solutions are strictly diagonal for $\xi=\zeta=0$. Therefore, the band-diagonal structure may be viewed as a deviation from the diagonal matrix due to finite $\xi$ and $\zeta$.

extracting the time evolution {#SS:Time_evolution}
-----------------------------

Next we discuss how we extract the time evolution from a given classical solution. The band-diagonal structure discussed in the previous section plays a crucial role here.

Let us define an $n\times n$ block $\bar{X}_i(t)$ by $$(\bar{X}_i (t))_{rs}
= (X_{i})_{k+r, k+s}
\quad (r, s=1,2,\ldots,n) \ ,$$ where $k=0,1,\ldots,N_{\rm X}-n$ and the argument $t$ represents the time defined by $$t
=\frac{1}{n}\sum_{r=1}^{n}\alpha_{k+r}  \ .$$ This block matrix $\bar{X}_i(t)$ represents the space structure at time $t$, from which one can obtain the time evolution of the 3-dimensional space. For instance, we define the extent of space at $t$ by $$R^{2}\left(t\right)
=\frac{1}{n}\sum_{i=1}^{3}
\mathrm{tr}\left(\bar{X}_{i} (t)\right)^2 \ ,$$ where the symbol “${\mbox{tr}}$” represents the trace over an $n \times n$ matrix. In Fig. \[fig:R2\], we plot $R^{2}(t)$ for the typical solution discussed in section \[SS:Config\_X\]. We find that $R^{2}(t)$ is roughly symmetric under $t\mapsto -t$ reflecting the symmetry of the model under $A_0 \mapsto - A_0$. In particular, an expanding behavior is seen at $t>0$, which is consistent with the results of simulations [@Nishimura:2019qal] 