Suppression of discontinuous phase transitions by particle diffusion

We investigate the phase transitions of the $q$-state Brownian Potts model in two dimensions (2D) comprising Potts spins that diffuse like Brownian particles and interact ferromagnetically with other spins within a fixed distance. With extensive Monte Carlo simulations we find a continuous phase transition from a paramagnetic to a ferromagnetic phase even for $q>4$. This is in sharp contrast to the existence of a discontinuous phase transition in the equilibrium $q$-state Potts model in 2D with $q>4$. We present detailed numerical evidence for a continuous phase transition and argue that diffusion generated dynamical positional disorder suppresses phase coexistence leading to a continuous transition.

Figure 1

Sketch illustrating the Brownian Potts model. The filled symbols represent the Brownian particles whose Potts spin states are color coded. Each particle interacts ferromagnetically with all particles within the distance $r_0$. The dashed circle denotes the interaction range of the particle $i$. After spin flips, particles perform a random jump of length $v_0$ to the position denoted by open symbols.

Figure 2

Order parameter (top) and the Binder cumulant (bottom) of the Brownian Potts model for $q=3$ [(a) and (c)] and $q=6$ [(b) and (d)].

Figure 3

Histograms of the order parameter $x=|{\mathit{m}}_v|$ (blue, filled curves) and spin fractions $x=n_{\sigma }$ with $\sigma =1,2,\cdots ,q$ (solid lines). Top row [(a), (b), and (c)] shows the histograms for the equilibrium Potts model and the bottom row [(d), (e), and (f)] for the Brownian Potts model. Data are for $q=6$, $L=128$, and $K\simeq K_c$. Note that $K_c=ln(1+\sqrt{6})\simeq 1.2382$ for the equilibrium six-state Potts model and $K_c\sim 4.08\left(3\right)$ for the Brownian Potts model. The dotted vertical lines indicate the value $x=1/q$.

Figure 4

(a) Order parameter of the Brownian Potts model as function of $L$ for $q=3$ at different coupling constants $K$. [(b)–(h)] Effective exponent $\mathrm{\Theta }$ as a function of $1/L$ near the critical points for different values of $q$.

Figure 5

Scaling plots of $m,\chi$, and $U_4$ at $q=5$ in (a), (c), and (e) and at $q=7$ in (b), (d), and (f) using the numerical estimates listed in Table 1.

Figure 6

(a) Spin-spin correlation function at $t=4^4,...,4^8$ evaluated at the critical point for $q=8$. Data points are aligned along a single curve, which indicates that the correlation function is isotropic. (b) Scaling plots according to Eq. (7) at all values of $2\le q\le 8$. For a better visualization, we shift the data sets vertically by multiplying $2^{q-6}$. All the data are obtained from the ensemble average over 100 samples. The system size is $L=4096$.

Figure 7

(a) Two-time correlation functions at $t=4^1,...,4^5$ evaluated at the critical point for $q=3$ and $L=512$ averaged over 100 samples. (b) Scaling plot according to Eq. (9) with the correlation time exponent $Z_C=2.0$ and $\eta =0.21$.

Figure 8

Ferromagnetic domain (filled blue area) inside a steady-state background. The length scales associated with particle diffusion and spin ordering are represented with the black and red arrows, respectively.

Figure 9

Log-log plots for the peak values of the specific heat. At $q=2$, $C_{\max }\sim lnL$. On the other hand, the curvature for $q>3$ indicates a crossover to a power-law scaling. For comparison, dashed straight lines with slope $2/5$ and 1 are shown, which are representative for the equilibrium Potts model with $q=3$ and 4, respectively.