | Torus algebra and logical operators at low energy |
| topic | Strongly Correlated Electrons Mathematical Physics |
| url | https://arxiv.org/abs/2403.01577 |