Spatial Modulation and Conductivities in Effective Holographic Theories

TL;DR

This work analyzes low-energy thermo-electric transport in bottom-up holographic Einstein-Maxwell-dilaton models with translational symmetry breaking implemented as a boundary lattice Φ1(x) = C cos(kx). By numerically constructing inhomogeneous backgrounds and performing linear perturbation analysis, the authors extract DC and AC conductivities via horizon data and perturbative methods, characterizing phases as coherent metal, insulator, or incoherent metal across a tunable parameter υ and lattice wavenumber k. They derive analytic DC conductivity formulas from hydrodynamic and membrane-paradigm perspectives and reveal phase diagrams where incoherent metals arise in restricted regions, with mid-IR scaling of the optical conductivity only in narrow parameter windows. The results highlight deviations from the Wiedemann-Franz law, the role of IR scalar dynamics in transport, and provide a phenomenological laboratory for diffusion-dominated transport in incoherent metals within holographic theories. The study lays groundwork for further investigations into quasinormal modes, near-horizon geometry signatures of transport, and broader explorations of V(Φ) and Z(Φ) parameter spaces.

Abstract

We analyze a class of bottom-up holographic models for low energy thermo-electric transport. The models we focus on belong to a family of Einstein-Maxwell-dilaton theories parameterized by two scalar functions, characterizing the dilaton self-interaction and the gauge coupling function. We impose spatially inhomogeneous lattice boundary conditions for the dilaton on the AdS boundary and study the resulting phase structure attained at low energies. We find that as we dial the scalar functions at our disposal (changing thus the theory under consideration), we obtain either (i) coherent metallic, or (ii) insulating, or (iii) incoherent metallic phases. We chart out the domain where the incoherent metals appear in a restricted parameter space of theories. We also analyze the optical conductivity, noting that non-trivial scaling behaviour at intermediate frequencies appears to only be possible for very narrow regions of parameter space.

Figures (24)

• Figure 1: Plots of the DC electrical and thermoelectric conductivities for a range of models against T for various theories parameterized by $\upsilon$, with $C =1.5, k=1$ held fixed. We clearly see the existence of the metallic and insulating regimes separated by an intermediate region.
• Figure 2: Plots of both forms of the thermal conductivity, $\kappa$ and $\bar{\kappa}$. We note both the qualitative similarity of the two quantities and the insensitivity to the variation in the $\upsilon$ parameter. In all cases that we have examined, including those associated with the data used to construct Fig. \ref{['fig:met_insul_trans_plane']}, the thermal conductivities were seen to increase monotonically with temperature.
• Figure 3: Plots of the Lorenz factors associated with $\kappa$ and $\bar{\kappa}$. The fact that these factors are neither constant as a function of temperature nor $\upsilon$ indicates that the Wiedemann-Franz law is violated. This is in accord with the lack of a phase transition in the thermal conductivity.
• Figure 4: The phase plot in the $(\upsilon,k)$ plane illustrating the metal insulator transition. The green data points correspond to regimes which are clearly metallic in character with monotonically increasing DC electrical conductivity at low temperatures. Likewise the red data points correspond to insulating phases characterized by a monotonically decreasing DC electrical conductivity at low temperatures. The intermediate region between these corresponds to the transition region where a turning point is still evident in the profile at low temperatures.
• Figure 5: Setting $\upsilon$ to be strictly zero means that the IR evolution of the scalar field flattens out. As this behaviour of the scalar controls many aspects of the IR physics qualitative changes in the response functions are to be expected.