Thermalization of the spectral function in strongly coupled two dimensional conformal field theories

Abstract

Using Wigner transforms of Green functions, we discuss non-equilibrium generalizations of spectral functions and occupation numbers. We develop methods for computing time-dependent spectral functions in conformal field theories holographically dual to thin-shell AdS-Vaidya spacetimes.

Figures (9)

• Figure 1: Spectral function of a quenched harmonic oscillator ($\omega_i=2$, $\omega_f=1$) with Gaussian time window $\rho_\sigma(T, \omega)$ at $T=0$ for $\sigma =2$ (left) and $\sigma \approx 1.48$ (right).
• Figure 2: Time evolution of $\rho_\sigma(T, \omega)$ (with $\omega_i=2$, $\omega_f=1$) for $\sigma = 7$ and $T = -12, -7.2, -2.4, 2.4, 7.2, 12$ from left to right. At early (late) times the spectral function is the one of a time-independent harmonic oscillator with frequency $\omega_i$ ($\omega_f$).
• Figure 3: The plot shows the occupation number for frequency $\omega = \omega_f = 1$ as a function of average time (here $T$ is multiplied by a factor of 2 compared with the convention in the text) for a quenched harmonic oscillator, with final frequency $\omega_f = 1$ and initial frequency $\omega_i = 2$, with smearing parameter $\sigma = \sqrt{8}$. The limiting values agrees with a conventional computation of the final occupation number in terms of a Bogoliubov transformation of the initial vacuum.
• Figure 4: Panel (A) plots the numerical results for the spectral function integrated over momenta $\rho(T,\omega)$ for different values of the mean time $T$ (with $\Delta=9/4$, $R=15$). When $T\ll -1/R$, the curve reduces to the the vacuum result (in blue dotted), while for $T\gg 1/R$, to the thermalised one (in red continuous). (The reference case $T=0$ is also shown in purple.) Close to the origin, the spectral function grows linearly in frequency. The slopes $\beta$ of this regime of linear growth are plotted in panel (B) as a function of time and for various values of $\Delta$. Similar results are obtained for larger values of $\Delta$. The oscillations in the slope characterize how the spectral function interpolates between the vacuum and thermal results.
• Figure 5: Spectral function integrated over momenta (A) and slopes of the regime of linear growth (B) as in Fig. \ref{['fig:spectralfunction']}, with time window set by $\sigma=2.9/R$.