Crossover Scaling of Binder Cumulant and its application in Non-reciprocal Sandpiles

Abstract

In this letter, we unveil a robust, pre-asymptotic scaling regime for the Binder cumulant $U_L$, a central finite-size scaling tool, demonstrating $U_L\sim N^{-1} |t|^{-dν}$ (disordered phase) and $\frac{2}{3}-U_L\sim N^{-1} |t|^{-dν}$ (ordered phase), with $t$ being the reduced control parameter, and $N$, $d$, $ν$ represent the total number of sites, the dimensionality, and correlation length exponent, respectively. Leveraging this result, we resolve a fundamental question on the stability of universality classes under the breaking of microscopic reciprocity. For the conserved Manna sandpile, we show that reciprocal biases preserve its universality class, merely shifting the critical point. In striking contrast, any non-reciprocal interaction acts as a relevant perturbation, decisively driving the system's critical exponents to flow from their non-mean-field values towards the mean-field related ones. This flow establishes non-reciprocity as a generic mechanism inducing mean-field criticality in conserved, non-equilibrium systems.

Figures (11)

• Figure 1: Discovery and validation of the pre-asymptotic Binder cumulant scaling in the 2D Ising model and Ising model on complete graph (CG). The results indicate that $U_L\sim |t|^{- d \nu}$ for $T>T_c$ and $\frac{2}{3}-U_L \sim |t|^{-d \nu}$ for $T<T_c$, where $d\nu=2$ for the 2D Ising model and $d_c \nu_{\text{MF}}=2$ for Ising model on complete graph (mean-field results). The dash lines represent the mean-field values of $\frac{2}{3}-U_{L\rightarrow \infty}$ and $U_{L\rightarrow \infty}$ at the critical point (see Eq. (8)).
• Figure 2: (a) The log-log plot of $\frac{2}{3}-U_L$v.s.$\rho-\rho'$ for $L=256$. By adjusting $\rho'$, we obtain the critical density $\rho_c$ when $\frac{2}{3}-U_L$ exhibits power-law behavior verse $\rho-\rho'$. (b) Critical density $\rho_c$ versus bias strength $\delta$ for Reciprocal Bias (RB, blue bullet), Non-reciprocal Bias A (NR-A, red down triangle), and Non-reciprocal Bias A (NR-B, green up triangle).
• Figure 3: The effective exponents (a) $\nu_{\text{eff}}$ and (b) $\beta_{\text{eff}}$ for different $\delta$ and bias protocols. Horizontal lines mark the 2D Manna class ("..- -"), the mean-field related values (".- -"), and exponents for 1D Manna model ("- -"), respectively. Reciprocal bias (RB) preserves the Manna universality class (except $\delta=0.25$ where the system shrink into the 1D conserved Manna model), while both non-reciprocal protocols (NR-A and NR-B) drive a flow to mean-field criticality.
• Figure 4: The log-log plot of $\chi'$v.s.$|t|$ for NR-A (a) and NR-B (b) with $L=256$ and different $\delta$. It shows that the exponent of fluctuation $\gamma'$ reduces to $0$ when increasing $\delta$, which indicating that the non-reciprocal interactions suppressing large scale fluctuations.
• Figure 5: Log-log plot of $\frac{2}{3}-U_L$ and $U_L$v.s.$|t| L^{1/d\nu}$ for three-dimensional Ising model (Left) and 2D site percolation system (Right).