Out-of-equilibrium scaling of the energy density along the critical relaxational flow after a quench of the temperature (2024)

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  Figure 1

  The ratio $R\equiv \xi /L$ along the critical relaxational flow vs $\mathrm{\Theta }\equiv t/L^z$ for $N=4$, at fixed $\mathrm{{\rm Y}}=0.2$ (bottom) and $\mathrm{{\rm Y}}=0.4$ (top). The statistical errors are very small, and practically invisible in the plots. The insets show the large-$L$ convergence for some fixed values of $\mathrm{\Theta }$, plotting the corresponding data vs $L^{-\omega }$, which is the expected behavior of the scaling corrections.

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  Figure 2

  The ratio $R\equiv \xi /L$ along the critical relaxational flow vs $\mathrm{\Theta }\equiv t/L^z$ for $N=3$, at fixed $\mathrm{{\rm Y}}=0.2$ (bottom) and $\mathrm{{\rm Y}}=0.1$ (top). The inset of the bottom figureshows the large-$L$ convergence for some fixed values of $\mathrm{\Theta }$ vs their expected leading $O\left(L^{-\omega }\right)$ suppression. The inset of the top figureshows the data up to a relatively large value $\mathrm{\Theta }=1$, for lattice sizes up to $L=24$, demonstrating the large-time convergence to the corresponding equilibrium value at the critical point, which, in turn, converges to the critical value [55] $R^{*}=0.56404\left(2\right)$ in the large-$L$ limit (indicated by the dashed line).

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  Figure 3

  Out-of-equilibrium FSS of the subtracted energy density $E_s$ along the critical relaxational flow for $N=4$, at fixed $\mathrm{{\rm Y}}=0.2$ (bottom) and $\mathrm{{\rm Y}}=0.4$ (top). The insets show the large-$L$ convergence for some fixed values of $\mathrm{\Theta }$, indicating that the scaling corrections are small, and consistent with $O\left(L^{-\omega }\right)$. The statistical errors are very small, and practically invisible in the plots. Note that the data at $t=0$ do not scale; see Fig.5.

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  Figure 4

  Out-of-equilibrium FSS of the $N=3$ subtracted energy density $E_s$ along the critical relaxational flow, at fixed $\mathrm{{\rm Y}}=0.2$ (bottom) and $\mathrm{{\rm Y}}=0.1$ (top). The inset of the bottom figureshows the large-$L$ convergence for some fixed values of $\mathrm{\Theta }$. The inset of the top figureshows the data up to a relatively large value of $\mathrm{\Theta }$, i.e., $\mathrm{\Theta }=1$, at fixed $\mathrm{{\rm Y}}=0.1$ and for lattice sizes up to $L=24$, demonstrating the large time convergence to the corresponding equilibrium value at the critical point. Note that the data at $t=0$ do not scale, like the $N=4$ cases.

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  Figure 5

  The subtracted energy $E_s$ at $t=0$ for $N=4$ and some values of $\mathrm{{\rm Y}}$ (the errors of the data are practically invisible in the plot). To make evident the expected equilibrium behavior reported in Eqs.(16) and (17), we plot $L^{y_r}{E}_s(t=0)$ vs $L^{\alpha /\nu }$ with $\alpha /\nu =2/\nu -d\approx -0.19$, which is the relative power law of the scaling term with respect to the background contribution at equilibrium. The data show the behavior reported in Eq.(29). The lines show linear fits of the data for the larger lattices, which have acceptable ${\chi }^2$. We stress that this scaling behavior does not match that observed at fixed $\mathrm{\Theta }>0$. Analogous power-law behaviors are observed for $N=3$.

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  Figure 6

  Log-log plot of data of the subtracted energy density, highlighting the behavior at small $\mathrm{\Theta }$, for $N=3$ (top) and $N=4$ (bottom), at $\mathrm{{\rm Y}}=0.2$. The dashed lines show linear fits of the data of $L^{d-y_r}{E}_s$ at small $\mathrm{\Theta }$, i.e., $\mathrm{\Theta }\lesssim 0.002$, to the asymptotic $\mathrm{\Theta }\to 0$ behavior $A{\mathrm{\Theta }}^{-\kappa }+B$ with $\kappa =-\alpha /\left(z\nu \right)$. Analogous results are obtained for other values of $\mathrm{{\rm Y}}$, such as $\mathrm{{\rm Y}}=0.4$ for $N=4$ and $\mathrm{{\rm Y}}=0.1$ for $N=3$.

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