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arXiv:1505.05271 · Physical Review E

Critical phenomena of a single defect

Xintian Wu, Yangyang Zhang

Published Sep 8, 2015• 9 citations
arXivHTMLPDF

Abstract

We consider the critical system with a point defect and study the variation of thermodynamic quantities, which are the differences between those with and without the defect. Within renormalization group theory, we show generally that the critical exponent of the internal energy variation is the specific heat exponent of a pure system, and the critical exponent of the heat capacity variation is that for the temperature derivative of specific heat of a pure system. This conclusion is valid for the isotropic systems with a short-range interaction. As an example we solve the two-dimensional Ising model with a point defect numerically. The variations of the free energy, internal energy, and specific heat are calculated with the bond propagation algorithm. At the critical point, the internal energy variation diverges with the lattice size logarithmically and the heat capacity variation diverges with size linearly. Near the critical point, the internal energy variation behaves as ln|t| and the heat capacity variation behaves as |t|^{-1}, where t is the reduced temperature.

DOI
10.1103/physreve.92.032108

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We consider the critical system with a point defect and study the variation of thermodynamic quantities, which are the differences between those with and without the defect. Within renormalization group theory, we show generally that the critical exponent of the internal energy variation is the specific heat exponent of a pure system, and the critical exponent of the heat capacity variation is that for the temperature derivative of specific heat of a pure system. This conclusion is valid for the isotropic systems with a short-range interaction. As an example we solve the two-dimensional Ising model with a point defect numerically. The variations of the free energy, internal energy, and specific heat are calculated with the bond propagation algorithm. At the critical point, the internal energy variation diverges with the lattice size logarithmically and the heat capacity variation diverges with size linearly. Near the critical point, the internal energy variation behaves as ln|t| and the heat capacity variation behaves as |t|^{-1}, where t is the reduced temperature.
We discuss the universality classes of the local critical behavior of a bulk system near a free surface and near an internal plane of defects. A phase diagram and plausible renormalization-group flow properties for a general model which encompasses both cases are described qualitatively. Several of the crossover exponents are related to bulk critical exponents or semi-infinite surface critical exponents. The picture which is proposed accomodates systems with nonuniversal local critical exponents near a plane of defects, such as the two-dimensional Ising model near a defect line, in a natural way. Necessary conditions for nonuniversal local critical exponents and for logarithmic corrections are derived. The singularities of the energy densities near a free surface and near an internal defect plane on approaching the bulk critical temperature are analyzed.
We study the energy fluctuations in 3D Ising model near the phase transition point. Specific heat is a relevant quantity which is directly related to the mean squared amplitude of the energy fluctuations in the system. We have made extensive Monte Carlo simulations in 3D Ising model to clarify the character of the singularity of the specific heat Cv based on the finite-size scaling of its maximal values Cvmax depending on the linear size of the lattice L. An original iterative method has been used which automatically finds the pseudocritical temperature corresponding to the maximum of Cv. The simulations made up to L ≤ 128 with application of the Wolff's cluster algorithm allowed us to verify the possible power-like as well as logarithmic singularity of the specific heat predicted by different theoretical treatments. The most challenging and interesting result we have obtained is that the finite-size scaling of Cvmax in 3D Ising model is well described by a logarithmic rather than power-like ansatz, just like in 2D case. Another modification of our iterative method has been considered to estimate the critical coupling of 3D Ising model from the Binder cumulant data within L ε [96; 384]. Furthermore, the critical exponent β has been evaluated from the simulated magnetization data within the range of reduced temperatures t ≥ 0.000086 and system sizes L ≤ 410.
The two-dimensional Ising model with a slit is studied. The slit free energy is defined, in which the bulk term, edge terms, and corner terms other than that of the slit are canceled. The bond propagation algorithm is used to calculate the free energy, internal energy, and heat capacity numerically. At the critical point, the slit free energy has a logarithmic correction L^{-1}lnL, in addition to the correction lnL, where L is the typical length of the system. The fitted coefficients of these terms agree with the conformal field theory very well. The integral constant and extrapolation length, which are pending in the conformal field theory, are obtained. In the slit internal energy, the leading term is proportional to L. There is a correction term, ln^{2}L, that has a geometry-independent coefficient, the exact value of which is conjectured to be -sqrt[2]/(2π^{2}). The leading term in the slit heat capacity is proportional to L^{2}. There is also a ln^{2}L term; however, its coefficient depends on the geometry. In the infinitely long strip limit, its exact value is conjectured to be 5/π^{2}. Near the critical point, for different size lattices, the rescaled slit free energies, internal energies, and heat capacities collapse onto the corresponding universal curves. According to the scaling and conformal field theory, the leading term of the slit free energy should be -c/8ln|t| for L|t|≫1 and |t|≪1, where t is the reduced temperature. The amplitude c/8 is universal, and c is the central charge. Our results validate these predictions qualitatively. For L|t|≫1 and |t|≪1, the slit internal energy and heat capacity scale as t^{-1} and t^{-2}, respectively.
In this paper, the influence of point defects such as vacancies on the thermodynamic properties of the two-dimensional Ising model is investigated. The main attention is paid to comparing two approaches to describing the system: the traditional analysis of internal energy E and a more complete analysis of the Helmholtz free energy F and entropy S. Using Monte Carlo methods - the Metropolis algorithm and the Wang - Landau algorithm - the temperature dependences of the changes in these quantities upon the introduction of defects are calculated. It is established that the interaction between defects is attractive with a effective radius of d ~ 3. It is shown that in the phase transition region, the free energy behaves as a continuous monotonically decreasing function of temperature, while the internal energy exhibits a singularity caused by the behavior of entropy. The possible dynamics of defects is investigated by comparing the energy of different configurations; a tendency towards defect clustering at temperatures above the critical one is observed. A comparative analysis of the accuracy and computational efficiency of both approaches is carried out.
Critical amplitudes ${A}_{\ifmmode\pm\else\textpm\fi{}}$ associated with the temperature (t) variation of the heat capacity (C\ensuremath{\sim}${A}_{\ifmmode\pm\else\textpm\fi{}}$\ensuremath{\Vert}t${\ensuremath{\Vert}}^{\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$) are analyzed by means of renormalization-group techniques in both position and momentum spaces. We describe a mechanism according to which the amplitudes ${A}_{\ifmmode\pm\else\textpm\fi{}}$ diverge as the critical exponent \ensuremath{\alpha} approaches a nonpositive integer. In between two consecutive divergences at least one amplitude vanishes at least once. The coefficient P in the expansion A+{A}_{+}A+​/${A}_{\mathrm{\ensuremath{-}}}$\ensuremath{\approxeq}1-P\ensuremath{\alpha}+O(${\ensuremath{\alpha}}^{2}$) is computed by means of \ensuremath{\epsilon} expansion and Migdal-Kadanoff renormalization-group technique. Systems for which the critical exponent \ensuremath{\alpha} is negative but larger than -1 exhibit either a cusped heat capacity if A+{A}_{+}A+​/${A}_{\mathrm{\ensuremath{-}}}$>0 or a smooth maximum in the heat capacity at a temperature other than the critical one (T\ensuremath{\ne}Tc{T}_{c}Tc​) and an infinite slope at Tc{T}_{c}Tc​ if A+{A}_{+}A+​/${A}_{\mathrm{\ensuremath{-}}}$0. Implications of this observation for the interpretation of experiments on random-bond systems such as ${\mathrm{Fe}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$Znx{\mathrm{Zn}}_{\mathrm{x}}Znx​F2{\mathrm{F}}_{2}F2​ are discussed.
Based on the foundations of thermodynamics and the equilibrium conditions for the coexistence of two phases in a magnetic Ising-like system, we show, first, that there is a critical point where the isothermal susceptibility diverges and the specific heat may remain finite, and second, that near the critical point the entropy of the system, and therefore all free energies, do obey scaling. Although we limit ourselves to such a system, we elaborate about the possibilities of finding universality, as well as the precise values of the critical exponents using thermodynamics only.