Higher bracket structure of density operators in Weyl fermion systems and topological insulators
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arXiv
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| Formato: | Preprint |
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2024
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| _version_ | 1866913668962189312 |
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| author | Langmann, Edwin Ryu, Shinsei Shiozaki, Ken |
| author_facet | Langmann, Edwin Ryu, Shinsei Shiozaki, Ken |
| contents | We study the algebraic structure of electron density operators in gapless Weyl fermion systems in $d=3,5,7,\cdots$ spatial dimensions and in topological insulators (without any protecting symmetry) in $d=4,6,8,\cdots$ spatial dimensions. These systems are closely related by the celebrated bulk-boundary correspondence. Specifically, we study the higher bracket -- a generalization of commutator for more than two operators -- of electron density operators in these systems. For topological insulators, we show that the higher-bracket algebraic structure of density operators structurally parallels with the Girvin-MacDonald-Platzman algebra (the $W_{1+\infty}$ algebra), the algebra of electron density operators projected onto the lowest Landau level in the quantum Hall effect. By the bulk-boundary correspondence, the bulk higher-bracket structure mirrors its counterparts at the boundary. Specifically, we show that the density operators of Weyl fermion systems, once normal-ordered with respect to the ground state, their higher bracket acquires a c-number part. This part is an analog of the Schwinger term in the commutator of the fermion current operators. We further identify this part with a cyclic cocycle, which is a topological invariant and an element of Connes' noncommutative geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_09683 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Higher bracket structure of density operators in Weyl fermion systems and topological insulators Langmann, Edwin Ryu, Shinsei Shiozaki, Ken Strongly Correlated Electrons Mesoscale and Nanoscale Physics High Energy Physics - Theory We study the algebraic structure of electron density operators in gapless Weyl fermion systems in $d=3,5,7,\cdots$ spatial dimensions and in topological insulators (without any protecting symmetry) in $d=4,6,8,\cdots$ spatial dimensions. These systems are closely related by the celebrated bulk-boundary correspondence. Specifically, we study the higher bracket -- a generalization of commutator for more than two operators -- of electron density operators in these systems. For topological insulators, we show that the higher-bracket algebraic structure of density operators structurally parallels with the Girvin-MacDonald-Platzman algebra (the $W_{1+\infty}$ algebra), the algebra of electron density operators projected onto the lowest Landau level in the quantum Hall effect. By the bulk-boundary correspondence, the bulk higher-bracket structure mirrors its counterparts at the boundary. Specifically, we show that the density operators of Weyl fermion systems, once normal-ordered with respect to the ground state, their higher bracket acquires a c-number part. This part is an analog of the Schwinger term in the commutator of the fermion current operators. We further identify this part with a cyclic cocycle, which is a topological invariant and an element of Connes' noncommutative geometry. |
| title | Higher bracket structure of density operators in Weyl fermion systems and topological insulators |
| topic | Strongly Correlated Electrons Mesoscale and Nanoscale Physics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2401.09683 |