Timelike Entanglement Entropy and Renormalization Group Flow Irreversibility

TL;DR

This paper introduces a timelike c-function in holography that captures irreversible RG flow, applicable to various quantum matter phases, and establishes conditions ensuring its monotonicity, thus extending holographic c-theorems to complex RG flows.

Contribution

It develops a new timelike c-function framework for holographic RG flows, demonstrating its monotonicity under broad conditions and extending the scope of holographic c-theorems.

Findings

Timelike c-functions are monotonic in diverse quantum matter phases.

Null energy condition and stability guarantee c-function monotonicity.

A geometric upper bound constrains the rate of RG flow changes.

Abstract

We study holographic c-theorems based on timelike entanglement entropy and show that a timelike c-function captures irreversible renormalization group (RG) flow. We demonstrate that timelike c-functions are applicable to both relativistic and non-relativistic quantum matter in nematic phases with broken rotational symmetry, and that they remain monotonic even under anisotropic RG flows, thereby passing some of the most stringent consistency tests. Across all classes of theories examined, we find that the null energy condition, thermodynamic stability, and a constraint on an effective spatial dimensionality are jointly sufficient to guarantee monotonicity of the timelike c-function along the RG flow. Moreover, we identify a geometric upper bound on the rate of change of the timelike c-function, which constrains how rapidly effective degrees of freedom can be coarse-grained along the RG flow whenever a timelike c-theorem applies. The applicability of holographic c-theorems is thus extended to highly nontrivial RG flows and points toward a new information-theoretic diagnostic of holographic RG dynamics.