Simple holographic dual of the Maxwell-Cattaneo model & the fate of KMS symmetry for non-hydrodynamic modes

TL;DR

The diffusion equation’s parabolic nature fails at short times, which the Maxwell-Cattaneo model fixes by introducing a finite relaxation time $\tau$, yielding hyperbolic dynamics with $\tau \partial_t^2 \Phi + \partial_t \Phi = D \nabla^2 \Phi$. The authors construct a simple holographic dual with a bulk $ (F_{MN}F^{MN})^2 $ term controlled by $\alpha$, and demonstrate, through numerical quasinormal-mode analysis and analytical matching, that the boundary theory reproduces MC transport with a relaxation time related to holographic data via $\tau = X(u_h)Y(u_h)$; they also derive a DC conductivity $\sigma_0$ and recall the MC dispersion $\omega^2 + i\omega/\tau = v_0^2 k^2$ with $v_0^2 = D/\tau$. A holographic Schwinger-Keldysh effective action is constructed, revealing that non-hydro modes generate a boundary EFT that obeys a generalized KMS symmetry, while chemical shift symmetry remains intact; canonical KMS does not hold for non-hydro modes. Together, these results validate a simple gravitational dual of MC dynamics and reveal the fate of KMS symmetry in the presence of non-hydrodynamic degrees of freedom, with potential applications to strongly coupled fluids and condensed-matter systems.

Abstract

Diffusion, as described by Fick's laws, governs the spreading of particles, information, data, and even financial fluctuations. However, due to its parabolic structure, the diffusion equation leads to an unphysical prediction: any localized disturbance instantaneously affects the entire system. The Maxwell-Cattaneo (MC) model, originally introduced to address relativistic heat conduction, refines the standard diffusion framework by incorporating a finite relaxation time $τ$, associated with the onset of local equilibrium. This modification yields physically relevant consequences, including the emergence of propagating shear waves in liquids and second sound in solids. Holographic methods have historically provided powerful tools for describing the hydrodynamics of strongly correlated systems. However, they have so far failed to capture the dynamics governed by the MC model, limiting their ability to model intermediate time-scale phenomena. In this work, we construct a simple holographic dual of the Maxwell-Cattaneo model and rigorously establish its equivalence through a combination of analytical and numerical techniques. As an important byproduct of our analysis, and contrary to previous ad-hoc assumptions, we find that effective field theories featuring non-hydrodynamic modes exhibit a generalized form of Kubo-Martin-Schwinger (KMS) symmetry, which reduces to the canonical form only in the hydrodynamic limit.

Figures (6)

• Figure 1: Quasinormal modes in the longitudinal sector at zero wave-vector $k=0$ upon dialing the dimensionless parameter $\hat{\alpha}\equiv \alpha \rho^2$ from $\hat{\alpha}=0$ (darker color) to $\hat{\alpha}\approx31$ (green color). The black star $\bigstar$ indicates the hydrodynamic diffusive mode.
• Figure 2: Relative error $\mathcal{E}_{\text{rel}}(\omega)\equiv |G^R_{nn,\text{numeric}}(\omega)-G^R_{nn,\text{MC}}(\omega)|/G^R_{nn,\text{MC}}(0)$ as a function of $\omega$ for different values of $\hat{\alpha}$. (a)$k/(\pi T)=0.1$, (b)$k/(\pi T)=0.4$.
• Figure 3: Dispersion relation $\omega(k)$ of the lowest QNMs for three different values of $\hat{\alpha}$. Panels (a)-(b) refer respectively to the real and imaginary part of the frequency. Symbols are the results of the numerical computation while dashed lines are the fits to the theoretical dispersion obtained from Eq. \ref{['mo']}.
• Figure 4: (a) The relaxation time extracted from the imaginary part of the imaginary QNM (open circles in Fig. \ref{['fig1']}) compared with the theoretical prediction (dashed line) from Eq. \ref{['tautheory']} as a function of $\hat{\alpha}$. (b) The real part of the optical conductivity $\mathrm{Re}[\sigma(\omega)]$ as a function of frequency with $\hat{\alpha}=1000$. The symbols are the numerical data and the dashed lines the theoretical predictions using the Drude formula with the relaxation time defined in Eq. \ref{['tautheory']} and the DC conductivity taken from Eq. \ref{['appB:sigmaDC']}.
• Figure 5: Charge density retarded correlator $G^R_{nn}(\omega,k)$ as a function of $\omega/(\pi T)$ with $\hat{\alpha}=1000$ and several values of $k/(\pi T)$. The dashed lines denote the analytic form, Eqs. \ref{['G1']}-\ref{['G2']}, with $\sigma_0=D \chi$ and $\tau$ analytically computed.