Free energy and Green function for the multi-coupled spin system

The free energy and Green function of the multi-coupled spin chain in low-e nergy theory are given by using the functional bosonization method. We dedu ce that 2(m) coupled spinless chains (m = 0, 1, 2,...), by using a series o f unitary transformations, can be reduced to some uncoupled chains, but for other numbers of coupled spinless chains we need to solve a series of inve rse matrices generally; for either case, the system can be formally separat ed into the charge and spin parts.