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Inhomogeneity simplified

Marika Taylor, William Woodhead

TL;DR

The paper develops a set of holographic models where translational symmetry is broken by inhomogeneous matter fields that preserve bulk isotropy and homogeneity. It analyzes both polynomial and square-root scalar actions, revealing a close link to massive gravity and to tensionless brane limits, and derives analytic expressions for DC conductivity and low-frequency optical response, complemented by numerical AC-conductivity studies. The square-root sector yields marginal operators in the dual field theory and produces transport features reminiscent of heavy fermion systems, including a linear-in-T DC resistivity in 3D and finite-frequency minima in the optical conductivity, while not displaying standard Drude behavior. Overall, the work provides a tractable, bottom-up holographic framework for momentum relaxation with robust connections to brane dynamics and massive gravity, offering insights into non-Fermi liquid phenomenology and potential top-down realizations.

Abstract

We study models of translational symmetry breaking in which inhomogeneous matter field profiles can be engineered in such a way that black brane metrics remain isotropic and homogeneous. We explore novel Lagrangians involving square root terms and show how these are related to massive gravity models and to tensionless limits of branes. Analytic expressions for the DC conductivity and for the low frequency scaling of the optical conductivity in phenomenological models are derived, and the optical conductivity is studied in detail numerically. The square root Lagrangians are associated with linear growth in the DC resistivity with temperature and also lead to minima in the optical conductivity at finite frequency, suggesting that our models may capture many features of heavy fermion systems.

Inhomogeneity simplified

TL;DR

The paper develops a set of holographic models where translational symmetry is broken by inhomogeneous matter fields that preserve bulk isotropy and homogeneity. It analyzes both polynomial and square-root scalar actions, revealing a close link to massive gravity and to tensionless brane limits, and derives analytic expressions for DC conductivity and low-frequency optical response, complemented by numerical AC-conductivity studies. The square-root sector yields marginal operators in the dual field theory and produces transport features reminiscent of heavy fermion systems, including a linear-in-T DC resistivity in 3D and finite-frequency minima in the optical conductivity, while not displaying standard Drude behavior. Overall, the work provides a tractable, bottom-up holographic framework for momentum relaxation with robust connections to brane dynamics and massive gravity, offering insights into non-Fermi liquid phenomenology and potential top-down realizations.

Abstract

We study models of translational symmetry breaking in which inhomogeneous matter field profiles can be engineered in such a way that black brane metrics remain isotropic and homogeneous. We explore novel Lagrangians involving square root terms and show how these are related to massive gravity models and to tensionless limits of branes. Analytic expressions for the DC conductivity and for the low frequency scaling of the optical conductivity in phenomenological models are derived, and the optical conductivity is studied in detail numerically. The square root Lagrangians are associated with linear growth in the DC resistivity with temperature and also lead to minima in the optical conductivity at finite frequency, suggesting that our models may capture many features of heavy fermion systems.

Paper Structure

This paper contains 23 sections, 2 theorems, 298 equations, 5 figures.

Table of Contents

  1. Introduction
  2. The simplest models of explicit translational symmetry breaking
  3. Polynomial Lagrangians
  4. Relation to massive gravity
  5. Relation to branes
  6. Square root models
  7. Holographic renormalisation for square root models
  8. Thermodynamics of brane solutions
  9. Two point functions
  10. Phenomenological models
  11. Linearized perturbations
  12. DC conductivity
  13. Parameter Space Restrictions
  14. DC Conductivity Temperature Dependence
  15. Finite frequency behaviour at low temperature
  16. ...and 8 more sections

Key Result

Theorem 1

Let $F(z)$ be the smooth polynomial obeying the Einstein equation and the constraints $F(0) = 1$, $F(z_0) = 0$, $F'(z_0) = -4\pi T \leq 0$, with $\tilde{\beta} > 0$ and $\tilde{\alpha}/(\mu z_0) > 0$. If $z_c$ is a root of $F(z)$ in the open interval $(0, z_0)$ then $F'(z_c) < 0$.

Figures (5)

  • Figure 1: Plots of $\sigma_{DC}/\mu^{d-3}$ against $T/\mu$ in $d=3$ for the given values of $\tilde{\alpha}$ and $\tilde{\beta}$. Solid lines denote results for the $\mu > 0$ branch, dashed lines denote results for the $\mu < 0$ branch. Note that $\sigma_{DC}$ decreases with $T$ for $\tilde{\alpha}$ non-zero.
  • Figure 2: Plots of $\sigma_{DC}/\mu^{d-3}$ against $T/\mu$ in $d=4$ for the given values of $\tilde{\alpha}$ and $\tilde{\beta}$. Note that $\sigma_{DC}$ is strictly increasing in $T$.
  • Figure 3: AC conductivity in $d = 3$ for $\tilde{\alpha} = 0, \tilde{\beta} = 2$.
  • Figure 4: AC conductivity in $d = 3$ for $\tilde{\alpha} = 1, \tilde{\beta} = 2$.
  • Figure 5: AC conductivity in $d = 3$ for $\tilde{\alpha} = 1, \tilde{\beta} = 0$.

Theorems & Definitions (4)

  • Theorem
  • proof
  • Lemma
  • proof : Proof of lemma