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arXiv:2512.16499

Timelike Entanglement Entropy and Renormalization Group Flow Irreversibility

Giataganas, Dimitrios

Published Dec 18, 2025• 0 citations
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Abstract

We study holographic $c$-theorems based on timelike entanglement entropy and show that timelike $c$-functions capture irreversible renormalization group (RG) flow. We demonstrate that timelike $c$-functions are applicable to both relativistic and nonrelativistic quantum matter in nematic phases with broken rotational symmetry, and that they remain monotonic along holographic anisotropic RG flows under explicit geometric conditions, thereby passing some of the most stringent consistency tests. We derive the exact flow of the composite timelike $c$-function in general homogeneous anisotropic geometries and establish the $c$-theorem for adiabatic flows. Across all classes of scaling backgrounds examined, consistent with the null energy condition and thermodynamic stability, the monotonicity condition universally reduces to a constraint on a running geometric effective dimension. We further construct a second $c$-function purely from the imaginary part of the timelike entanglement entropy and prove its monotonicity under the local effective-dimension condition, without requiring adiabaticity. For Poincaré-invariant flows, this condition follows directly from the null energy condition. Moreover, we identify a geometric upper bound on the rate of change of the timelike $c$-functions, which constrains how rapidly effective degrees of freedom can be coarse-grained. Our results suggest that capturing RG irreversibility in nonrelativistic flows requires observables that are intrinsically sensitive to the temporal direction. 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.

DOI
10.48550/arxiv.2512.16499

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We study holographic ccc-theorems based on timelike entanglement entropy and show that timelike ccc-functions capture irreversible renormalization group (RG) flow. We demonstrate that timelike ccc-functions are applicable to both relativistic and nonrelativistic quantum matter in nematic phases with broken rotational symmetry, and that they remain monotonic along holographic anisotropic RG flows under explicit geometric conditions, thereby passing some of the most stringent consistency tests. We derive the exact flow of the composite timelike ccc-function in general homogeneous anisotropic geometries and establish the ccc-theorem for adiabatic flows. Across all classes of scaling backgrounds examined, consistent with the null energy condition and thermodynamic stability, the monotonicity condition universally reduces to a constraint on a running geometric effective dimension. We further construct a second ccc-function purely from the imaginary part of the timelike entanglement entropy and prove its monotonicity under the local effective-dimension condition, without requiring adiabaticity. For Poincaré-invariant flows, this condition follows directly from the null energy condition. Moreover, we identify a geometric upper bound on the rate of change of the timelike ccc-functions, which constrains how rapidly effective degrees of freedom can be coarse-grained. Our results suggest that capturing RG irreversibility in nonrelativistic flows requires observables that are intrinsically sensitive to the temporal direction. The applicability of holographic ccc-theorems is thus extended to highly nontrivial RG flows and points toward a new information-theoretic diagnostic of holographic RG dynamics.
The integration of high-energy degrees of freedom along the renormalization group (RG) flow in Poincaré-invariant theories is captured by a monotonic c -function. For such theories, holographic c -functions have been constructed using entanglement entropy. However, in theories with broken Lorentz invariance, these constructions generally fail, reflecting both the violation of entanglement RG monotonicity and its limitations in capturing essential features of nonrelativistic RG flows. Since many quantum many-body systems lack Lorentz invariance, it is important to identify a quantity that captures the decrease of effective degrees of freedom along such flows. We show that the recently introduced holographic timelike entanglement entropy naturally gives rise to a new c -function applicable to all such theories, which moreover obeys a geometric upper bound on its rate of change. We further demonstrate the existence of this c -function in theories with Lifshitz and hyperscaling-violating fixed points, showing that, provided the null energy condition and thermodynamic stability are satisfied, it obeys the expected RG monotonicity.
A bstract The recently introduced concept of timelike entanglement entropy has sparked a lot of interest. Unlike the traditional spacelike entanglement entropy, timelike entanglement entropy involves tracing over a timelike subsystem. In this work, we propose an extension of timelike entanglement entropy to Euclidean space (“temporal entanglement entropy”), and relate it to the renormalization group (RG) flow. Specifically, we show that tracing over a period of Euclidean time corresponds to coarse-graining the system and can be connected to momentum space entanglement. We employ Holography, a framework naturally embedding RG flow, to illustrate our proposal. Within cutoff holography, we establish a direct link between the UV cutoff and the smallest resolvable time interval within the effective theory through the irrelevant TT‾T\overline{T }TT deformation. Increasing the UV cutoff results in an enhanced capability to resolve finer time intervals, while reducing it has the opposite effect. Moreover, we show that tracing over a larger Euclidean time interval is formally equivalent to integrating out more UV degrees of freedom (or lowering the temperature). As an application, we point out that the temporal entanglement entropy can detect the critical Lifshitz exponent z in non-relativistic theories which is not accessible from spatial entanglement at zero temperature and density.
Entanglement entropy has proven to be a powerful tool for probing renormalization group (RG) flows in quantum field theories, with c-functions derived from it serving as candidate measures of the effective number of degrees of freedom. While the monotonicity of such c-functions is well established in many settings, notable exceptions occur in theories with a mass scale. In this work, we investigate entanglement c-functions in the context of holographic RG flows, with a particular focus on flows across dimensions induced by circle compactifications. We argue that in spacetime dimensions d≥4d \geq 4d≥4, standard constructions of c-functions, which rely on higher derivatives of the entanglement entropy of either a ball or a cylinder, generically lead to non-monotonic behavior. Working with known dual geometries, we argue that the non-monotonicity stems not from any pathology or curvature singularity, but from a transition in the holographic Ryu--Takayanagi surface. In compactifications from four to three dimensions, we propose a modified construction that restores monotonicity in the infrared, although a fully monotonic ultraviolet extension remains elusive. Furthermore, motivated by entanglement entropy inequalities, we conjecture a bound on the cylinder entanglement c-function, which holds in all our examples.
A bstract Entanglement entropy has proven to be a powerful tool for probing renormalization group (RG) flows in quantum field theories, with c-functions derived from it serving as candidate measures of the effective number of degrees of freedom. While the monotonicity of such c-functions is well established in many settings, notable exceptions occur in theories with a mass scale. In this work, we investigate entanglement c-functions in the context of holographic RG flows, with a particular focus on flows across dimensions induced by circle compactifications. We argue that in spacetime dimensions d ≥ 4, standard constructions of c-functions, which rely on higher derivatives of the entanglement entropy of either a ball or a cylinder, generically lead to non-monotonic behavior. Working with known dual geometries, we argue that the non-monotonicity stems not from any pathology or curvature singularity, but from a transition in the holographic Ryu-Takayanagi surface. In compactifications from four to three dimensions, we propose a modified construction that restores monotonicity in the infrared, although a fully monotonic ultraviolet extension remains elusive. Furthermore, motivated by entanglement entropy inequalities, we conjecture a bound on the cylinder entanglement c-function, which holds in all our examples.
Abstract The vacuum entanglement entropy in quantum field theory provides nonperturbative information about renormalization group flows. Most studies so far have focused on the universal terms, related to the Weyl anomaly in even space-time dimensions, and the sphere free energy F in odd dimensions. In this work we study the entanglement entropy on a sphere of radius R in a large radius limit, for field theories with gravity duals. At large radius the entropy admits a geometric expansion in powers of R; the leading term is the well-known area term, and we also consider the subleading contributions. These terms can be physical, they contain information about the full renormalization group flow, and they reproduce known monotonicity theorems in particular cases. We set up an efficient method for calculating them using the Hamilton-Jacobi equation for the holographic entanglement entropy. We first reproduce the known result for the area term, the coefficient multiplying Rd−2 in the entanglement entropy. We then obtain the holographic result for the Rd−4 term and establish its irreversibility. Finally, we derive the Rd−6 coefficient for holographic theories, and also establish its irreversibility. This result goes beyond what has been proved in quantum field theory based on strong subadditivity, and hints towards new methods for analyzing the monotonicity of the renormalization group in space-time dimensions bigger than four.
The concept of renormalization group (RG) flow is one of the most novel and broad-reaching aspects of quantum field theory (QFT). The RG flow is implemented by constructing effective descriptions of a QFT at decreasing energy scales. One reason that RG flow is useful is that often one is interested in low-energy properties of theories with complicated short-distance structures. RG flows are subject to C theorems in relativistic QFT. The C theorems order the space of Lorentz-invariant QFTs. RG flows generically begin at scale-invariant fixed points known as conformal field theories (CFTs) and end in trivial massive theories. With tuning, the RG flows may end at non-trivial CFTs. Each CFT has an associated dimensionless C value. The C theorem states that under RG flow from a UV to an IR fixed point, the C value decreases. In this Dissertation I present the F-theorem, which is a C theorem in three spacetime dimensions. I show that the correct quantity to consider is the Euclidean free energy of the CFT conformally mapped to the three-sphere, known as the F value. After motivating the F-theorem, I develop tools for calculating the F value in a variety of CFTs, with and without supersymmetry, including free field theories and gauge theories with large numbers of flavors. I also show that the F value is itself a useful quantity for probing the gauge/gravity duality and understanding other aspects of CFT, such as the scaling dimensions of monopole operators. The F theorem is closely related to quantum entanglement entropy. At conformal fixed points, the F value is equal to minus the renormalized entanglement entropy (REE) in flat Minkowski space across a circle. Away from the fixed points, the REE is a monotonically decreasing function along the RG flow. I compute the REE in a variety of holographic and non-holographic theories. I conclude the Dissertation by discussing a somewhat surprising result: the REE is not stationary at conformal fixed points.