[Paper Review] A simple model of momentum relaxation in Lifshitz holography

Tomás Andrade|arXiv (Cornell University)|Feb 1, 2016

Black Holes and Theoretical Physics5 citations

TL;DR

This paper proposes a holographic model of momentum relaxation in Lifshitz spacetimes by introducing axion fields with linear dependence on boundary spatial coordinates, breaking translational invariance homogeneously. Using Einstein-Proca theory coupled to massless scalars, it derives analytic black brane solutions with Lifshitz asymptotics and shows via linearized perturbations that a dual Ward identity governs momentum dissipation, with finite DC thermal conductivity confirmed numerically.

ABSTRACT

We expand the holographic studies of momentum relaxation to include non-relativistic scaling symmetries in the ultraviolet. We do so by constructing black branes with Lifshitz asymptotics dressed with axions which explicitly depend on the boundary directions. Such configurations arise as analytic solutions of the Einstein-Proca theory coupled to massless scalar fields in arbitrary dimensions. Studying linear perturbations on these backgrounds, we conclude that there is a dual Ward identity which accounts for the dissipation of momentum in the system. In addition, we numerically compute the frequency dependent thermal conductivity of the branes and verify that its DC limit is finite.

Motivation & Objective

• To extend holographic momentum relaxation to non-relativistic, Lifshitz-symmetric field theories with anisotropic scaling.
• To construct analytic black brane solutions in Einstein-Proca theory coupled to massless scalar fields with linear spatial dependence (axions) to break translations homogeneously.
• To study linearized perturbations on these backgrounds to derive the dual Ward identity for momentum dissipation.
• To numerically compute the frequency-dependent thermal conductivity and verify its finite DC limit.

Proposed method

• Constructing black brane solutions in Einstein-Proca theory coupled to massless scalar fields in arbitrary dimensions, with axions that depend linearly on boundary spatial coordinates to break translational invariance.
• Using the Lifshitz metric as the asymptotic boundary geometry, with dynamical exponent z ≠ 1, to model non-relativistic field theories.
• Performing linearized perturbation analysis on the background to study the response of the system to external sources.
• Deriving the symplectic flux formalism to identify sources and vacuum expectation values (vevs) of dual operators, particularly for momentum and scalar operators.
• Imposing boundary conditions that set gauge-invariant sources to zero and enforce finiteness of the symplectic flux, ensuring well-defined dynamics.
• Numerically solving the linearized equations of motion to compute the frequency-dependent thermal conductivity and extract its DC limit.

Experimental results

Research questions

• RQ1 How can momentum relaxation be consistently incorporated into Lifshitz holography with anisotropic scaling symmetry?
• RQ2 What is the structure of the dual Ward identity that accounts for momentum dissipation in non-relativistic holographic systems?
• RQ3 Can analytic black brane solutions with Lifshitz asymptotics and homogeneous momentum relaxation be constructed in arbitrary dimensions?
• RQ4 Does the thermal conductivity in such systems exhibit a finite DC limit, indicating incoherent transport?
• RQ5 How do the symplectic flux and boundary conditions constrain the physical interpretation of sources and vevs in the dual field theory?

Key findings

• The model realizes momentum relaxation via axion fields with linear spatial dependence, breaking translations homogeneously while preserving homogeneity in the bulk stress tensor.
• A dual Ward identity is derived that accounts for the dissipation of momentum in the boundary field theory, consistent with the presence of momentum relaxation.
• The DC thermal conductivity is found to be finite, confirming the system's incoherent transport behavior and absence of perfect Drude peak.
• The symplectic flux formalism identifies the sources and vevs of the momentum and scalar operators, with the requirement to set s^(0) = 0 and h_tx^(1) = 0 to ensure finiteness and well-defined dynamics.
• The numerical computation of the frequency-dependent thermal conductivity confirms the expected incoherent metallic behavior with a finite DC limit.
• The use of gauge-invariant variables ensures that the boundary conditions do not overdetermine the system, preserving consistency of the linearized equations.