Finite-size Scaling Considerations on the Ground State Microcanonical Temperature in Entropic Sampling Simulations

Abstract

In this work we discuss the behavior of the microcanonical temperature ∂S(E) ∂E obtained by means of numerical entropic sampling studies. It is observed that in almost all cases the slope of the logarithm of the density of states S(E) is not infinite in the ground state, since as expected it should be directly related to the inverse temperature 1 T . Here we show that these finite slopes are in fact due to finite-size effects and we propose an analytic expression a ln(bL) for the behavior of ∆S ∆E when L → ∞. To test this idea we use three distinct two-dimensional square lattice models presenting second-order phase transitions. We calculated by exact means the parameters a and b for the two-states Ising model and for the q = 3 and 4 states Potts model and compared with the results obtained by entropic sampling simulations. We found an excellent agreement between exact and numerical values. We argue that this new set of parameters a and b represents an interesting novel issue of investigation in entropic sampling studies for different models.

Critical Behavior of the Spin-1/2 Baxter-Wu Model: Entropic Sampling Simulations

“…Just as in the cases discussed in Ref. [31] the inverse temperature of the Baxter-Wu model diverges in the ground state as a ln(bL), yielding a = 1 6 and b = 1. For estimating the coefficients a and b from our simulational data we added four smaller sizes, L = 8, 16, 20, 26.…”

“…Nevertheless for any discrete model, as in the Baxter-Wu model, ∆E = const. Therefore the limit becomes exact only if L → ∞ (E min → −∞), where L is the linear lattice size [31]. In the Baxter-Wu model the ground state has four configurations, one with all spins positive and three more where two sub-lattices are negated successively.…”

“…As in Refs. [30,31] to calculate the mean value of these ten results we adopted a single averaging instead of an average with unequal uncertainties, since the values fall into a Gaussian distribution and in most cases they do not agree within the error bars. Thus,performing the second procedure would lead to an unrealistic small error bar.…”

An entropic simulational study of the spin-$1$ Baxter-Wu model in a crystal field

“…Entropic simulations [11][12][13][14] are excellent to study phase transitions and critical phenomena. The results obtained by this technique have revealed important characteristics regarding different models [10,[15][16][17]. The estimation of the joint density of states [14,17,18] brings a range of additional information, besides making it possi-ble to obtain thermodynamic properties for any temperature and coupling constants, allowing the construction of the phase diagram.…”