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## A. Phase transition points: Analysis of φ 1 

Employing the method in Ref. [13], we find the transi- tion point determined by the order parameter φ$_{1}$ . Note that the case studied in Ref. [13] corresponds to h$_{o}$ = h$_{e}$ = 0 in our setting. To be self-contained, we explain the simulation method [13]: 

Step 1: We prepare for the checkerboard configuration Eq. (37). 

Step 2: We choose a site randomly and increase energy at that site by 1, then toppling begins. 

Step 3: We wait until either the system falls into an ab- sorbing state or all sites have toppled at least once. 

Step 4–1: If the system falls into an absorbing state, we go back to step 2 with the present absorbing state. 

Step 4–2: If all sites have toppled at least once, the sim- ulation is terminated and we go back to step 1. 

The energy density e = E/V at step 4–2 is a random vari- able we are interested in and we calculated its average, denoted by ζ$_{h}$$_{o}$$_{,h}$$_{e}$ , and the standard deviation, denoted by σ$_{h}$$_{o}$$_{,h}$$_{e}$ . 

Simulation results for certain combinations of h$_{o}$ and h$_{e}$ are summarized in Table I. The error of ζ$_{h}$$_{o}$$_{,h}$$_{e}$ is esti- mated as 3 σ$_{h}$$_{o}$$_{,h}$$_{e}$$_{/}$√$_{N}$$_{E}$$_{, where}$ N$_{E}$ is the number of re- alizations for the given parameters. We also studied the case h$_{o}$ = h$_{e}$ = 0 and obtained a consistent result with Ref. [13] (data not shown). As can be seen from Table I, transition points indeed depend on the initial condition. 

The behavior of ζ$_{2}$$_{,}$$_{2}$ is quite distinct from the other cases in two respects. First, ζ$_{2}$$_{,}$$_{2}$ does not show any symp- tom of saturation up to L = 2 12 unlike the other cases, which makes it difficult to estimate the transition point accurately from this type of study. Second, as shown in Fig. 3, the probability distribution of the energy density e is well approximated by a Gaussian for other cases, 

while that for h$_{o}$ = h$_{e}$ = 2 deviates significantly from a Gaussian. 

The configuration Eq. (37) with h$_{o}$ = h$_{e}$ = 2 is actually a recurrent configuration of the BTW model. Now, we will argue that φ$_{1}$ in the infinite-size limit is nonzero for any nonzero ζ$_{0}$ if h$_{x}$ ’s in Eq. (35) form a recurrent configuration of the BTW model. 

Whether a stable configuration of the BTW model is recurrent is determined by the burning algorithm [6]. If all sites in a configuration are burnt by the burning al- gorithm, adding energy to every boundary site by the amount of dissipated energy per toppling there will make all sites topple once and come back to the original con- figuration. Therefore, in the context of the BTW-FES, the burning algorithm implies that the increase of en- ergy density by 4 /L is enough to make the system active if h$_{x}$ ’s form a recurrent configuration. 

Since the boundary is a kind of an infinite percolating cluster, a random energy distribution with small ζ$_{0}$ has no chance to have such a percolating cluster of sites with added energy, in the infinite-size limit. The burning- algorithm argument can at best provide a crude possibil- ity that a negligible density increase might be enough to make the system active. 

A more plausible scenario is provided by Ref. [18], where it is shown that the addition of a finite, but large, amount of energy (note that the density change is still zero) to the configuration (37) with h$_{o}$ = h$_{e}$ = 2 in the infinite L limit is enough for toppling to continue for- 

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