Two-stage melting of an inter-component Potts long-range order in two dimensions

TL;DR

This paper rigorously analyzes a 2D hexatic-nematic XY model with $Z_3$ degrees of freedom, revealing a two-stage melting process of inter-component order driven by topological defects, using matrix product state methods.

Contribution

The study introduces a rigorous matrix product state approach to exactly solve the 2D model and characterizes the two-stage melting of $Z_3$ long-range order.

Findings

Existence of inter-component $Z_3$ Potts long-range order at low temperatures.

Identification of a two-stage melting process with an intermediate Potts liquid phase.

Proliferation of domain walls and vortices drives the phase transitions.

Abstract

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.