Criticality of the random field Ising model in and out of equilibrium: A nonperturbative functional renormalization group description

We show that, contrary to previous suggestions based on computer simulations or erroneous theoretical treatments, the critical points of the random-field Ising model out of equilibrium, when quasistatically changing the applied source at zero temperature, and in equilibrium are not in the same universality class below some critical dimension $d_{DR}\approx 5.1$. We demonstrate this by implementing a nonperturbative functional renormalization group for the associated dynamical field theory. Above $d_{DR}$, the avalanches, which characterize the evolution of the system at zero temperature, become irrelevant at large distance, and hysteresis and equilibrium critical points are then controlled by the same fixed point. We explain how to use computer simulation and finite-size scaling to check the correspondence between in and out of equilibrium criticality in a far less ambiguous way than done so far.

Figure 1

RFIM at zero temperature: Schematic illustration of the hysteresis loop (black) and of the equilibrium curve (blue) in the magnetization ($m$ or $\varphi$) versus applied magnetic field ($H$ or $J$) representation. For the chosen value of the disorder strength, the system has two (symmetric) out-of-equilibrium critical points (black dots) on the ascending and the descending branches, but a first-order transition in equilibrium (the equilibrium critical random-field strength is larger than the out-of-equilibrium one [5, 6, 9]).

Figure 2

Finite-size scaling functions $\widehat{f}\left(\phi \right)$ for the connected susceptibility (a) and $\tilde{f}\left(\phi \right)$ for the disconnected susceptibility (b) for equilibrium and hysteresis criticality, as predicted by the present NP-FRG theory from fixed-point functions in $d=3.6$. (We consider here the ascending branch of the hysteresis loop.) For the hysteresis the maxima are located at a nonzero offset field ${\phi }_0$, as a result of the lack of $Z_2$ symmetry, whereas for equilibrium ${\phi }_0=0$. See Sec. 10 for more details.

Figure 3

Ratio of scaling functions $\tilde{f}\left(\phi \right)/\widehat{f}{\left(\phi \right)}^2$ for equilibrium and hysteresis criticality, as predicted by the present NP-FRG theory from the fixed-point dimensionless second cumulant of the renormalized random field in $d=3.6$. See Sec. 10 for more details.

Figure 4

Critical behavior on the ascending branch of the hysteresis curve in $d=5.5$: RG flows of the anomalous dimensions ${\eta }_k$ and ${\overline{\eta }}_k$ starting from a variety of initial non-$Z_2$ cuspy conditions. By dichotomy one can fine tune the initial conditions to reach the fixed point as $s=ln(k/\mathrm{\Lambda })\to -\infty$. Except for a short transient regime the two anomalous dimensions become indistinguishable (and are then equal to their equilibrium counterparts). The arrow indicates the direction of the RG flow.

Figure 5

Critical behavior on the ascending branch of the hysteresis curve in $d=5.5$: (a) Dimensionless second cumulant ${\breve{\delta }}_k(\phi ,0)$ for a variety of values of the RG time $s=ln(k/\mathrm{\Lambda })$ and initial non-$Z_2$ cuspy conditions that are fine tuned to lead to the fixed point. (b) Same for the cusp amplitude ${\breve{\delta }}_{k,\mathrm{cusp}}\left(\phi \right)$. The initial cusp amplitude was put to a constant value of 1 and is not shown for practical reasons. The green curve is for $s=-1$ and the amplitude has already dropped by a factor of 200. The arrows indicate the direction of the RG flow.

Figure 6

Critical behavior on the ascending branch of the hysteresis curve for $d<d_{DR}$: Fixed-point functions $j_{*}\left(\phi \right), z_{*}\left(\phi \right), {\breve{\delta }}_{*}(\phi ,0)$, and ${\breve{\delta }}_{*,\mathrm{cusp}}\left(\phi \right)$ plotted versus $\phi -{\phi }_0$ for several values of $d$ between 5 and 3.5.

Figure 7

Comparison of the equilibrium and hysteresis fixed-point functions $j_{*}\left(\phi \right)$ (a) and ${\breve{\delta }}_{*}(\phi ,0)$ (b) for $d=4$ and $d=5$ [versus $\phi -{\phi }_0$ with ${\phi }_0=0$ and $j_{*}\left({\phi }_0\right)=0$ for equilibrium].

Figure 8

Anomalous dimensions $\eta$ and $\overline{\eta }$ versus $d$ for the equilibrium and hysteresis critical fixed points. This also includes the extrapolated points for hysteresis obtained from the $\lambda$-modified FRG equations as described in the text. The symbols represent the results of ground-state computations for equilibrium [64, 65, 66] and computer simulations for hysteresis [6]. (The error bars are not shown but one should note that for hysteresis the uncertainty on $\eta$ is very large [6].)

Figure 9

Solution of the $\lambda$-modified FRG equations (see text) in $d=3.55$: Fixed-point function ${\breve{\delta }}_{*}(\phi ,0|\lambda )$ for several values of the parameter $\lambda \in [0,1]$.

Figure 10

Correlation length exponent $\nu$ versus $d$ for equilibrium and hysteresis critical points. This also includes the extrapolated points for hysteresis obtained from the $\lambda$-modified FRG equations as described in the text. The symbols represent the results of ground-state computations for equilibrium [64, 65, 66] and computer simulations for hysteresis [6, 40]. (The error bars are not shown, but one should note that the uncertainty on the value of $\nu$ for hysteresis, which varies between 1.2 [40] and 1.4 [6] in $d=3$, is very large.)

Figure 11

Dynamical critical exponent $z$ versus $d$ for the hysteresis critical point(s). The symbols represent the results of a computer simulation [6] (the error bars are not shown but the uncertainty is quite large). Note that close to $d=6$ when dimensional reduction applies, $z$ should be given by the $\epsilon$ expansion of the pure model: The result [67] obtained at order $\mathrm{O}({\epsilon }^3$) is shown as the red dashed line.

Figure 12

Anomalous dimension $\eta \left(\lambda \right)$ obtained from the solution of the $\lambda$-modified FRG equations for several values of $d$ between 3 (top) and 4 (bottom).

Figure 13

Correlation length exponent $\nu \left(\lambda \right)$ obtained from the solution of the $\lambda$-modified FRG equations for several values of $d$ between 3 (top) and 4 (bottom).