Critical exponents from cluster coefficients

For a large class of repulsive interaction models, the Mayer cluster integrals can be transformed into a tridiagonal real symmetric matrix $R_{mn}$, whose elements converge to two constants. This allows for an effective extrapolation of the equation of state for these models. Due to a nearby (nonphysical) singularity on the negative real $z$ axis, standard methods (e.g., Padé approximants based on the cluster integrals expansion) fail to capture the behavior of these models near the ordering transition, and, in particular, do not detect the critical point. A recent work [E. Eisenberg and A. Baram, Proc. Natl. Acad. Sci. U.S.A. 104, 5755 (2007)] has shown that the critical exponents $\sigma$ and ${\sigma }^{\prime }$, characterizing the singularity of the density as a function of the activity, can be exactly calculated if the decay of the $R$ matrix elements to their asymptotic constant follows a $1/n^2$ law. Here we employ renormalization group (RG) arguments to extend this result and analyze cases for which the asymptotic approach of the $R$ matrix elements toward their limiting value is of a more general form. The relevant asymptotic correction terms (in RG sense) are identified, and we then present a corrected exact formula for the critical exponents. We identify the limits of usage of the formula and demonstrate one physical model, which is beyond its range of validity. The formula is validated numerically and then applied to analyze a number of concrete physical models.

Figure 1

(Color online) Hard hexagons matrix elements, fitted to form (2), with $A=\sqrt{125}/4$, $a=0.0027$, $a^{\prime }=-0.063$, $B=5.5$, $b=0.627$, $b^{\prime }=0.129$, and $q=0.36$.

Figure 2

(Color online) Critical exponents as measured from equation of state (19), compared with exact prediction (3) for various choices of $\text{cos}\left(qn\right)/n$ corrections.

Figure 3

(Color online) Triangular N2 matrix elements. The difference $A_{n+1/2}-B_n+A_{n+1/2}$ extrapolates to 0.107(1), much lower than $1/z_t=0.1726$. The slope with respect to $n^{-2}$ is 3.61(1), much larger than $A/4$. These two observations are consistent with a spectrum edge shift.

Figure 4

(Color online) N4 matrix elements. The diagonal terms are fitted to functional form (2): $B=17.0121$, $b=0.19$, $b^{\prime }=0.029$, and $q=0.295$. The off-diagonal term fit well Eq. (1) with an added cubic correction $a^{″}/n^3$: $2A=17.0316$, $2a=1.634$, $2a^{\prime }=0$, and $2a^{″}=0.0957$.

Figure 5

(Color online) N5 matrix elements. Diagonal and off-diagonal terms fit well [Eq. (1)] with an added cubic correction $b^{″}/n^3$ $\left(a^{″}/n^3\right)$: $B=20.547$, $b=-0.0166$, $b^{″}=-0.706$. $2A=20.553$, $2a=2.28$, and $2a^{″}=0.143$.