^{1}

^{1}

^{*}

We have carried out micro-canonical Monte Carlosimulations of a planar rotator model in 30 × 30 lattice using periodic boundary conditions. The energy distribution of the rotator in the lattice shows features that can be associated with spin wave and vortex excitations. The results supplement the first-order transition observed in canonicalMonte Carlosimulation, due to vortex nucleation. We also see features that can be associated with the in-homogeneity of vortex charge in the critical region.

Simulations of statistical systems have gained much interest, recently [1-4]. The spin-models with continuous symmetry, such as the planar rotator model [_{KT}. But the temperature dependence of the specific heat shows a peak at a temperature, which is about 15% higher than T_{KT}.

It has been found that the vortex unbinding transition temperature (T_{v}) increases by reducing the potential well width. On the other hand, local disorder sets in at temperatures near the potential well height

.

Therefore, for a sufficiently narrow potential well T_{D} < T_{v}, we expect the continuous transition to yield to a first-order vacancy condensation transition. Several studies have been carried out to understand the first order transition in this system [28-48]. Here we report the study of the nature of the first order transition in a finite isolated system, which supplements the results observed in canonical Monte Carlo simulations [28,29].

The Hamiltonian of the planar rotator model with modified potential is given by:

J in Equation 1 denotes the interaction strength (>0 for the ferromagnetic case) and the sum is over all the nearest neighbors [^{2} is the controlling parameter and q > 0. As q is raised, it has an increasingly narrow well of width and for it is essentially constant at V(π) = 2J. For q = 1, the Hamiltonian gives rise to the Kosterlitz-Thouless transition and for large value of q, the transition is first-order in nature.

We performed micro-canonical Monte Carlo (MC) simulations [49,50] on 30 × 30 rotator system using the Hamiltonian given in Equation (1). There are also other simulation methods which have been reported recently [51,52]. We used periodic boundary conditions. It is known from renormalization group theory that fluctuations at all wave lengths are equally important around the phase transition [

where k_{B} is the Boltzmann constant. (Hereafter we replace k_{B}T/J by T and E/J by E for simplicity). In these simulations the total energy (E) is an input parameter and the temperature (T) is determined from the simulations. The system was heated in steps across the first order transition during which energy was added to the rotator system through the demon. ^{5} Monte Carlo step per rotator (MCSR) for equilibration and 5 × 10^{6} MCSR for averaging. We performed block averages consisting of 5 × 10^{4} MCSR each and then found the standard deviation of the block averages. The standard deviation of the estimated temperature is less than about 0.5 percent. It is seen that the temperature dependence of energy shows a van der Waals-like loop at the first order transition, which occurs at T = 1.0. It is well known that meta-stability gives rise to van der Waals-like loop [_{β}_{,L}(E) by the following equation [

We studied the system (was studied) for various energy values with 5 × 10^{6} MCSR for equilibration and 5 × 10^{6} MCSR for averaging. The standard deviation of the estimated temperature in this set, is also less than about 0.5 percent. We studied the energy distribution of a rotator in lattice after equilibration. Figures 2-4 show the energy distribution of the rotator in the lattice for various values of total system energy. In

We attribute this as due to the spin waves. In _{s} = 7.9 develops gradually apart from the broad low temperature peak which reduces in magnitude. There are also several intermediate peaks. In

absent and the peak at E_{s} = 7.9 continues to increase, along with the intermediate peaks. We attribute the high energy peak as due to the vortices. The height (or intensity) of the peaks which arises due to spin waves and vortices is shown as a function of system energy in

It is known that, the spin waves dominate at low temperatures [12-14]. The broad peak at low temperatures (_{s} = 7.9, is close to the value of energy needed to create a vortex-anti-vortex pair. The energy needed to create a vortex-anti-vortex pair (2μ) can be estimated from the expected exponential temperature dependence of vortex density: V~e^{–}^{2μ/T} [_{s} = 7.9, can be attributed as due to vortices.

Simulations have shown that the vortex charge distribution is inhomogeneous [

of the vortices, certain features can be understood as follows. In the low temperature (energy) insulating phase the vortices are bound tightly which results in a small value of the vortex in-homogeneity. In the high temperature (energy) Debye-Hűckel regime, the in-homogeneity is also small due to the presence of a large number of free charges in the liquid phase. We speculate that cluster of vortices in the critical region are responsible for the peak in the vortex charge in-homogeneity. The intermediate peaks seen in Figures 3 and 4 can be attributed as due to the vortex-charge in-homogeneity.

In conclusion, we have studied the energy distribution of the rotator in the lattice. The energy distribution of the rotator, shows features that can be associated with spin wave and vortex excitations. The results substantiate the first-order transition observed in canonical Monte Carlo simulation, due to vortex nucleation. We also see features that can be associated with the in-homogeneity of vortex charge in the critical region.

The program for this simulation was developed at the Max-Planck-Institut für Festkörperforschung, Stuttgart, Germany. One of us (S. O.) acknowledges the financial support of the Department of Science and Technology, New Delhi.