Timelike Entanglement Entropy and Renormalization Group Flow Irreversibility

TL;DR

Timelike entanglement entropy provides a robust holographic c-function that tracks irreversible RG flow, extending c-theorems to strongly anisotropic and nonrelativistic settings under NEC and thermodynamic stability. The authors derive a directional c-function framework with an explicit derivative formula, show equivalence to boundary-temporal formulations, and establish a universal upper bound on the rate of degrees-of-freedom decimation along RG flows. They test the construction in isotropic Poincaré, Lifshitz-like, and hyperscaling-violating backgrounds, confirming monotonicity across a broad class of physically reasonable theories. The results offer a new information-theoretic diagnostic of holographic RG dynamics and propose constraints that any consistent holographic theory must satisfy.

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.

Figures (1)

• Figure 1: The parameter space of the anisotropic hyperscaling violation theory with $d_1=2 d_2=2$. The NEC plus stability region is covered by the union of these three parametric colored regimes. We consider the regimes where all spatial elements $g_{ii}$ diverge at the boundary. In the parametric region A) where $\alpha_i>0$ and $\delta>-1$ the c-theorem holds and this covers the orange-shaded bottom right rectangle. In the parametric region B) where $\alpha_i<0$ and $\delta<-1$ the c-theorem holds and it covers the bottom right green-shaded region. In the parametric region C) where $\alpha_i>0$ and $\delta<-1$ the c-theorem holds and covers the top left red-shaded region. The shaded area of any color matches the parametric space allowed by the NEC and the thermodynamic stability conditions.