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Physical Review Letters

Singularities and Scaling Functions at the Potts-Model Multicritical Point

Michael Nauenberg, Douglas J. Scalapino

Published Mar 31, 1980• 170 citations
Publisher

Abstract

Differential renormalization equation for the $q$-state Potts model are proposed, and the critical behavior of the model near $q={q}_{c}$ discussed. The equations give rise to critical and tricritical fixed points which merge at $q={q}_{c}$ when a dilution field becomes marginal, to an essential singularity in the latent heat as a function of $q={q}_{c}$, in accordance with the exact result of Baxter, and, for $q={q}_{c}$, to a logarithm correction to the power-law behavior of the free energy as a function of $T\ensuremath{-}{T}_{c}$.

DOI
10.1103/physrevlett.44.837

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Differential renormalization equation for the qqq-state Potts model are proposed, and the critical behavior of the model near q=qcq={q}_{c}q=qc​ discussed. The equations give rise to critical and tricritical fixed points which merge at q=qcq={q}_{c}q=qc​ when a dilution field becomes marginal, to an essential singularity in the latent heat as a function of q=qcq={q}_{c}q=qc​, in accordance with the exact result of Baxter, and, for q=qcq={q}_{c}q=qc​, to a logarithm correction to the power-law behavior of the free energy as a function of T\ensuremath−TcT\ensuremath{-}{T}_{c}T\ensuremath−Tc​.
A theory of the scaling behavior near the qc=4{q}_{c}=4qc​=4 state multicritical point of the two-dimensional Potts lattice gas model is developed. Proceeding from the assumption that a dilution field becomes marginal at the multicritical point while the thermal and ordering fields are relevant, a set of differential renormalization group (RG) equations for these fields are constructed. Keeping terms through second order we find that these equations are characterized by five universal parameters which we evaluate using exact as well as conjectured results. Based upon these RG equations, we investigate the physical properties of the two-dimensional Potts lattice gas for qqq near qc{q}_{c}qc​. For the pure Potts model with q=qcq={q}_{c}q=qc​ we find logarithmic temperature corrections to the specific heat and the spontaneous magnetization. At Tc{T}_{c}Tc​ we find ln(r)\mathrm{ln}(r)ln(r) corrections to the power law behavior of the spin-spin correlation function. For the dilute Potts model with q=qcq={q}_{c}q=qc​ we find that the latent heat, the discontinuity in the magnetization, and the discontinuity in the coexisting densities vanish with an essential singularity as TTT approaches the multicritical point from the first-order side. Results for q>qcq>{q}_{c}q>qc​ and q<qcq<{q}_{c}q<qc​ are also given.
Critical properties of the QQQ-state Potts model for dimensions $3\ensuremath{\le}d\ensuremath{\le}6$ are calculated by means of Wilson's exact momentum-space renormalization-group equation. The scaling-field method of Golner and Riedel is used to approximate the functional differential equation by a set of 11 ordinary coupled differential equations. For $d=4\ensuremath{-}\ensuremath{\epsilon}$, lines of critical and tricritical Potts fixed points are found as functions of QQQ that annihilate as QQQ approaches a critical value ${Q}_{c}=\frac{2+{\ensuremath{\epsilon}}^{2}}{a+O({\ensuremath{\epsilon}}^{3})}$. For Q>QcQ>{Q}_{c}Q>Qc​, the Potts transition is first order. Along these fixed lines the critical and tricritical exponents (upper and lower sign, respectively) are to leading order: $\frac{1}{\ensuremath{\nu}}=2\ensuremath{-}\frac{1}{6}[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]$, $\frac{\ensuremath{\varphi}}{\ensuremath{\nu}}=\ensuremath{\mp}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}$, and $\ensuremath{\eta}=\frac{{[\ensuremath{\epsilon}\ifmmode\pm\else\textpm\fi{}{({\ensuremath{\epsilon}}^{2}\ensuremath{-}a\ensuremath{\delta})}^{\frac{1}{2}}]}^{2}}{216}+b\ensuremath{\delta}$, where $\ensuremath{\epsilon}=4\ensuremath{-}d$, $\ensuremath{\delta}=Q\ensuremath{-}2$, and $\ensuremath{\delta}\ensuremath{\le}{\ensuremath{\delta}}_{c}=\frac{{\ensuremath{\epsilon}}^{2}}{a}+O({\ensuremath{\epsilon}}^{3})$. While the form of the $\ensuremath{\epsilon}$ and $\ensuremath{\delta}$ dependences is exact, the coefficients aaa and bbb cannot be obtained systematically by $\ensuremath{\epsilon}$ expansion, since the upper critical dimensionality of the Potts model is six when $Q\ensuremath{\ne}2$. In our truncation, a=6.52a=6.52a=6.52 and b=0.065b=0.065b=0.065. The results have been extended to dimensions $3.4\ensuremath{\le}d\ensuremath{\lesssim}4$ by solving the renormalization-group equations numerically. The percolation limit of the Potts model, Q=1Q=1Q=1, is also investigated and the critical exponents ${\ensuremath{\nu}}^{P},{\ensuremath{\varphi}}^{P}$, and ${\ensuremath{\eta}}^{P}$ determined as functions of dimension for $3\ensuremath{\le}d\ensuremath{\le}6$.
The effect of quenched impurities on systems undergoing first-order phase transitions is studied within the framework of the qqq-state Potts model. For large qqq a mapping to the random-field Ising model explains the absence of any latent heat in 2D, and suggests that for d>2d>2d>2 such systems exhibit a tricritical point with an exponent $\ensuremath{\nu}$ related to those of the random-field model by $\ensuremath{\nu}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\nu}}_{\mathrm{RF}}/(2\ensuremath{-}{\ensuremath{\alpha}}_{\mathrm{RF}}\ensuremath{-}{\ensuremath{\beta}}_{\mathrm{RF}})$. In 2D we analyze the model using finite-size scaling and conformal invariance, and find a continuous transition with a ratio $\ensuremath{\beta}/\ensuremath{\nu}$ which varies continuously with qqq, and a weakly varying exponent $\ensuremath{\nu}\ensuremath{\approx}1$. We find strong evidence for the multiscaling of the correlation functions.
We consider the 3-state Potts model in d ≥ 2 dimensions. For d less than the upper critical dimension d crit , the model has a critical and a tricritical fixed point. In d = 2 , these fixed points are described by minimal models, and so are exactly solvable. For d > 2 , however, strong coupling makes them difficult to study and there is no consensus on the value of d crit . We use the numerical conformal bootstrap to compute critical exponents of both the critical and tricritical fixed points for general d . In d = 2 our results match the expected values, and as we increase d we find that the critical exponents of each fixed point get closer until they merge near d crit ≲ 2.5 , which is an upper bound on d crit .
The phase diagram and critical behavior of the spin-1 Baxter-Wu model with a crystal field in two dimensions is explored by renormalization group, conventional finite-size scaling, and conformal invariance techniques. We found that the phase diagram of this model is qualitatively the same as that of the dilute 4-states Potts model, presenting a multicritical point for a finite value of the crystal field, in disagreement with previous work based on finite-size calculations. However, our results indicate that the critical exponents vary continuously along the second-order transition line, differently from the expected behavior of the dilute 4-states Potts model.