Phase Transitions in the Blume–Capel Model with Trimodal and Gaussian Random Fields

Abstract

We study the effect of different symmetric random field distributions: trimodal and Gaussian on the phase diagram of the infinite range Blume–Capel model. For the trimodal random field, the model has a very rich phase diagram. We find three new ordered phases, multicritical points like tricritical point (TCP), bicritical end point (BEP), critical end point (CEP) along with some multi-phase coexistence points. We also find re-entrance at low temperatures for some values of the parameters. On the other hand for the Gaussian distribution the phase diagram consists of a continuous line of transition followed by a first order transition line, meeting at a TCP. The TCP vanishes for higher strength of the random field. In contrast to the trimodal case, in Gaussian case no new phase emerges.

Appendix A

1.1 Rate Function for the RFBC

The probability of a spin configuration $C_N$ with magnetization $x_1 = \frac{\sum \limits _i s_i}{N}$ and quadrupole moment $x_2 = \frac{\sum \limits _i s_i^2}{N}$ is proportional to $e^{- \beta {\mathcal {H}}}$, where ${\mathcal {H}}$ is the Hamiltonian given in Eq. 2. This via large deviation principle (LDP) in the limit of $N \rightarrow \infty$ goes as $P(C_N) \sim e^{- N I(x_1, x_2)}$. The function $I(x_1, x_2)$ here is the rate function which is like the generalized free energy functional. To calculate $I(x_1, x_2)$ we use two steps :

• 1. Calculate the rate function $R(x_1, x_2)$ corresponding to the probability $P_{{\mathcal {H}}_{ni}} (C_N) \sim e^{- N R(x_1, x_2)}$. Here ${\mathcal {H}}_{ni}$ is the non-interacting part of the Hamiltonian i.e ${\mathcal {H}}_{ni} = \Delta \sum _i s_i ^2 - \sum _i (h_i+H) s_i$. The function $R(x_1, x_2)$ is calculated using the Gärtner–Ellis (GE) theorem [49]. GE theorem states that $R(x_1, x_2)$ is given by the Legendre–Fenchel transformation of the scaled cumulant generating function $\uplambda (k_1, k_2)$, provided $\uplambda (k_1, k_2)$ is differentiable. The expression of $R(x_1, x_2)$ is

  $$\begin{aligned} R(x_1,x_2)= & {} \sup _{k_1,k_2} \Bigg [x_1k_1+x_2k_2 - \uplambda (k_1, k_2) \Bigg ] \nonumber \\ \end{aligned}$$

  (25)

  The function $\uplambda (k_1, k_2) = \lim \limits _{N \rightarrow \infty } \frac{1}{N} \uplambda _N(k_1, k_2)$ where $\uplambda _N (k_1, k_2)$ is the logarithmic cumulant generating function of $x_1$ and $x_2$ w.r.t the probability $P_{{\mathcal {H}}_{ni}}$. The $\uplambda (k_1, k_2)$ for the random variables $x_1$ and $x_2$ is given by

  $$\begin{aligned} \uplambda (k_1, k_2) = \Bigg < \log (1+2 e^{k_2-\beta \bigtriangleup } \cosh (k_1+ \beta H+ \beta h_i))\Bigg >_h \nonumber \\ \end{aligned}$$

  (26)

  $\langle \rangle _h$ represents the average over the random field distribution. Minimization of the expression $x_1k_1+x_2k_2 - \uplambda (k_1, k_2)$ in Eq. 25 w.r.t $k_1$ and $k_2$ gives the following equations for the supremum ($k_1^*$, $k_2^*$) as a function of $x_1$ and $x_2$

  $$\begin{aligned} x_1= & {} \Bigg < \frac{ 2 e^{k_2^* -\beta \Delta } \sinh (\beta h_i+ \beta H +k_1^*)}{1+ 2 e^{k_2^* -\beta \Delta } \cosh (\beta h_i+ \beta H + k_1^*)}\Bigg >_h \end{aligned}$$

  (27)
  $$\begin{aligned} x_2= & {} \Bigg < \frac{2 e^{k_2^* -\beta \Delta } \cosh (\beta h_i+ \beta H +k_1^*)}{1+ 2 e^{k_2^* -\beta \Delta } \cosh (\beta h_i+ \beta H + k_1^*)}\Bigg >_h \end{aligned}$$

  (28)

• 2. The full rate function of the interacting Hamiltonian can be calculated via tilted LDP [57]. This principle allow us to calculate the rate function $I(x_1, x_2)$ from the old rate function ($R(x_1, x_2)$) using a change in measure by integrating against an exponential of a continuous function $G(x_1, x_2)$ which in our case is the interacting part of the Hamiltonian, $G = \frac{\beta x_1^2}{2}$. The rate function $I(x_1, x_2)$ is given by (see [50, 51] for more details)

  $$\begin{aligned} I(x_1, x_2) = R(x_1, x_2) - \frac{\beta x_1^2}{2} - \inf \limits _{y_1, y_2} \Big (R(y_1, y_2) - \frac{\beta y_1^2}{2} \Big ) \end{aligned}$$

  (29)

  After substituting $R(x_1, x_2)$ we get

  $$\begin{aligned} I(x_1, x_2)= & {} x_1 k_1^* + x_2 k_2^* - \frac{\beta x_1^2}{2} - \,\, \Bigg < \log (1+2 e^{k_2^*-\beta \bigtriangleup } \cosh (k_1^*+ \beta H+ \beta h_i))\Bigg >_h\nonumber \\ \end{aligned}$$

  (30)

  here ($k_1^*, \,\, k_2^*$) are given by the solutions of Eqs. 27 and 28. Minimizing the full rate-function w.r.t the order parameters ($x_1$, $x_2$) we get $k_1^*= \beta m$ and $k_2^*=0$. The variables m and q represent the minimum of $x_1$ and $x_2$ respectively. On substituting $k_1^*$ and $k_2^*$ in Eq. 30 we get the free energy functional to be

  $$\begin{aligned} f(m) = \frac{\beta m^2}{2}- \,\, \Bigg < \log (1+2 e^{-\beta \bigtriangleup } \cosh { \beta (m + H + h_i)} \,\, )\Bigg >_h \end{aligned}$$

  (31)