Holographic Operator Mixing and Quasinormal Modes on the Brane

TL;DR

This work develops a general framework to compute holographic retarded Green's functions for systems with coupled bulk fields, showing that the matrix spectral function ρ(k) = i(G^R(k) - G^A(k)) is radial-coordinate independent and can be interpreted as a matrix of bulk Noether currents. The authors apply the formalism to the D3/D7 system at finite baryon density, where longitudinal vector perturbations couple to brane-embedding scalars, and derive a numerical prescription to extract G^R, its poles (quasinormal modes), and residues. They explore the resulting hydrodynamic and quasiparticle regimes, crossovers between diffusion and reactive behavior, and the dispersion relations of the modes, linking the spectral features to the induced horizon on the D7-branes. The study provides a robust, gauge-invariant method for analyzing operator mixing in holographic setups and offers a practical route to connect holographic predictions with transport phenomena at finite density and temperature.

Abstract

We provide a framework for calculating holographic Green's functions from general bilinear actions and fields obeying coupled differential equations in the bulk. The matrix-valued spectral function is shown to be independent of the radial bulk coordinate. Applying this framework we improve the analysis of fluctuations in the D3/D7 system at finite baryon density, where the longitudinal perturbations of the world-volume gauge field couple to the scalar fluctuations of the brane embedding. We compute the spectral function and show how its properties are related to the quasinormal mode spectrum. We study the crossover from the hydrodynamic diffusive to the reactive regime and the movement of quasinormal modes as functions of temperature and density. We also compute their dispersion relations and find that they asymptote to the lightcone for large momenta.

Figures (8)

• Figure 1: Normalized induced horizon area on the $D7$-branes as a function of the quark mass and the baryon density. Red lines mark regions of equal induced area with the corresponding value of $\psi_0$ specified. For ${\tilde{d}}\leq 0.00315$ the curve is multivalued close to $m=1.3$, signaling a first order phase transition. Furthermore, it was recently shown in munichpaper that an unstable quasinormal mode with positive imaginary part of the frequency exists in that region. We will however not consider it in the current paper.
• Figure 2: Example of the position of the quasinormal modes with positive real part (red points, scale in negative axis) and the corresponding finite temperature contribution to the component of the spectral function associated with the longitudinal electric field propagator (continuous line, scale in positive axis). Notice that in figure (a) only half of the poles seem to contribute to the spectral function, this is because the other half have a small residue. In figure (b) we plot a detailed version of a spectral function where all the poles have an observable contribution. These plots are for $m=0.01$, ${\tilde{d}}=2$ and ${\mathfrak{q}}=0.2$ and ${\mathfrak{q}}=2.2$ respectively.
• Figure 3: Crossover from the diffusive to the reactive regime in terms of quasinormal modes. Two different mechanisms of how this crossover happens can be seen. At small density (a) the hydrodynamic mode crosses the imaginary part of the lowest non-hydrodynamic mode. At large density (c) the hydrodynamic mode pairs up with another purely imaginary mode and moves off the imaginary axes as a pair with non-vanishing real frequencies. In between (b) the three imaginary parts of the modes meet at a single value of momentum.
• Figure 4: Position of the quasinormal modes with positive real part as we vary $m$ for fixed ${\tilde{d}}=0.01$ and ${\mathfrak{q}}=0.01$. The massless quark limit corresponds to the lower points on the graphs and we evolve up to $m=2.01$. We see that when the quark mass is increased the pole gets closer to the real axis hardly changing the value of $\Omega_n$. From a given value of the quark mass it changes completely the behaviour, approaching the real axis asymptotically in $\Omega_n$. The very large frequency limit can be read as a $T\to 0$ limit, therefore the poles should sit exactly on the real axis. The poles for different values of the parameters evolve in the same qualitative way.
• Figure 5: Position of the first eight quasinormal modes with positive real part as we vary ${\tilde{d}}$ for fixed $m=2$ and ${\mathfrak{q}}=3$. The red points mark the values (beginning at the top) ${\tilde{d}}=0.01$, $0.012$, $0.026$, $0.063$, $0.135$, $0.254$, $0.432$, $0.680$, $1.01$, $1.43$, $1.96$, $3.38$, $5.37$, $8.01$, $11.4$, $15.6$. Between any two consecutive red points there are ten data points. When ${\tilde{d}}$ is increased all the quasinormal modes begin to orbit a certain point, but this happens beyond the quasiparticle regime, we have not investigated whether this is a numerical issue.