Perfectly Spherical Bloch Hyper-spheres from Quantum Matrix Geometry
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866929400904155136 |
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| author | Hasebe, Kazuki |
| author_facet | Hasebe, Kazuki |
| contents | Exploiting analogies between the precessing quantum spin system and the charge-monopole system, we construct Bloch hyper-spheres with $\it{exact}$ spherical symmetries in arbitrary dimensions. Such Bloch hyper-spheres are realized as a collection of the orbits of a precessing quantum spin. The geometry of Bloch hyper-spheres is exactly equal to the quantum Nambu geometry of higher dimensional fuzzy spheres. The stabilizer group symmetry of the Bloch hyper-sphere necessarily introduces degenerate spin-coherent states, giving rise to the Wilczek-Zee geometric phase of non-Abelian monopoles associated with the hyper-sphere holonomy. The degenerate spin-coherent states induce matrix-valued quantum geometric tensors. While the minimal spin Bloch hyper-spheres exhibit similar properties in even and odd dimensions, their large spin counterparts differ qualitatively depending on the parity of the dimensions. Exact correspondences between spin-coherent states and monopole harmonics in higher dimensions are established. We also investigate density matrices described by Bloch hyper-balls and elucidate their corresponding statistical and geometric properties, such as von Neumann entropies and Bures quantum metrics. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_07149 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Perfectly Spherical Bloch Hyper-spheres from Quantum Matrix Geometry Hasebe, Kazuki Quantum Physics Materials Science Statistical Mechanics High Energy Physics - Theory Exploiting analogies between the precessing quantum spin system and the charge-monopole system, we construct Bloch hyper-spheres with $\it{exact}$ spherical symmetries in arbitrary dimensions. Such Bloch hyper-spheres are realized as a collection of the orbits of a precessing quantum spin. The geometry of Bloch hyper-spheres is exactly equal to the quantum Nambu geometry of higher dimensional fuzzy spheres. The stabilizer group symmetry of the Bloch hyper-sphere necessarily introduces degenerate spin-coherent states, giving rise to the Wilczek-Zee geometric phase of non-Abelian monopoles associated with the hyper-sphere holonomy. The degenerate spin-coherent states induce matrix-valued quantum geometric tensors. While the minimal spin Bloch hyper-spheres exhibit similar properties in even and odd dimensions, their large spin counterparts differ qualitatively depending on the parity of the dimensions. Exact correspondences between spin-coherent states and monopole harmonics in higher dimensions are established. We also investigate density matrices described by Bloch hyper-balls and elucidate their corresponding statistical and geometric properties, such as von Neumann entropies and Bures quantum metrics. |
| title | Perfectly Spherical Bloch Hyper-spheres from Quantum Matrix Geometry |
| topic | Quantum Physics Materials Science Statistical Mechanics High Energy Physics - Theory |
| url | https://arxiv.org/abs/2402.07149 |