Isospin diffusion in thermal AdS/CFT with flavor

TL;DR

This work investigates a finite-temperature gauge theory with finite isospin chemical potential using a holographic setup of two coincident D7-branes in AdS-Schwarzschild. By turning on a constant A_0^3=\mu in the SU(2) flavor sector and analyzing fluctuations, the authors derive and solve the non-Abelian equations of motion to obtain current-current Green functions. They find a frequency-dependent diffusion coefficient D(\omega) = \sqrt{\omega/(2\mu)}/(2\pi T), indicating transport beyond linear response and a non-analytic diffusion behavior arising from the non-Abelian structure. The results offer qualitative insights into isospin diffusion in strongly coupled plasmas, with potential implications for neutron-rich matter and heavy-ion physics, while highlighting the need for numerical methods to capture the Abelian limit and full momentum range.

Abstract

We study the gauge/gravity dual of a finite temperature field theory at finite isospin chemical potential by considering a probe of two coincident D7-branes embedded in the AdS-Schwarzschild black hole background. The isospin chemical potential is obtained by giving a vev to the time component of the non-Abelian gauge field on the brane. The fluctuations of the non-Abelian gauge field on the brane are dual to the SU(2) flavor current in the field theory. For the embedding corresponding to vanishing quark mass, we calculate all Green functions corresponding to the components of the flavor current correlator. We discuss the physical properties of these Green functions, which go beyond linear response theory. In particular, we show that the isospin chemical potential leads to a frequency-dependent isospin diffusion coefficient.

Figures (4)

• Figure 1: Real (left plot) and imaginary part (right plot) of the correlator $G_{0\widetilde{0}}$ as a function of frequency $\omega/T$ at different chemical potential values $\mu/T=0.5$ (solid line), $\mu/T=0.3$ (short-dashed line) and $\mu/T=0.1$ (long-dashed line). The corresponding plots for the correlator $G_{0\widetilde{0}}$ would look like the mirror image of the ones given. The real part would be reflected about the vertical axis at $\omega=0$, the imaginary part would be reflected about the origin. All dimensionful quantities are given in units of temperature. The numerical values used for the parameters are $q/T=0.1$, $N_c=100$.
• Figure 2: In the left plot the sum of both correlators in $0 0$-directions is split into its imaginary (dashed line) and real (solid line) part and plotted against frequency. For comparison the right plot shows the corresponding real and imaginary parts for the $G_{A_0^3A_0^3}$. It is qualitatively similar to the Abelian correlator in $i=0$ Lorentz direction computed from the Super-Maxwell action in Policastro:2002se. Note the different frequency scales in the two plots. The curves in $a=1,\,2$-directions are much narrower due to their square root dependence on $\omega$. Furthermore they have a much larger maximum amplitude. All dimensionful quantities are given in units of temperature. The numerical values used for the parameters are, as in Fig. \ref{['fig:reImGX0X0tMu']}, $q/T=0.1$, $N_c=100$ and only in the left plot $\mu/T=0.2$.
• Figure 3: These plots show the real and imaginary part of the function $F(u)$ which is part of $X_\alpha=(1-u)^\beta F(u)$. The solid line depicts the analytical approximation, obtained in this paper. As a check we solved the equations of motion for $F(u)$ numerically. They are drawn as dashed lines. In this example we used $T=1$. The numerical solution was chosen to agree with the analytical one at the horizon and boundary.
• Figure 4: Here the thermal spectral functions in distinct Lorentz- and flavor-directions are plotted against frequency $\omega/T$ in units of temperature. In the left plot the chemical potential was chosen to be $\mu/T=0.7$, in the right one $\mu/T=0.2$. Flavor-directions $a=1,\,2$ are summed and displayed as one curve. The frequency-dependence of $00$- (solid red line) and $03$-Lorentz-directions (short-dashed blue line) is shown. By the dotted line we denote the spectral curve in $11$- or $22$-directions. This curve was scaled by a factor 100 in order to make it visible in these plots. The third flavor-direction is only plotted for the spectral function in Lorentz-directions $00$ (long-dashed green curve). We do not show the $33$-direction spectral function which has a square root dependence and is comparable in size with the $11$-direction.