Momentum dissipation and effective theories of coherent and incoherent transport

TL;DR

The paper investigates heat transport in systems without momentum conservation, contrasting coherent transport—characterized by a slowly decaying, long-lived mode and a Drude-like AC peak—with incoherent transport, where diffusion dominates and no such pole is near the origin. It develops a hydrodynamic framework and validates it against a neutral holographic axion model with tunable momentum relaxation, revealing a coherent/incoherent crossover controlled by the ratio $\Gamma/\Lambda$ (with $\Lambda\sim T$). A key result is the exact Green's functions at a self-dual point ($m/T=\sqrt{8}\,\pi$) in the holographic theory, where an emergent SL$(2,\mathbb{R})\times$SL$(2,\mathbb{R})$ symmetry renders all correlators solvable; a related gravitational self-duality yields a frequency-independent heat conductivity $\kappa(\omega)$. The study also extends the coherent/incoherent classification to charge transport in non-Maxwell holographic theories, including probe branes and higher-derivative couplings, demonstrating coherent transport can arise without an obvious almost-conserved current. Overall, the work provides a unified, long-wavelength description of heat (and charge) transport in strongly interacting systems with broken translational symmetry and highlights exact, symmetry-driven results at special points for holographic models.

Abstract

We study heat transport in two systems without momentum conservation: a hydrodynamic system, and a holographic system with spatially dependent, massless scalar fields. When momentum dissipates slowly, there is a well-defined, coherent collective excitation in the AC heat conductivity, and a crossover between sound-like and diffusive transport at small and large distance scales. When momentum dissipates quickly, there is no such excitation in the incoherent AC heat conductivity, and diffusion dominates at all distance scales. For a critical value of the momentum dissipation rate, we compute exact expressions for the Green's functions of our holographic system due to an emergent gravitational self-duality, similar to electric/magnetic duality, and SL(2,R) symmetries. We extend the coherent/incoherent classification to examples of charge transport in other holographic systems: probe brane theories and neutral theories with non-Maxwell actions.

Figures (7)

• Figure 1: Left: Schematic depiction of the poles of the AC heat conductivity, showing the parametric separation between the Drude-like pole (in light blue) with $\text{Im}(\omega)\sim \Gamma$ and the other poles at $\text{Im}(\omega)\sim\Lambda$ (in brown). Right: Schematic depiction of the motion of poles of $\kappa(\omega,k)$ as $\Gamma$ is increased (at fixed $k$) in the coherent regime $\Gamma\ll\Lambda$. Two sound poles (in light blue) exist when $\Gamma=0$. As $\Gamma\ll k$ is increased, these sound-like poles move deeper in the lower half plane and start getting closer to the imaginary axis. When $\Gamma\sim k$, they collide and produce two purely imaginary poles (shown in red), one of which is diffusive and moves up the imaginary axis as $\Gamma$ is increased further.
• Figure 2: Plots of the AC heat conductivity at low (left panel) and high (right panel) values of $m/T$. Values of $m/T$ (given in the plot headings) increase from top to bottom. The dots are the numerical results of our holographic model, and the solid lines are the analytical results \ref{['eq:coherenthydroopticalconductivity']} (left panel) and \ref{['eq:incoherenthydroopticalconductivity']} (right panel) of the hydrodynamic model. The agreement with the Drude-like conductivity \ref{['eq:coherenthydroopticalconductivity']} is better at lower values of $m/T$ (coherent regime, left), while the constant conductivity \ref{['eq:incoherenthydroopticalconductivity']} is better at higher values of $m/T$ (incoherent regime, right). Note also the change of slope from peak (left) to valley (right).
• Figure 3: Poles of the AC conductivity of the holographic model, as $m/T$ is increased from the coherent to incoherent regimes. The left panel shows the purely imaginary pole closest to the origin in our holographic model (dots), compared to the prediction $\omega=-im^2/4\pi T$ (solid line) for a state whose momentum slowly dissipates at a constant rate $m^2/4\pi T$. There is excellent agreement when $m\ll T$, indicating that this is a coherent state with the aforementioned dissipation rate, but there is disagreement for $m\gtrsim T$ as the crossover to an incoherent regime begins. When $m\sim9.5T$, the pole is no longer purely imaginary and we have definitively exited the coherent regime. The right panel shows the location of the poles of the holographic model in the complex frequency plane. The arrows show the movements of the poles as $m$ is increased. At small $m$, there is only a Drude-like pole near the origin. As $m$ increases, this moves down the imaginary axis before colliding with another purely imaginary pole when $m\sim9.5T$, producing two off-axis (non-Drude-like) poles.
• Figure 4: The crossover between diffusive transport ($k\ll\Gamma\ll T$) and sound-like transport ($\Gamma\ll k\ll T$) in the coherent regime. The left panel shows the imaginary part of the dispersion relation of the longest lived excitation of the holographic model (dots), and the corresponding dispersion relation (\ref{['eq:coherentcombinedexcitationholo']}) in the hydrodynamic model (solid line), when $m/T=1/2$. There is excellent agreement, indicating diffusion (\ref{['eq:holodiffusionanalytic']}) at small $k$ and a sound-like excitation (\ref{['eq:holosoundlikeanalytic']}) at large $k$. The right panel shows the movement of the long-lived poles of the holographic model in the complex $\omega$ plane, as $m$ is increased (within the coherent regime) at fixed $k/T=1/100$. The arrows show the motion of the poles as $m$ increases: at small $m$ (i.e. $\Gamma\ll k$), there are sound-like excitations near the real axis. As $m$ increases, these poles approach each other and then collide at $\Gamma\sim k$, forming a diffusive excitation that moves up the imaginary axis and is long-lived when $k\ll\Gamma$.
• Figure 5: The imaginary part of the dispersion relation of the longest-lived pole in the incoherent regime with $m/T=100$ (the real part is zero). The dots denote the poles of the holographic model and the solid line is the analytic result (\ref{['eq:holodiffusionanalytic']}) of the hydrodynamic model. There is excellent agreement, showing that heat always diffuses in this regime.