Finite-size scaling in Ising-like systems with quenched random fields: Evidence of hyperscaling violation

In systems belonging to the universality class of the random field Ising model, the standard hyperscaling relation between critical exponents does not hold, but is replaced with a modified hyperscaling relation. As a result, standard formulations of finite-size scaling near critical points break down. In this work, the consequences of modified hyperscaling are analyzed in detail. The most striking outcome is that the free-energy cost $\Delta F$ of interface formation at the critical point is no longer a universal constant, but instead increases as a power law with system size, $\Delta F\propto L^{\theta }$, with $\theta$ as the violation of hyperscaling critical exponent and $L$ as the linear extension of the system. This modified behavior facilitates a number of numerical approaches that can be used to locate critical points in random field systems from finite-size simulation data. We test and confirm the approaches on two random field systems in three dimensions, namely, the random field Ising model and the demixing transition in the Widom-Rowlinson fluid with quenched obstacles.

Figure 1

FSS in the $d=3$ pure Ising model in the critical regime. The linear dimension $L$ is given in units of the lattice spacing. (a) OPD $P_L\left(m\right)$ obtained at $T=T_{\text{c}}$, $H=H_{\text{c}}=0$, and for several system sizes. (b) The same data plotted versus the scaling variable, with the critical exponents taken from Table ; the data for different $L$’s overlap. (c) Demonstration of the cumulant intersection method to locate $T_{\text{c}}$. Plotted is $U_1$ versus $T$ for several system sizes. At the critical point, the curves for different $L$’s intersect. In the ordered (disordered) phase, $U_1\to 1$ $\left(U_1\to \pi /2\right)$ as $L$ increases, in accord with Eq. (27).

Figure 2

Finite-size effects in the $d=3$ pure Ising model in the ordered phase, where the transition is first order. Plotted is the scaled-and-shifted logarithm of the OPD, $W_L=\text{ln}{P}_L\left(m\right)$, for various system sizes at $T=3.33$, which is well below $T_{\text{c}}$. The peak height corresponds to the interfacial tension $f_{\text{int}}$. Note also the flat region unfolding between the peaks as $L$ increases.

Figure 3

Scaling of the free-energy barrier $\Delta F_L$ in the pure Ising model. By plotting $\Delta F_L$ versus $T$ for various $L$, the critical temperature appears as an intersection point. The horizontal line marks the (universal) value $\Delta F^{\star }\sim 0.9$ for the $d=3$ Ising model.

Figure 4

Schematic plot of the scaling function $\tilde{F}\left(x\right)$, defined by Eq. (39), describing the free-energy barrier $\Delta F_L$ in the RFIM versus the scaling variable $x=t{L}^{1/\nu }$. There occurs a smooth crossover from $\tilde{F}\left(x\right)\propto {\left|x\right|}^{\left(d-1-\theta \right)\nu }$ for $x⪡0$ to $\tilde{F}\left(x\right)\propto 1/{\left|x\right|}^p$ for $x⪢0$ (with $p>\nu \theta$).

Figure 5

Investigation of the typical shape of $P_{L,i}\left(m\right)$ in the RFIM at $T=T_{\text{c}}^{\text{NB}}$ using the symmetry path $l_S$. (a) Histograms $H\left(A\right)$ for various system sizes $L$. Note the logarithmic vertical scale. (b) Typical distribution $W_{L,i}=\text{ln}{P}_{L,i}\left(m\right)$ for $L=14$; for this distribution $A\sim 1$.

Figure 6

Cumulant analysis of the RFIM using the symmetry path $l_S$. For clarity, the curves are shifted by unity, and a logarithmic vertical scale is used. (a) $U_{1,\text{con}}$ versus $T$ for various $L$’s; the arrow marks $T_{\text{c}}^{\text{NB}}$. (b) The disconnected cumulant $U_{1,\text{dis}}$ versus $T$ for various $L$’s. Note the absence of intersections in $U_{1,\text{con}}$, and their presence in $U_{1,\text{dis}}$. This conforms to the modified hyperscaling scenario.

Figure 7

(a) The quenched-averaged distribution $Q_L\left(m\right)$ of the RFIM obtained at $T=T_{\text{c}}^{\text{NB}}$, $L=14$, and using the symmetry path $l_S$. The peak-to-peak distance is proportional to the order parameter $M$, while the peak widths $W$ reflect the sum of thermal and sample-to-sample fluctuations. (b) The leading cumulant of $Q_L\left(m\right)$ versus $T$ for various $L$’s.

Figure 8

(a) Variance $\left[H^2\right]-{\left[H\right]}^2$ of the “tuned” external fields $H_i$ versus $T$ using the path $l_{\Gamma }$ for several system sizes. (b) The variance multiplied by $L^d$.

Figure 9

Investigation of the typical shape of $P_{L,i}\left(m\right)$ in the RFIM using the path $l_{\Gamma }$. Shown are histograms $H\left(U_1\right)$ for various $L$’s obtained at (a) $T=T_{\text{c}}^{\text{NB}}$ and (b) $T=3.35$. The histograms peak at $U_1=1$ implying that most distributions are bimodal. Note the logarithmic vertical scales. (c) Typical distribution $W_{L,i}=\text{ln}{P}_{L,i}\left(m\right)$ obtained for $L=14$ and $T=T_{\text{c}}^{\text{NB}}$. Since $W_{L,i}$ is bimodal, a free-energy barrier $\Delta F_{L,i}$ can be extracted (vertical arrow).

Figure 10

FSS of the free-energy barrier $\Delta F_L$ in the RFIM using the path $l_{\Gamma }$. (a) $\Delta F_L$ versus $T$ for various system sizes. (b) The same data as in (a) but using scaled variables. The validity of the scaling form [Eq. (39)] is confirmed by the collapse of the data from the various system sizes onto a single curve.

Figure 11

FSS in the RFIM using the path $l_{\Gamma }$. (a) $\kappa$ versus $T$ for various $L$’s. Note the intersection point, which yields an estimate of $T_{\text{c}}$. (b) The disconnected cumulant [Eq. (37)] versus $T$ for various $L$’s. The intersections again yield $T_{\text{c}}$. For clarity, the cumulant curves are shifted by unity, and a logarithmic vertical scale is used.

Figure 12

${\mathcal{P}}_L\left(⟨\left|m\right|⟩\right)$ for the RFIM at $T=T_{\text{c}}^{\text{VFB}}$, using the path $l_{\Gamma }$, and for various system sizes.

Figure 13

(Color online) Cumulant plot for the WRM without quenched disorder. Plotted is $U_1$ versus $z_B$ for different $L$’s. The intersection yields $z_{\text{crit}}=0.9377\left(4\right)$ and $U_1^{\star }=1.228\left(5\right)$.

Figure 14

(Color online) Extrapolation of $z_L\left(x\right)$, $x∊\left\{\chi ,{\chi }_4^{-},{\chi }_4^{+}\right\}$, according to Eq. (45) for the WRM with quenched disorder. From the linear fits $z_{\text{crit}}=0.9376\left(5\right)$ is obtained.

Figure 15

(Color online) Cumulant plot for the WRM with quenched obstacles. Plotted is $U_{1,\text{con}}$ [Eq. (35)] versus $z_B$ for different $L$’s. The intersection of system sizes $L<10$ is attributed to a crossover effect from Ising universality to RFIM universality. Curves for $L\ge 10$ no longer intersect at this point indicating that for these system sizes the crossover to RFIM universality has largely completed.

Figure 16

(Color online) Variation of $\kappa$ with $z_B$ for various $L$’s for the WRM with quenched obstacles [analog of Fig. 11a]. The intersection of the curves indicates a critical point around $z_B\sim 1.35$.

Figure 17

(Color online) Scaling plot of the free-energy barrier, according to Eq. (39), for the WRM with quenched obstacles [analog of Fig. 10b].

Figure 18

(Color online) Extrapolation of $z_L\left(x\right)$, $x∊\left\{\left[\chi \right],{\left[{\chi }_4\right]}^{-},{\left[{\chi }_4\right]}^{+}\right\}$, via Eq. (45) for the WRM with quenched disorder. The plot uses $\nu =2.1$ (as obtained in Fig. 17), and predicts $z_{\text{crit}}=1.37\left(2\right)$ in the limit $L\to \infty$.

Figure 19

“Running averages” of the connected and the disconnected susceptibility versus the number of random field samples $K$. The data were obtained for the RFIM using the symmetry path $l_S$, $L=14$, and $T=T_{\text{c}}^{\text{VFB}}$. Note the logarithmic horizontal scale.

Figure 20

Demonstration of an alternative method to extract the free-energy barrier $\Delta F_L$. The data in this figure were obtained for the RFIM using $L=14$. (a) The quenched-averaged free-energy distribution $W_L\left(m\right)$ constructed with the recursion relation of Eq. (A6) at $T=T_{\text{c}}^{\text{VFB}}$; a free-energy barrier $\Delta F_L$ can be meaningfully extracted. (b) The corresponding variation of $\Delta F_L$ versus $T$, compared to the “original” method, where $\Delta F_L$ is averaged over individual samples using the path $l_{\Gamma }$.