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Physical review. B, Condensed matter

Renormalization-group analysis of heat-capacity critical amplitudes

Scott I. Chase, Miron Kaufman

Published Jan 1, 1986• 23 citations
Publisher

Abstract

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 ${A}_{+}$/${A}_{\mathrm{\ensuremath{-}}}$\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 ${A}_{+}$/${A}_{\mathrm{\ensuremath{-}}}$>0 or a smooth maximum in the heat capacity at a temperature other than the critical one (T\ensuremath{\ne}${T}_{c}$) and an infinite slope at ${T}_{c}$ if ${A}_{+}$/${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}}$${\mathrm{Zn}}_{\mathrm{x}}$${\mathrm{F}}_{2}$ are discussed.

DOI
10.1103/physrevb.33.239

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Critical amplitudes A\ifmmode±\else\textpm\fi{A}_{\ifmmode\pm\else\textpm\fi{}}A\ifmmode±\else\textpm\fi​ associated with the temperature (t) variation of the heat capacity (C\ensuremath{\sim}A\ifmmode±\else\textpm\fi{A}_{\ifmmode\pm\else\textpm\fi{}}A\ifmmode±\else\textpm\fi​\ensuremath{\Vert}t\ensuremath∥\ensuremath−\ensuremathα{\ensuremath{\Vert}}^{\mathrm{\ensuremath{-}}\ensuremath{\alpha}}\ensuremath∥\ensuremath−\ensuremathα) 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±\else\textpm\fi{A}_{\ifmmode\pm\else\textpm\fi{}}A\ifmmode±\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\ensuremath−{A}_{\mathrm{\ensuremath{-}}}A\ensuremath−​\ensuremath{\approxeq}1-P\ensuremath{\alpha}+O(\ensuremathα2{\ensuremath{\alpha}}^{2}\ensuremathα2) 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\ensuremath−{A}_{\mathrm{\ensuremath{-}}}A\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\ensuremath−{A}_{\mathrm{\ensuremath{-}}}A\ensuremath−​0. Implications of this observation for the interpretation of experiments on random-bond systems such as {\mathrm{Fe}}_{1\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Zn}}_{\mathrm{x}}$${\mathrm{F}}_{2} are discussed.
The real-space renormalization-group transformation based on the cumulant expansion is modified for systems with free-surface boundary conditions. A discussion is given of the accuracy and limitations of the method and of the extension to higher orders in the cumulant expansion. Free energies and heat capacities of ferromagnetic strips ($n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$) and slabs ($n\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}\ifmmode\times\else\texttimes\fi{}\ensuremath{\infty}$) are determined over a wide temperature range. Exact results are obtained by a transfer matrix technique for strips with n< 8n<~8n< 8, and compared to the renormalization results. The critical or pseudocritical behavior is found to approach the bulk limit as $n\ensuremath{\rightarrow}\ensuremath{\infty}$ according to a simple power law $|{T}_{c}(n)\ensuremath{-}{T}_{c}(\ensuremath{\infty})|\ensuremath{\propto}{n}^{\ensuremath{-}\ensuremath{\lambda}}$, where the shift exponent $\ensuremath{\lambda}$ is equal to the reciprocal of the critical length exponent. Renormalization equations are used for the first time to calculate the heat capacity of the three-dimensional Ising model over a wide temperature range. In addition, the effect of surface anisotropy is considered for ferromagnetic strips, and procedures are outlined for obtaining analytic derivatives of the free energy with respect to the renormalized parameters.
The critical temperature, thermal and magnetic exponents, and the graphs of dTeffdl\frac{d{T}_{\mathrm{eff}}}{\mathrm{dl}}dldTeff​​ versus Teff{T}_{\mathrm{eff}}Teff​ (where Teff{T}_{\mathrm{eff}}Teff​ denotes "effective" temperature of the system and lll the renormalization-group parameter) for the d=3d=3d=3, n=2,3n=2, 3n=2,3 models are obtained using the Migdal-Kadanoff scheme. Results of finite-lattice approximations for the d=2d=2d=2, n=2n=2n=2 model are also presented.
The extent to which the renormalization of critical-point behavior should be visible experimentally is investigated on the basis of detailed numerical calculations for a three-dimensional soluble model (a mobile-electron Ising ferromagnet). If a dilution parameter xxx is defined such that the change in critical temperature from the "pure" or unrenormalized system is $|{T}_{c}(x)\ensuremath{-}{T}_{c}^{0}|=\frac{x{T}_{c}}{f}$, where $f=0.6\ensuremath{-}0.9$, then we conclude that the effective exponents ${\ensuremath{\beta}}_{\mathrm{fit}}(x)$ and ${\ensuremath{\gamma}}_{\mathrm{fit}}(x)$ which will be observed experimentally, vary roughly as ${\ensuremath{\beta}}_{\mathrm{fit}}\ensuremath{\approx}\ensuremath{\beta}+x\ensuremath{\Delta}{\ensuremath{\beta}}_{X}$ and ${\ensuremath{\gamma}}_{\mathrm{fit}}\ensuremath{\simeq}\ensuremath{\gamma}+\frac{1}{5}x\ifmmode\times\else\texttimes\fi{}(1+2{x}^{2})\ensuremath{\Delta}{\ensuremath{\gamma}}_{X}$. Here $\ensuremath{\beta}$ and $\ensuremath{\gamma}$ are the ideal exponents for the order parameter and total fluctuation or susceptibility of the pure system, while $\ensuremath{\Delta}{\ensuremath{\beta}}_{X}={\ensuremath{\beta}}_{X}\ensuremath{-}\ensuremath{\beta}$ and $\ensuremath{\Delta}{\ensuremath{\gamma}}_{X}={\ensuremath{\gamma}}_{X}\ensuremath{-}\ensuremath{\gamma}$, in which ${\ensuremath{\beta}}_{X}=\frac{\ensuremath{\beta}}{(1\ensuremath{-}{\ensuremath{\alpha}}^{\ensuremath{'}})}$ and ${\ensuremath{\gamma}}_{X}=\frac{\ensuremath{\gamma}}{(1\ensuremath{-}\ensuremath{\alpha})}$ are the fully renormalized exponents, while $\ensuremath{\alpha}$ and ${\ensuremath{\alpha}}^{\ensuremath{'}}$ (assumed positive) describe the divergence of the specific heats of the pure system. [Theoretically the true limiting asymptotic behavior at the transition is described by $\ensuremath{\beta}(x)={\ensuremath{\beta}}_{X}$ and $\ensuremath{\gamma}(x)={\ensuremath{\gamma}}_{X}$ for all x>0x>0x>0.] The renormalized specific heats are found to be sensitive to xxx but their true renormalized behavior is not evident until $x\ensuremath{\gtrsim}0.3$. Various techniques of data analysis such as logarithmic, semilogarithmic, Heller-Benedek, and Kouvel-Fisher plots have been tested.
A cumulant expansion is used to calculate the transition temperature of Ising models with random-bond defects. For a concentration, x, of missing interactions in the simple-square Ising model the author finds -T c -1 dT c /dx mod x=0 =1.329 compared with the mean-field value of one. If the interactions are independent random variable with a width delta J/J identical to epsilon , the result is -T c -1 dT c /d epsilon 2 mod epsilon =0 =0.312 compared with the mean-field results of zero. An approximation yields the specific heat in the critical regime as C approximately C 0 /(1+x gamma 2 C 0 ), where gamma is a constant and C 0 is the unperturbed specific heat at a renormalized temperature. Thus, the specific heat divergence is broadened over a temperature interval Delta T, with Delta T/T c approximately x (1 alpha )/, where alpha is the critical exponent for the specific heat, and a maximum value of order x -1 is attained. Heuristic arguments show that this smoothing effect occurs if alpha >0.