Microscopic field theories of the quantum skyrmion Hall effect

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 $\delta$ 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.