Inhomogeneous thermal quenches

We describe holographic thermal quenches that are inhomogeneous in space. The main characteristic of the quench is to take the system far from its equilibrium configuration. Except in special extreme cases, the problem has no analytic solution. Using the numerical holography methods, we study different observables that measure thermalization such as the time evolution of the apparent horizon, two-point Wightman function and entanglement entropy (EE). Having an extra nontrivial spacial direction allows us to study this peculiar generalization since we categorize the problem based on whether we do the measurements along this special direction or perpendicular to it. Exciting new features that are absent in the common computations appear in the literature; the appearance of negative EE valleys surrounding the positive EE hills and abrupt quenches that occupy the whole space at their universal limit are some of the results of this paper. Physical explanation is given, and connections to Cardy’s idea of thermalization are discussed.

Figure 1

Figure (a) is the profile of $p_0$ that is being sent into the black hole. The rest of the plots are time evolutions of variations in the radial position of the horizon. In (b), (c), (d) and (e), plots are drawn for a fixed value of $\sigma =\sqrt{L_x}$ with $L_x=10$ the length of the domain in the $x$ direction. The varying parameters are correspondingly $\alpha \in \{1,\frac{1}{2},\frac{1}{4},\frac{1}{8}\}$. In (f), these parameters are $\alpha =1$ and $\sigma =\sqrt{1.5L_x}$. The interpolations are based on $N_x=N_{\rho }=20–30$, the number of Chebyshev points along the inhomogeneous direction $x$ and radial direction $\rho$. The number of time steps used for the fourth-order Runge-Kutta varies between 7810 and 17,560.

Figure 2

The disturbance drawn in red pen is that of a Gaussian function, representing the inhomogeneity. We are interested in the correlation of points off this plane, i.e. points A and B in case I. Similarly, in case II, the correlation between C and D will be studied. Note the resemblance of the setup to the elliptic flow in heavy-ion collisions.

Figure 3

Time evolution of the two-point Wightman functions for operators with large conformal dimensions. Figures in (a), (b), (c) and (d) are plotted for $\alpha \in \{1,\frac{1}{2},\frac{1}{4},\frac{1}{8}\}$ and $\sigma =\sqrt{L_x}$. The interpolation of points are based on $N_x=N_{\rho }=20-30$ along the inhomogeneity direction $x$ and radial direction $\rho$. The number of time steps for 4 step Runge–Kutta methods (RK4) is $7810-17$, 560.

Figure 4

Changing the value of $\sigma$ from $\sqrt{L_x}$ to $\sqrt{1.5L_x}$ from left to right causes the distributions to rescale. This factor must be a nontrivial function of the dynamics under study. On the left-hand side $N_x=N_{\rho }=20$ and on the right-hand side $N_x=N_{\rho }=30$ Chebyshev points have been used.

Figure 5

Time evolution of the two-point Wightman function for operators with large conformal dimension. In case II, the correlations are measured by an observer along the plane of reactions. Plots in (a), (b), (c) and (d) are for a fixed value of $\sigma =\sqrt{L_x}$ while varying $\alpha \in \{1,\frac{1}{2},\frac{1}{4},\frac{1}{8}\}$. Instead, in (e), $\alpha =1$ with $\sigma =\sqrt{1.5L_x}$. All these figures are deduced for geodesics with the deepest bulk penetration which is given by the choice ${\rho }_m=0.999{\rho }_h$ in our setup. Other parameters of the simulations are similar to ones used in the previous sections.

Figure 6

Plots of the time evolution of the variation of the entanglement entropy at $\mathcal{O}(l^2)$. In case I, the correlating region is orthogonal to the plane of reaction. (a) is the source on the boundary, and (b)–(e) are the corresponding plots for the EE as we vary ${\rho }_m$ for fixed $\alpha =1$ and $\sigma =L_x$. The numerical setup is identical to the previous sections.

Figure 7

Corresponding plots for the EE as we reduce $\alpha$ in (a)–(c) for $\sigma =\sqrt{L_x}$. In (d)–(f), we increase $\sigma$ with fixed $\alpha =1$. We are also assuming ${\rho }_m=0.999{\rho }_h$ and $L_x=10$ in the above plots.

Figure 8

In the above figures, time evolutions of $S_{\mathrm{\Sigma }(2)}$ for case II, are depicted. From (a)–(d), we increase the value of ${\rho }_m$ to reach the maximum thermalization. Fixed tuning parameters such as $\alpha =1$ and $\sigma =\sqrt{L_x}$ together with $N_x=N_{\rho }=20$ Chebyshev points have been used. The number of time steps for the fourth-order Runge-Kutta has been 7810.

Figure 9

Plots of the time evolution of the entanglement entropy in case II. In (a)–(c), the value of $\alpha$ has been reduced, while in (d)–(f), we are increasing the tuning parameter $\sigma$. The numerical setup is identical to the last figure.

Figure 10

A tensor product grid; there are two spacial directions. $x$ is the direction of the inhomogeneity, and $\rho$ is the bulk radius. The numbers at each site represent the lexicographic representation of the grid points while doing the operation as a tensor grid.

Figure 11

Plots of $\varphi (\tau ,\rho ,x)$ at two specific times. On the left-hand sides, the quench has not been switched on. Specifically, these plots show the configuration at $\tau =-3.75$. At some time long after the quench, for instance, $\tau =12$, the profiles for $\varphi (\tau ,\rho ,x)$ are shown on the right-hand sides. Dimensions of lattices from the first to the last row are respectively given by $20\times 20$, $30\times 30$, $40\times 40$ and $50\times 50$. Fixed parameters are $\alpha =1$ and $\sigma =\sqrt{L_x}$.

Figure 12

The effect of the numerical instabilities coming from the RK method shown for the two-point function. Here, in both diagrams, $N_x=N_{\rho }=20$. The time step for the left-hand diagram is $\mathrm{\Delta }\tau$ and half of this value for the plot on the right-hand side.

Figure 13

Evolution of the normalizable mode of the scalar field on various lattice sizes. Black color for $N_x=N_{\rho }=20$, blue for $N_x=N_{\rho }=30$, green for $N_x=N_{\rho }=40$ and red for $N_x=N_{\rho }=50$.