The interplay of topology and competing interactions can induce enriched phases and phase transitions at finite temperatures. We consider a weakly coupled two-dimensional hexatic-nematic XY model with a relative
Z3 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 intercomponent
Z3 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 intercomponent
Z3 Potts LRO appears, accompanying the binding of nematic vortices. In the nematic regime, however, the intercomponent
Z3 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
Z3 Potts variable, respectively. Our results thus provide a prototype example of two-stage melting of a two-dimensional LRO driven by multiple topological defects.
Proliferation of defects is a mechanism that allows for topological phase transitions. Such a phase transition is found in two dimensions for the XY model, which lies in the Berezinskii-Kosterlitz-Thouless (BKT) universality class. The transition point can be found using renormalization group analysis. We apply renormalization group arguments to determine the nature of BKT transitions for the three-dimensional plaquette-dimer model, which is a model that exhibits fractonic mobility constraints. We show that an important part of this analysis demands a modified dimensional analysis that changes the interpretation of scaling dimensions upon coarse-graining. Using this modified dimensional analysis, we compute the beta functions of the model and predict a finite critical value above which the fractonic phase melts, proliferating dipoles. Importantly, the transition point is found through a renormalization group analysis that accounts for the phenomenon of UV/IR mixing, characteristic of fractonic models.
In this chapter, we introduce topological defects as excitations complementary to the elementary Nambu–Goldstone modes and relate their classification to the presence of broken symmetries. We show that the proliferation of topological defects can lead to the suppression of long-range correlations and the emergence of algebraic long-range order. The Berezinskii–Kosterlitz–Thouless phase transition is discussed both generally and in the specific example of the one-dimensional Ising model. In the same context, the Kramers–Wannier duality between topological defects and elementary modes is introduced.