A simple holographic model of momentum relaxation

Abstract: We consider a holographic model consisting of Einstein-Maxwell theory in (d+1) bulk spacetime dimensions with (d-1) massless scalar fields. Momentum relaxation is realised simply through spatially dependent sources for operators dual to the neutral scalars, which can be engineered so that the bulk stress tensor and resulting black brane geometry are homogeneous and isotropic. We analytically calculate the DC conductivity, which is finite. In the d=3 case, both the black hole geometry and shear-mode current-current correlators are those of a sector of massive gravity.

• The paper introduces a simple holographic framework that employs spatially dependent massless scalar fields to induce momentum relaxation without explicit lattice structures.
• The model extends the Einstein-Maxwell setup to yield homogeneous black brane solutions, enabling analytical calculations of finite DC conductivity across dimensions.
• Analytical results in d=3 reveal parallels with massive gravity, providing new insights into transport phenomena in condensed matter systems.

A Simple Holographic Model of Momentum Relaxation: An Expert Overview

This paper presents a straightforward holographic model to address momentum relaxation using an Einstein-Maxwell setup coupled with massless scalar fields. The authors, Tomás Andrade and Benjamin Withers, propose a mechanism where spatially dependent sources induce momentum relaxation in a holographic context. The model operates in a $d+1$ dimensional bulk spacetime with $d-1$ massless scalars, yielding homogeneous and isotropic black brane solutions. This allows for the analytical calculation of finite DC conductivity, providing insights into how holography can simulate real-world condensed matter systems exhibiting finite conductivities.

Key Contributions

• Model Formulation: The paper extends the Einstein-Maxwell framework by incorporating massless scalar fields sourced spatially, deviating from traditional holographic lattices. Momentum dissipation is achieved without explicit lattice structures; instead, scalar fields act as reservoirs absorbing momentum from charge carriers.
• Analytical Results: In the case where $d=3$, the model demonstrates equivalence with massive gravity sectors, specifically in shear-mode current-current correlators and black hole geometries, elucidating parallels between these seemingly disparate approaches.
• Finite DC Conductivity: The introduction of spatially varying scalar fields allows for a finite DC conductivity calculation across different dimensions. The result is particularly impactful for $d=3$, echoing findings from massive gravity formulations, yet with a clearer field theory interpretation.

Numerical and Theoretical Implications

The finite DC conductivity outcome, expressed succinctly for general dimensions, is a central achievement of this work:

$$\sigma_{DC} = r_0^{d-3}\left( 1 + (d-2)^2 \frac{\mu^2}{\alpha^2}\right)$$

Here, $r_0$ denotes the black brane horizon, $\mu$ the chemical potential, and $\alpha$ a scalar source gradient. The DC conductivity's dimensional dependence has practical implications for understanding temperature-dependent conductivities across various physical systems.

Future Directions and Speculations

The model paves the way for extensions incorporating additional bulk terms or exploring low-entropy limits at zero temperature, akin to recent advancements in holographic memory effects and transport phenomena. Additionally, while providing clear benefits over massive gravity models, potential string-theoretic embeddings could offer refinements, shedding light on the dual field theory dynamics underlying these holographic insights.

In summary, the paper articulates a coherent framework for holographic momentum relaxation, broadening the toolkit available for simulating condensed matter phenomena using gravitational duals. It establishes a significant connection between scalar field dynamics and conductivity, challenging conventional limits observed in purely massive gravity-based interpretations without lapses in mathematical or conceptual integrity.