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Modular transformation and anyonic statistics of multi-component fractional quantum Hall states

We investigate the response to modular transformations and the fractional statistics of Abelian multi-component fractional quantum Hall (FQH) states. In particular, we analytically derive the modular matrices encoding the statistics of anyonic excitations for general Halperin states using the conformal field theories (CFTs). We validate our theory by several microscopic examples, including the spin-singlet state using anyon condensation picture and the Halperin (221) state in a topological flat-band lattice model using numerical calculations. Our results, uncovering that the modular matrices and associated fractional statistics are solely determined by the $K$-matrix, further strengthens the correspondence between the 2D CFTs and (2+1)D topological orders for multi-component FQH states.