Pseudospectra of holographic diffusion: gauge fields breaking free from the master scalar

TL;DR

This work investigates the pseudospectra of quasinormal frequencies (QNFs) and complex linear momenta (CLMs) for a U(1) gauge field in Schwarzschild–AdS black branes. It develops and compares two formulations: a direct gauge-field (GF) approach and the conventional master-scalar (MS) method, clarifying their relationship via Hodge duality and boundary-term contributions to the energy norm. The study finds that hydrodynamic QNFs are spectrally stable, while hydrodynamic CLMs exhibit enhanced instability due to a zero-frequency pole collision, a signature of exceptional points and nontrivial spectral sensitivity. Overall, GF and MS formulations yield equivalent pseudospectra when boundary terms are handled correctly, reinforcing the robustness of the hydrodynamic description and highlighting the value of energy-norm pseudospectra for holographic transport analyses; the framework also opens paths to analyzing pseudospectra in other backgrounds and for gravitational perturbations.

Abstract

We study pseudospectra of quasinormal frequencies and complex linear momenta of a U(1) gauge field in a Schwarzschild black branes in Anti-de Sitter. We present a novel approach for computing the pseudospectra which uses directly the gauge field variables and contrast it to a conventional master scalar field approach. Upon clarifying a subtlety in the energy norm of the master scalar we show that the pseudospectra of both approaches conincide. In the hydrodynamic regime we find that the hydrodynamic quasinormal frequency, the diffusive mode, is spectrally stable to a very good approximation. On the other hand hydrodynamic complex linear momenta show enhanced spectral instability as a consequence of a pole-collision at zero frequency.

Figures (22)

• Figure 1: Pseudospectrum of $A\mathcal{C}-\lambda\mathcal{C}$ in \ref{['eq:toymodel_eigenvaleq_rectangular']}. The red dots correspond to the eigenvalues, the white lines represent the boundaries of various $\varepsilon$-pseudospectra, and the heat map represent to the logarithm in base 10 of the inverse of the norm of the resolvent.
• Figure 2: (Left) $\mathbb{C}$LM$\,$ pseudospectrum at $\omega=10^{-4}$ for the SAdS$_{5+1}$ black brane. The white lines denote the boundaries of different $\varepsilon$-pseudospectra and the heat map corresponds to the logarithm in base 10 of the inverse of the norm of the resolvent. The green circles correspond to the boundaries of the $\varepsilon=1$-pseudospectra and the dashed red circles are circles of radius $1$ centered on the $\mathbb{C}$LMs$\,$. For the higher $\mathbb{C}$LMs$\,$ these coincide denoting spectral stabilty. (Right) Heatmap of the percentage difference between MS and GF frameworks. The red dots correspond to the $\mathbb{C}$LMs$\,$.
• Figure 3: (Left) $\mathbb{C}$LM$\,$ pseudospectrum at $\omega=1$ for the SAdS$_{5+1}$ black brane. The white lines denote the boundaries of different $\varepsilon$-pseudospectra and the heat map corresponds to the logarithm in base 10 of the inverse of the norm of the resolvent. The green circles correspond to the boundaries of the $\varepsilon=1$-pseudospectra and the dashed red circles are circles of radius $1$ centered on the $\mathbb{C}$LMs$\,$. Notably, the $\mathbb{C}$LMs$\,$ showcase mild spectral instability. (Right) Heatmap of the percentage difference between MS and GF frameworks. The red dots correspond to the $\mathbb{C}$LMs$\,$.
• Figure 4: (Left) $\mathbb{C}$LM$\,$ pseudospectrum at $\omega=10$ for the SAdS$_{5+1}$ black brane. The white lines denote the boundaries of different $\varepsilon$-pseudospectra and the heat map corresponds to the logarithm in base 10 of the inverse of the norm of the resolvent. The green circles correspond to the boundaries of the $\varepsilon=1$-pseudospectra and the dashed red circles are circles of radius $1$ centered on the $\mathbb{C}$LMs$\,$. Notably, the $\mathbb{C}$LMs$\,$ showcase spectral instability. (Right) Heatmap of the percentage difference between MS and GF frameworks. The red dots correspond to the $\mathbb{C}$LMs$\,$.
• Figure 5: (Left) Condition numbers for the hydro $\mathbb{C}$LM$\,$ and the first two non-hydro $\mathbb{C}$LMs$\,$ as a function of the frequency $\omega$ for the SAdS$_{5+1}$ black brane. (Right) Plot of the hydrodynamic $\mathbb{C}$LM$\,$ condition number at small frequency $\omega$ for the SAdS$_{5+1}$ black brane. The red dots denote the numerical values of the condition numbers and the black line is the fit $\log_{10}(\kappa_0)=-0.130-0.500\log_{10}(\omega)$.