\documentclass[12pt]{article} % % \RequirePackage[a4paper] \usepackage{mathptmx,graphicx} \begin{document} \begin{center} {\Large\bfseries Quenched disorder effect on a first-order phase transition in 3D\par} \vspace{3ex} {\bfseries L.~A.~Fern\'andez$^{1,3}$, \underline{A.~Gordillo-Guerrero}$^{2,3}$, \\V.~Mart\'\i{}n-Mayor$^{1,3}$, J.~J.~Ruiz-Lorenzo$^{2,3}$\par} {\footnotesize\itshape 1. Departamento de F\'\i{}sica Te\'orica I, Universidad Complutense, 28040 Madrid, Spain.\2. Departamento de F\'{\i}sica, Universidad de Extremadura, 06071 Badajoz, Spain.\3. Instituto de Biocomputaci\'on y F\'{\i}sica de Sistemas Complejos (BIFI), 50009 Zaragoza, Spain.\par}

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Although there exist in Nature a good number of systems undergoing first order phase transitions, in which the properties of the systems such as energy, pressure or magnetization change roughly, not much is known about the consequences of adding quenched impurities to such systems. Studies on this question are compulsory given that most of natural systems are not pure in some degree. While annealed disorder (impurities are allowed to move) is necessary to describe lots of typical systems that present first order phase transitions (such as water), the quenched disorder approximation is much more appropriate to describe magnetic transitions like the one we study. The question we want to answer is: what happens to a system which undergoes a first-order phase transition on a perfectly pure sample if one increasingly deteriorates the quality of the sample introducing quenched impurities? This is still an open problem in Statistical Mechanics but also one with practical implications in fields such as highly correlated electron systems (as high temperature superconductors or colossal magnetoresistance oxides) where phase coexistence and chemical disorder play crucial roles~\cite{MANGA}).

% We %have partially give answers to this question, see~\cite{OURPRL}.

The question is solved in two-dimensions~\cite{Aize89}: even the most insignificant amount of impurities is enough to switch the phase transition from first-order to second-order. In $D\!=\!3$ an analytical background is provided by the Cardy-Jacobsen conjecture~\cite{Card90}: a critical concentration is expected to exist, $1>p_\mathrm{t}>0$, such that the phase transition is of the first-order for spin concentration $p>p_\mathrm{t}$ and of the second order for $p<p_\mathrm{t}$ (at $p_\mathrm{t}$ one has a {\em tricritical

  point}).  The main objection to this argument is that the

Cardy-Jacobsen conjecture relies on a mapping from the (large number of states) disordered Potts model~\cite{WU} onto the RFIM (two unsolved models in $D=3$).

The problem has already been numerically studied in the past~\cite{Ball00,Chat05}. Large regions of the critical line $T_\mathrm{c}(p)$ were found to be second order. Unfortunately, the study of the tricritical point as well as that of the first-order part of the critical line seemed beyond hope due to two factors: long-tailed probability distribution functions and Exponential Critical Slowing Down (the simulation time of a sample of linear size $L$ grows exponentially with $L^{D-1}$). These factors limited previous works~\cite{Chat05} to $L\leq 25$.

We propose~\cite{OURPRL} an alternative method to perform the sample average (over different impurities distributions), reproducing the correct thermodynamic limit and avoiding diverging-variance probability distribution functions. This method takes the sample average of the system entropy at fixed energy, rather than the sample average of the free-energy at fixed temperature. Moreover, the method requires less accuracy in each sample. On the other hand, we exploit a novel microcanonical Monte Carlo method~\cite{Mart07} which allows to study directly the system entropy. This combination has permitted us to study systems of size up to $L\!=\!128$, also making possible to perform a Finite-Size Scaling study of the tricritical point.

To specify, we study the four states Potts model with site dilution and periodic boundary conditions. We consider quenched impurities, which is sensible for magnetic systems provided the relatively slow dynamic of the impurities (full atoms without magnetic character) compared with which the spin one (electrons of the outermost orbital of each atom showing the magnetic character).

From the latent-heat and the surface tension we found that the location of the tricritical point (i.e. the point where both quantities vanish) shifts quickly to larger $p$ for growing $L$. With this information we were not able to safely allocate the tricritical point. To determine $p_\mathrm{t}$ we consider the correlation-length, making a finite size scaling study of the crossing points of this observable for different lattice sizes. We find that a really small degree of dilution (just a 5\%) is enough to smooth the transition to the point of becoming second order.

We claim that (quenched) disordered first-order transitions do exist in $D\!=\!3$, although the quenched disorder is unreasonably effective in smoothing the transition. Our results have been made possible by the new definition of the quenched average and by the use of a recently introduced microcanonical Monte Carlo method~\cite{Mart07}.

\begin{thebibliography}{99}

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\bibitem{Aize89} M. Aizenman and J. Wehr, Phys. Rev. Lett. {\bf 62,} 2503 (1989); K. Hui and A.N. Berker {\em ibid} {\bf 62,} 2507 (1989).

\bibitem{Card90} J. Cardy and J.L. Jacobsen, Phys. Rev. Lett. {\bf 79,} 4063 (1997); Nucl. Phys. B, {\bf 515}, 701 (1998).

\bibitem{WU} F.Y. Wu, Rev. Mod. Phys. {\bf 54,} 235 (1982).

\bibitem{Ball00} H. G. Ballesteros, L. A. Fern\'andez, V. Martin-Mayor, A. Mu\~noz Sudupe, G. Parisi, J. J. Ruiz-Lorenzo, Phys. Rev. B {\bf 61,} 3215 (2000).

\bibitem{Chat05} C. Chatelain, B. Berche, W. Janke, and P.-E. Berche, Nucl. Phys. B {\bf 719,} 275 (2005).

\bibitem{Mart07} V. Martin-Mayor, Phys. Rev. Lett. {\bf 98,} 137207 (2007).

\bibitem{OURPRL} L.~A.~Fern\'andez, A.~Gordillo-Guerrero, V.~Mart\'{i}n-Mayor and J.~J.~Ruiz-Lorenzo, Phys. Rev. Lett. {\bf 100}, 057201 (2008).

\end{thebibliography}

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