Berezinskii-Kosterlitz-Thouless transition close to the percolation threshold
As it is known, in diluted models there is no an order parameter like the magnetization in an order-disorder phase transition. The phase transition is mediated by the unbinding of point like defects and characterized by a qualitative change in the behavior of the correlation function \(G(r)\). In the Berezinskii-Kosterlitz-Thouless \((BKT)\) model the correlation function \(G(r)\) behaves as \(r\) in the exponent of \(-(d-2+\eta)\), where d is the dimension of space. The exponent \(\eta(T)\) is not universal depending continuously on temperature, \(T\), i.e. there is a low temperature phase, where the model is critical anywhere. This letter presents numerical Monte Carlo calculation of the critical behavior of the \(BKT\) transition in a diluted two-dimensional model close to the percolation threshold \(p\approx p_{c}\). Close to \(p_{c}\), it is expected that many non-percolating clusters appear in the system and when a percolating cluster appears its structure is fractal. Here, any cluster techniques becomes useless, since the cluster size is very small. Due to, the authors use a simulated annealing to reach the equilibrium in each simulation conducted. Simulations are performed in square lattice of sizes \(L\bullet L\) with \(16<L<640\). The obtained numerical results show that the \(BKT\) transition in the diluted two-dimensional \(XY\) model in a square lattice remains until the percolation threshold, when the \(BKT\) phase is extinguished. There is no any uncommon behavior in the classical model. When the percolation threshold is approached, the \(BKT\) temperature goes to zero. In this regime, quantum fluctuations become important.
scientific article; zbMATH DE number 6306758