Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

$c$-Theorem for Anisotropic RG Flows from Holographic Entanglement Entropy

Chong-Sun Chu, Dimitrios Giataganas

TL;DR

The paper extends the holographic c-theorem to quantum field theories with broken Lorentz and rotational symmetry by defining anisotropic c-functions built from slab entanglement entropies in distinct directions. It derives the necessary and sufficient geometric conditions for monotonic RG flow, showing that null energy conditions alone are insufficient and that UV boundary data (and FG expansion data) can guarantee monotonicity for at least one of the anisotropic c-functions. The work provides explicit sufficient conditions, analyzes Lifshitz-like and hyperscaling-violation examples, and discusses necessary conditions and UV/IR boundary criteria. The framework broadens the applicability of entropic c-functions to anisotropic systems and offers tools for constructing monotonic RG flows in holographic models, with potential connections to F-theorem-like statements and nonlocal observables.

Abstract

We propose a candidate $c$-function in arbitrary dimensional quantum field theories with broken Lorentz and rotational symmetry. For holographic theories we derive the necessary and sufficient conditions on the geometric background for these $c$-functions to satisfy the $c$-theorem. We obtain the null energy conditions for anisotropic background to show that do not themselves assure the $c$-theorem. By employing them, we find that is possible to impose conditions on the UV data that are enough to guarantee at least one monotonic $c$-function along the RG flow. These UV conditions can be used as building blocks for the construction of anisotropic monotonic RG flows. Finally, we apply our results to several known anisotropic theories and identify the region in the parameters space of the metric where the $c$-theorem holds for our proposed $c$-function.

$c$-Theorem for Anisotropic RG Flows from Holographic Entanglement Entropy

TL;DR

The paper extends the holographic c-theorem to quantum field theories with broken Lorentz and rotational symmetry by defining anisotropic c-functions built from slab entanglement entropies in distinct directions. It derives the necessary and sufficient geometric conditions for monotonic RG flow, showing that null energy conditions alone are insufficient and that UV boundary data (and FG expansion data) can guarantee monotonicity for at least one of the anisotropic c-functions. The work provides explicit sufficient conditions, analyzes Lifshitz-like and hyperscaling-violation examples, and discusses necessary conditions and UV/IR boundary criteria. The framework broadens the applicability of entropic c-functions to anisotropic systems and offers tools for constructing monotonic RG flows in holographic models, with potential connections to F-theorem-like statements and nonlocal observables.

Abstract

We propose a candidate -function in arbitrary dimensional quantum field theories with broken Lorentz and rotational symmetry. For holographic theories we derive the necessary and sufficient conditions on the geometric background for these -functions to satisfy the -theorem. We obtain the null energy conditions for anisotropic background to show that do not themselves assure the -theorem. By employing them, we find that is possible to impose conditions on the UV data that are enough to guarantee at least one monotonic -function along the RG flow. These UV conditions can be used as building blocks for the construction of anisotropic monotonic RG flows. Finally, we apply our results to several known anisotropic theories and identify the region in the parameters space of the metric where the -theorem holds for our proposed -function.

Paper Structure

This paper contains 15 sections, 66 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A $c$-Function for Theories with Broken Spacetime Symmetry
  3. The Null Energy Conditions for Anisotropic Theories
  4. Sufficient Conditions of Monotonicity of the $c$-function
  5. Sufficient Condition for Anisotropic Theories
  6. Sufficient Condition for Isotropic UV Fixed Points
  7. Asymptotics and Boundary Criteria of Monotonicity
  8. Boundary Condition on the Geometry
  9. UV Criteria on Fefferman-Graham Expansion
  10. Sufficient Conditions Applied on Certain Anisotropic Theories
  11. Lifshitz-like Anisotropic Symmetry
  12. Anisotropic Hyperscaling Violation Symmetry
  13. Necessary Conditions on Anisotropic RG flows
  14. Discussion
  15. An Alternative Form for the Sufficient Conditions of Monotonicity

Figures (2)

  • Figure 1: The parametric volume $(\theta,~z,~d)$ that is sufficient to guarantee RG flows with the right $c$-function monotonicity. The meshed lines on the surface represent the values of the integer spatial dimensions $d$. Notice how drastically the increase of the dimension $d$ shrinks the allowed region in the parametric space that gives a well behaved RG flow. The necessary conditions of section \ref{['sec:exact']} are more tolerant.
  • Figure 2: The window of parameters that lead always to well behaved RG flows for $d=3$ space-time dimensions. The different coloring represents where the different conditions are satisfied and the triangle is the common region in the parameter space that is sufficient to guarantee well behaved RG flow.