A recursion approach to thin cylinder approximants for fractional quantum Hall states

Aside from the beautiful many-body wave functions that have defined the field, recursive and/or second-quantized presentations of fractional quantum Hall states have played an increasingly important role in the past. Examples include the recent discovery of the MPS structure of many quantum Hall states, but also a recursion for the Laughlin state related to the “non-local order parameter’’ defined by Read, originally defined in a mixed first/second quantized manner. Such recursions have recently been generalized to all composite fermion states, and put into a purely second-quantized, “all guiding-center’’ form. Here we observe that these recursions commute with the expansion of cylinder fractional quantum hall states in a “thin-cylinder parameter’’. This allows one to define not only the full quantum Hall state recursively, but also any “thin cylinder approximant’’, to any order. Relations/differences with the MPS are discussed, and we speculate on broader implications for DMRG approaches to quantum Hall states, in particular the efficient evaluation of non-local order parameters.