Holographic Thermalization with Chemical Potential

Abstract

We study the thermalization of a strongly coupled quantum field theory in the presence of a chemical potential. More precisely, using the holographic prescription, we calculate non- local operators such as two point function, Wilson loop and entanglement entropy in a time- dependent background that interpolates between AdSd+1 and AdSd+1 -Reissner-Nordström for d = 3, 4. We find that it is the entanglement entropy that thermalizes the latest and thus sets a time-scale for equilibration in the field theory. We study the dependence of the thermalization time on the probe length and the chemical potential. We find an interesting non-monotonic behavior. For a fixed small value of T l and small values of μ/T the thermalization time decreases as we increase μ/T, thus the plasma thermalizes faster. For large values of μ/T the dependence changes and the thermalization time increases with increasing μ/T . On the other hand, if we increase the value of T l this non-monotonic behavior becomes less pronounced and eventually disappears indicating two different regimes for the physics of thermalization: non-monotonic dependence of the thermalization time on the chemical potential for T l << 1 and monotonic for T l >> 1.

Figures (28)

• Figure 1: The dependence of temperature on the chemical potential in (a) $d=3$ and (b) $d=4$. We are measuring the temperature and the chemical potential in units of the AdS-radius. We have also set $M=1$.
• Figure 2: Left panel (a): The case $d=3$. The blue curve corresponds to $\chi_{(3)} \approx 0.003$ and the red curve corresponds to $\chi_{(3)} \approx 22.4$. Right panel (b): The case $d=4$. The blue curve corresponds to $\chi_{(4)} \approx 0.002$ and the red curve corresponds to $\chi_{(4)} \approx 18.3$. In (c), we have shown how ${\cal L}_{\rm thermal}$ scales with $\chi_{(d)}$ for both $d=3$ with $(4\pi T) \ell = 0.6$ and $d=4$ with $(4\pi T)\ell = 2$. Here ${\cal L}_{\rm thermal}$ is measured in units of the AdS-radius, $L$.
• Figure 3: A schematic diagram of the Wilson loops of different shapes and the corresponding minimal area surfaces: rectangular in (a) and circular in (b).
• Figure 4: Left panel (a): The case $d=3$. The blue curve corresponds to $\chi_{(3)} \approx 0.003$ and the red curve corresponds to $\chi_{(3)} \approx 22.5$. Right panel (b): The case $d=4$. The blue curve corresponds to $\chi_{(4)} \approx 0.002$ and the red curve corresponds to $\chi_{(4)} \approx 18.3$. In (c), we have shown how ${\cal A}_{\rm thermal}$ scales with $\chi_{(d)}$ for both $d=3$ and $d=4$ for fixed $(4\pi T)\ell = 2$. ${\cal A}_{\rm thermal}$ has been measured in units of the AdS-radius.
• Figure 5: Left panel: The case $d=3$. The blue curve corresponds to $\chi_{(3)} \approx 0.003$ and the red curve corresponds to $\chi_{(3)} \approx 0.19$. Right panel: The case $d=4$. The blue curve corresponds to $\chi_{(4)} \approx 0.002$ and the red curve corresponds to $\chi_{(4)} \approx 0.24$. ${\cal A}_{\rm thermal}$ is measured in units of the AdS-radius.