Microscopic field theories of the quantum skyrmion Hall effect
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| Format: | Preprint |
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2025
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| _version_ | 1866908499330465792 |
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| author | Patil, Vinay Banerjee, Archi Cook, Ashley M. |
| author_facet | Patil, Vinay Banerjee, Archi Cook, Ashley M. |
| contents | We construct effective field theories of the quantum skyrmion Hall effect from matrix Chern-Simons theory for $N$ electrons, corresponding to matrix dimension $N$. We first consider a quantum Hall droplet within finite $N$ matrix Chern-Simons theory. Taking into account the differential geometry of the matrix Chern-Simons droplet for a partially-filled fuzzy two-sphere, we first generalize the quantization procedure by replacing the Poisson bracket, a classical Lie derivative, with a quantum counterpart, the Lie derivative for a deformed fuzzy sphere. This yields the topological invariant introduced in earlier works on the quantum skyrmion Hall effect and previously unidentified fusion rules. This is consistent with treatment of a spin $S$ of multiplicity $2S+1$ as a quantum Hall droplet within matrix Chern-Simons theory for $N=2S+1$ spinless electrons and a generalization of a Jain composite particle for a Laughlin state. We then construct $D$-dimensional arrays of coupled small $N$ matrix Chern-Simons droplets as effective field theories of the quantum skyrmion Hall effect. In higher-symmetry constructions, this yields what appears to be a D+1 dimensional $U(N)$ Yang-Mills theory, but actually contains $δ$ extra fuzzy dimensions from the finite $N$ MCS theory as well as deformations from $U(N)$ due to partial filling of the fuzzy spheres. In this construction, the Chern-Simons level is $k+1$ for each small $N$ droplet, while the entire array can be interpreted as an unbounded matrix Chern-Simons theory at level $k$. Such constructions at $k=2$ are consistent with earlier results for the multiplicative Chern insulator. We also formulate the quantum skyrmion Hall effect in terms of a Lagrangian for an array of potentially distinct, small $N$ droplets within anisotropic fuzzification. We discuss the relevance of these results to spin lattice models and lattice gauge theories. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_16547 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Microscopic field theories of the quantum skyrmion Hall effect Patil, Vinay Banerjee, Archi Cook, Ashley M. High Energy Physics - Theory Strongly Correlated Electrons Mathematical Physics Quantum Physics We construct effective field theories of the quantum skyrmion Hall effect from matrix Chern-Simons theory for $N$ electrons, corresponding to matrix dimension $N$. We first consider a quantum Hall droplet within finite $N$ matrix Chern-Simons theory. Taking into account the differential geometry of the matrix Chern-Simons droplet for a partially-filled fuzzy two-sphere, we first generalize the quantization procedure by replacing the Poisson bracket, a classical Lie derivative, with a quantum counterpart, the Lie derivative for a deformed fuzzy sphere. This yields the topological invariant introduced in earlier works on the quantum skyrmion Hall effect and previously unidentified fusion rules. This is consistent with treatment of a spin $S$ of multiplicity $2S+1$ as a quantum Hall droplet within matrix Chern-Simons theory for $N=2S+1$ spinless electrons and a generalization of a Jain composite particle for a Laughlin state. We then construct $D$-dimensional arrays of coupled small $N$ matrix Chern-Simons droplets as effective field theories of the quantum skyrmion Hall effect. In higher-symmetry constructions, this yields what appears to be a D+1 dimensional $U(N)$ Yang-Mills theory, but actually contains $δ$ extra fuzzy dimensions from the finite $N$ MCS theory as well as deformations from $U(N)$ due to partial filling of the fuzzy spheres. In this construction, the Chern-Simons level is $k+1$ for each small $N$ droplet, while the entire array can be interpreted as an unbounded matrix Chern-Simons theory at level $k$. Such constructions at $k=2$ are consistent with earlier results for the multiplicative Chern insulator. We also formulate the quantum skyrmion Hall effect in terms of a Lagrangian for an array of potentially distinct, small $N$ droplets within anisotropic fuzzification. We discuss the relevance of these results to spin lattice models and lattice gauge theories. |
| title | Microscopic field theories of the quantum skyrmion Hall effect |
| topic | High Energy Physics - Theory Strongly Correlated Electrons Mathematical Physics Quantum Physics |
| url | https://arxiv.org/abs/2508.16547 |