Effect of non-conformal deformation on the gapped quasi-normal modes and the holographic implications

Abstract

The spectral curve of quasinormal modes for a massive real scalar field in the background of a non-conformal black brane geometry has been obtained by utilizing a Frobenius type near-horizon expansion. The gauge/gravity duality maps this to the computation of spectral curve of a massive scalar operator $\mathcal{O}_φ$ for a large-$N$ conformal field theory with irrelevant type non-conformal deformation. In this context, non-conformality has been holographically introduced by using the Einstein-dilaton theory with Liouville type dilaton potential as the bulk theory. It has been observed that the obtained quasinormal modes are characterized by specific gapped dispersion relations. The pole-skipping points have also been computed and classified based upon different dispersion relations satisfied by them. The effect of non-conformality is evident from these results. The analytic form of the mentioned gapped dispersion relation have been obtained in the long-wavelength limit by setting the mass of the scalar field to zero. The coefficients of the momentum terms in the dispersion relation also capture the signature of non-conformal deformation. The radius of convergence of the derivative expansion in the momentum space is then computed from the critical points of the spectral curve. It has been observed that presence of non-conformality increases the domain of applicability of the derivative expansion in momentum space, as it increases the radius of convergence for a given conformal dimension. The comparison between the convergence radii and the absolute momenta corresponding to lowest order pole-skipping points also leads to some interesting findings.

Figures (6)

• Figure 1: Plot of the spectral curve of the massive scalar operator $\mathcal{O}_{\phi}$ for purely imaginary momenta and PS points obtained from upto fourth-order in near-horizon expansion. Here we consider $d=4$ and mass $m=2\sqrt{3}$.
• Figure 2: Effect of non-conformal parameter $\eta$ on the dispersion relations of QNMs where we set $d=4$ and $m=2\sqrt{3}$.
• Figure 3: Lowest four QNMs for zero momenta and zero mass of the scalar field. We have set $d=4$.
• Figure 4: We set $\Delta_{\phi}=2$ and plot the QNMs for $\tilde{k}^2=|\tilde{k}^2|e^{i\theta}$ with various values of $|\tilde{k}|$ and $0\leq\theta\leq 2\pi$. The left plot of the upper panel is for $|\tilde{k}|=0.55$, right plot of the upper panel is for $|\tilde{k}|=0.57$, left plot of the lower panel is for $|\tilde{k}|=0.97$ and the right plot in the lower panel is for $|\tilde{k}|=0.98$. Here, the red points depict the solutions for $\theta=0$ and green points represents the critical value of $\tilde{\omega}$ at which the QNMs collide.
• Figure 8: Comparison between $\tilde{k}_*$ (in dashed) and $\tilde{k}_c$ associated to both first order QNM (in red), second order QNM (in green). In this case $\eta=0$.