Phase diagram of the one-dimensional three-state Potts model with an additional mean-field interaction

TL;DR

We address the phase diagram of a one-dimensional three-state Potts model with a competing mean-field interaction by mapping it to a BEG spin-1 model and solving the canonical ensemble. The authors derive a two-parameter phase diagram in K and T with only first-order transitions and three notable points (TP1, TP2, MCP), demonstrating that symmetry breaking is partial and m = 0 in equilibrium. Analytical results include asymptotic forms for the transition lines at large |K| and the explicit coordinates of the MCP; the line to the right of TP2 is obtained analytically, and a large-N counting argument supports the large-negative K limit. The work highlights nontrivial long-range effects in a low-dimensional setting and suggests extensions to more spin states and finite-size analyses.

Abstract

We derive the phase diagram of the one-dimensional three-state Potts model with an additional mean-field interaction in the canonical ensemble. The free energy is obtained by mapping the model onto the spin-$1$ Blume-Emery-Griffiths model and solving it by using an Hubbard-Stratonovich transformation combined with the transfer matrix method. A complex structure with lines of first-order transitions, two triple points and a critical point appears at finite temperature. The phase diagram is two-dimensional, since there are two adjustable parameters, the nearest-neighbour coupling $K$ and the temperature $T$. We show that the phase diagram does not present second-order phase transition lines, due to the fact that the order parameter is not a symmetry-breaking one. Quite remarkably, we are able to determine analytically one of the first-order phase-transition lines. We also prove that, when the nearest-neighbour coupling $K$ is large and negative, the first-order transition temperature becomes asymptotically independent of $K$.

Figures (7)

• Figure 1: The $(K,T)$ phase diagram. In the left plot we show the complete phase diagram, while in the right plot we zoom in the region marked with the box in the left plot. The three points TP1, MCP and TP2 in this plot cannot be conveniently displayed at the scale of the left plot. All lines are first-order transition lines. The points TP1 and TP2 are triple points, with coordinates $(-0.19525,0.36820)$ and $(-0.17681,0.41328)$, respectively. The point MCP, with coordinates $(-0.19345,0.38486)$, is a critical point. The small dashed lines, on the bottom axis of both plots, mark the $K$ values for which we present, in the following figures, the curves of the quadrupole moment $q$ as a function of the temperature $T$, showing the first-order transitions. We remark that all three dashed lines on the left plot are at $K$ values outside the range shown in the right plot, and that the first and second dashed lines on the right plot are for $K$ values between that of TP1 and that of MCP, while the third one is for a $K$ value larger than that of MCP.
• Figure 2: The quadrupole moment $q$ as a function of the temperature $T$ for a fixed value of $K=-0.5$ (left plot) and $K=-0.22$ (right plot), showing the first-order transition with the corresponding jump of $q$. As expected from the phase diagram in figure \ref{['fig1']}, for $K=-0.5$ there is one phase transition, while there are two phase transitions for $K=-0.22$. Also, the plots show clearly that for all temperatures larger than that at the phase transition (or that at the last phase transition if there is more than one), $q$ is constant equal to $\frac{2}{3}$. This occurs for all values of $K$. We finally note that for $T\to 0$ the quadrupole moment tends to $1$ for $K=-0.5$, while it tends to $0$ for $K=-0.22$, coherently with the fact that the ground state has $q=1$ for $K<-\frac{1}{4}$, while it has $q=0$ for $K>-\frac{1}{4}$. For a better visualization the plots, as well the left plot in figure \ref{['fig3']} and the plots in figure \ref{['fig4']}, have a bottom and a top axis corresponding to values of $q$ slightly smaller than $0$ and slightly larger than $1$, although these are obviously unphysical values.
• Figure 3: The quadrupole moment $q$ as a function of the temperature $T$ for a fixed value of $K=-0.19475$. In agreement with the phase diagram, see figure \ref{['fig1']}, here we have three first-order phase transitions by increasing $T$. In the left plot there is the complete curve, while in the right plot there is a zoom of the region around the last phase transition by increasing $T$. This plot has also the purpose to show more clearly the (small) jump of $q$ occurring at the transition.
• Figure 4: The quadrupole moment $q$ as a function of the temperature $T$ for fixed values of $K=-0.18$ (left plot) and $K=-0.17$ (right plot). As expected from the phase diagram in figure \ref{['fig1']}, for $K=-0.18$ there are two phase transitions, while there is one phase transition for $K=-0.17$.
• Figure 5: The quadrupole moment $q$ as a function of the temperature $T$ for a fixed value of $K=-0.1938$, which is slightly smaller than the value at the point MCP. Then, according to the phase diagram in figure \ref{['fig1']}, we have three phase transitions. The right plot is a zoom of a quite narrow range in $q$ and $T$ (such that the space between the dots corresponding to the computed values is now visible), that shows more clearly that also the last transition is first-order, although with a very small jump in $q$.