In this paper, we use the Replica-Exchange Wang–Landau (REWL) technique to study the behavior of the two-point correlation function of the site diluted classical anisotropic Heisenberg (AH) model in two dimensions. The model has a topological Berezinskii–Kosterlitz–Thouless (BKT) transition for any dilution down to the percolation threshold, [Formula: see text]. Contrary to earlier results that predict the exponent of the correlation function, [Formula: see text], to decay linearly to zero as a function of temperature, we found that it decreases with temperature to a finite value [Formula: see text] at [Formula: see text] for any dilution [Formula: see text].
In this paper, we use the Replica-Exchange Wang–Landau (REWL) technique to study the behavior of the two-point correlation function of the site diluted classical anisotropic Heisenberg (AH) model in two dimensions. The model has a topological Berezinskii–Kosterlitz–Thouless (BKT) transition for any dilution down to the percolation threshold, [Formula: see text]. Contrary to earlier results that predict the exponent of the correlation function, [Formula: see text], to decay linearly to zero as a function of temperature, we found that it decreases with temperature to a finite value [Formula: see text] at [Formula: see text] for any dilution [Formula: see text].
The importance of topological excitations in physics, mainly in the description of phase transitions, has a long history beginning with the superconductivity phenomenon in the rst years of the last century. However, the ideas behind this new concept has been more accurately described by Kosterlitz and Thouless, about 40 years ago. Conductor-superconductor, uidsuperuid, rough transitions and many others are characterized as being due the appearance of topological excitations in the system. Another concept of great importance is that the universality in phase transitions . Close to a phase transition the correlations of the system become innity, so that, the details of the model become irrelevant. In this situation, systems originaly so dierent as superconductors and ferromagnets, can be described by the same theory. The critical expoents, that caracterize the transition, depend only on the system dimension, the range of the potential and of their symmetries. This work is dedicated to study of the anisotropic Heisenberg model in two dimensions. This model has a non-usual phase transition, with quasi-long range order, characterized by a change in the behavior of the spin-spin correlation function C(r). In low temperature, T < TBKT, C(r) decays as a power law with distance, r. For temperatures greater than TBKT the correlation function falls o exponencially. The free energy has all the continuous derivatives, for this reason the transition is known as innity order transition. The BKT teory assume that the transition is driven by a vortice-antivortice unbiding mechanism in the system. In our study we did extensive numerical simulations with the tecnique known as Replica Exchange Wang-Landau. This method allow us to estimate the density of states g(E) of the model. With g(E) we can calculated the relevant termodynamic functions of the system (energy, magnetization, suceptibility and correlations for instance). Using Finite Size Scaling tecniques we have determinade the transition temperature TBKT and the behavior of the correlation function for a large range of temperatures. The calculations were done for various dilutions of non-magnetic sites, p, with p = 0,0;0,20;0,30 e 0,35, that allowed us to describe with great precision the critical behavior of the model in particular the correlation lentgh
The Berezinskii-Kosterlitz-Thouless (BKT) transition is the paradigmatic example of a topological phase transition without symmetry breaking, where a quasiordered phase, characterized by a power-law scaling of the correlation functions at low temperature, is disrupted by the proliferation of topological excitations above the critical temperature T_{BKT}. In this Letter, we consider the effect of long-range decaying couplings ∼r^{-2-σ} on the BKT transition. After pointing out the relevance of this nontrivial problem, we discuss the phase diagram, which is far richer than the corresponding short-range one. It features-for 7/4<σ<2-a quasiordered phase in a finite temperature range T_{c}<T<T_{BKT}, which occurs between a symmetry broken phase for T<T_{c} and a disordered phase for T>T_{BKT}. The transition temperature T_{c} displays unique universal features quite different from those of the traditional, short-range XY model. Given the universal nature of our findings, they may be observed in current experimental realizations in 2D atomic, molecular, and optical quantum systems.
The Berezinskii-Kosterlitz-Thouless (BKT) phase transition in two-dimensional planar rotator and XY models on a square lattice, diluted by randomly placed vacancies, is studied here using hybrid Monte Carlo simulations that combine single spin flip, cluster, and over-relaxation techniques. The transition temperature Tc is determined as a function of vacancy density ${\ensuremath{\rho}}_{\mathrm{vac}}$ by finite-size scaling of the helicity modulus and the in-plane magnetic susceptibility. The results for Tc are consistent with those from the much less precise fourth-order cumulant of Binder. Tc is found to decrease monotonically with increasing ${\ensuremath{\rho}}_{\mathrm{vac}}$, and falls to zero close to the square lattice percolation limit, ${\ensuremath{\rho}}_{\mathrm{vac}}\ensuremath{\approx}0.41$. The result is physically reasonable: the quasi-long-range orientational order of the low-temperature phase cannot be maintained in the absence of sufficient spin interactions across the lattice.
We study the effect of a linear tunneling coupling between two-dimensional systems, each separately exhibiting the topological Berezinskii-Kosterlitz-Thouless (BKT) transition. In the uncoupled limit, there are two phases: one where the one-body correlation functions are algebraically decaying and the other with exponential decay. When the linear coupling is turned on, a third BKT-paired phase emerges, in which one-body correlations are exponentially decaying, while two-body correlation functions exhibit power-law decay. We perform numerical simulations in the paradigmatic case of two coupled XY models at finite temperature, finding evidences that for any finite value of the interlayer coupling, the BKT-paired phase is present. We provide a picture of the phase diagram using a renormalization group approach.
The spin-spin correlation function of the two-dimensional (2D) XY model decays as a power law at all temperatures below the Berezinskii-Kosterlitz-Thouless transition point with a temperature-dependent exponent $\ensuremath{\eta}=\ensuremath{\eta}(T/J)$ (J is the ferromagnetic coupling strength). It is known from computer experiments that in the 2D XY model with site or bond dilution this exponent depends on the concentration p of removed sites or bonds as well. Knowing the slope $\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p$ at point p=0, one can predict the value of the exponent for small dilution concentrations: ${\ensuremath{\eta}(p)\ensuremath{\simeq}\ensuremath{\eta}(0)+p(\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p)|}_{p=0}$. As shown in this paper, the spin-wave Hamiltonian allows one to obtain exact results for this slope: ${(\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p)|}_{p=0}=T/(2J)+O({(T/J)}^{2})$ and $T/(\ensuremath{\pi}J)+O({(T/J)}^{2})$ for site and for bond dilution, respectively.
The Berezinskii-Kosterlitz-Thouless (BKT) transition is the prototype of a phase transition driven by the formation and interaction of topological defects in two-dimensional (2D) systems. In typical models these are vortices: above a transition temperature TBKT vortices are free, below this transition temperature they get confined. In this work we extend the concept of BKT transition to quantum systems in two dimensions. In particular, we demonstrate that a zero-temperature quantum BKT phase transition, driven by a coupling constant can occur in 2D models governed by an effective gauge field theory with a diverging dielectric constant. One particular example is that of a compact U(1) gauge theory with a diverging dielectric constant, where the quantum BKT transition is induced by non-relativistic, purely 2D magnetic monopoles, which can be viewed also as electric vortices. These quantum BKT transitions have the same diverging exponent z as the quantum Griffiths transition but have nothing to do with disorder.