  Interplay of topology and competing interactions can induce new phases and
phase transitions at finite temperatures. We consider a weakly coupled
two-dimensional hexatic-nematic XY model with a relative $Z_3$ Potts degrees of
freedom,and apply the matrix product state method to solve this model
rigorously. Since the partition function is expressed as a product of
two-legged one-dimensional transfer matrix operator, an entanglement entropy of
the eigenstate corresponding to the maximal eigenvalue of this transfer
operator can be used as a stringent criterion to determine various phase
transitions precisely. At low temperatures, the inter-component $Z_3$ Potts
long-range order (LRO) exists, indicating that the hexatic and nematic fields
are locked together and their respective vortices exhibit quasi-LRO. In the
hexatic regime, below the BKT transition of the hexatic vortices, the
inter-component $Z_3$ Potts LRO appears, accompanying with the binding of
nematic vortices. In the nematic regime, however, the inter-component $Z_3$
Potts LRO undergoes a two-stage melting process. An intermediate Potts liquid
phase emerges between the Potts ordered and disordered phases, characterized by
an algebraic correlation with formation of charge-neutral pairs of both hexatic
and nematic vortices. These two-stage phase transitions are associated with the
proliferation of the domain walls and vortices of the relative $Z_3$ Potts
variable, respectively. Our results thus provide a prototype example of
two-stage melting of a two-dimensional long-range order, driven by multiple
topological defects.
