First-order to second-order phase transition changeover and latent heats of $q$-state Potts models in $d$=2,3 from a simple Migdal-Kadanoff adaptation

The changeover from first-order to second-order phase transitions in $q$-state Potts models is obtained at $q_c=2$ in spatial dimension $d=3$ and essentially at $q_c=4$ in $d=2$, using a physically intuited simple adaptation of the Migdal-Kadanoff renormalization-group transformation. This simple procedure yields the latent heats at the first-order phase transitions. In both $d=2$ and 3, the calculated phase transition temperatures, respectively compared with the exact self-duality and Monte Carlo results, are dramatically improved. The method, when applied to a slab of finite thickness, yields dimensional crossover.

Figure 1

From Ref. [33]: (a) Migdal-Kadanoff approximate renormalization-group transformation for the $d=3$ cubic lattice with the length-rescaling factor of $b=2$. (b) Construction of the $d=3,b=2$ hierarchical lattice for which the Migdal-Kadanoff recursion relation is exact.

Figure 2

Calculated transition temperatures $1/J$ of $q$-state Potts models. The top curve is obtained with the conventional Migdal-Kadanoff approximation. In $d=2$, the bottom curve is the exact transition temperatures obtained from self-duality. In $d=3$, the bottom curve is Monte Carlo results [30]. The intermediate curve is obtained with our simply improved Migdal-Kadanoff approximation. First- and second-order phase transitions are given with triangles and squares, respectively. The improved calculation gives the changeover from second to first order exactly (after $q=2$) in $d=3$ and very nearly (after $q=5$ instead of $q=4$) in $d=2$. In the latter case, the changeover can be brought down to $q=4$ by a simple physical argument and calculation, as seen in Fig. 4. Both in $d=2$ and 3, the values of the phase transition temperatures are dramatically improved with the improved calculation and join the exact results for $q\gtrsim 10$ and $q\gtrsim 5$, respectively.

Figure 3

Calculated $q$-state Potts energy densities in $d=2$ and 3. In each panel, the curves are, from right to left, for $q=2,3,4,5,6,7,8,20,50,100$. The latent-heat discontinuities of the first-order phase transitions are shown with the dashed lines. The second-order phase transitions are marked with $\times$.

Figure 4

Calculated energy density in $q=5$ and $d=2$. The left curve uses the $q-0.25$ for the local disorder multiplicity. The correct first-order phase transition is obtained (left curve) by simple, physically motivated adjustment.

Figure 5

Calculated energy densities for a two-dimensional slab of finite thickness $b^n$, for $q=3.$ The system is solved by performing $n$ renormalization-group transformations for $d=3$, reducing the system to $d=2$, followed by infinitely many renormalization-group transformations for $d=2$. Left panel: The temperature variation of the energy density. The curves are, from bottom to top on the left, $n=0,1,2,3$. The curves for $n\ge 3$ coincide to the accuracy of the figure. For the narrow slabs, $n=0,1$, the correlation length at the would-be $d=3$ first-order transition is larger than the slab thickness and the second-order phase transition of $d=2$ occurs. For thicker slabs, $n\ge 2$, the correlation length at the first-order phase transition of $d=3$ is less than the slab thickness and thus this first-order phase transition actually occurs. Right panel: Slab thickness dependence of the energy density. The curves are, from top to bottom, for temperatures $T=J^{-1}=0.5,1.0,1.5,...,5.5$. The first-order transition of $n\ge 2$ is seen by the gap opening. From both panels, it is seen that the latent heat does not depend on the slab thickness for $n\ge 3$, since the correlation length at the first-order phase transition is less than $b^n$.