Renormalization Group Flow Irreversibility Enables Constraints on Effective Spatial Dimensionality

The behaviour of matter under extreme conditions and its evolution over varying scales remains a fundamental challenge in physics, and recent work explores this through the lens of entanglement entropy. Dimitrios Giataganas from National Sun Yat-Sen University, along with colleagues, investigates how this concept, specifically ‘timelike entanglement entropy’, reveals the direction of change in complex systems. Their research demonstrates that a ‘timelike c-function’ accurately captures irreversible flows during ‘renormalization group’ transformations, which describe how physical systems simplify at larger scales. This achievement extends the applicability of established theoretical tools to more complex scenarios, including systems lacking rotational symmetry, and establishes a new diagnostic for understanding the dynamics of holographic systems, effectively providing a deeper understanding of how complexity arises and evolves in the universe.

Timelike Entanglement, Thermalization and Anisotropic Systems

This research investigates the connection between holographic duality and complex materials, particularly those lacking simplifying symmetries. Scientists explore timelike entanglement entropy (TLE), a measure of entanglement between regions separated in time, to understand how systems evolve towards equilibrium and how to describe materials where properties differ depending on direction. The study focuses on establishing a ‘timelike c-function’, analogous to a function in conformal field theory, that characterizes the ‘renormalization group’ (RG) flow of complex systems and signals transitions between different states of matter. The RG flow describes how a system changes as its energy scale is varied, and the c-function helps identify stable and predictable behaviors.

Researchers employed the AdS/CFT correspondence, linking gravity in a higher-dimensional space to quantum field theory on its boundary, to perform calculations. They calculated TLE using holographic techniques, effectively translating calculations between the gravitational and quantum realms, and proposed a definition for the timelike c-function, testing its behavior in non-conformal, anisotropic systems, and systems undergoing phase transitions. The results demonstrate that the proposed timelike c-function accurately indicates the RG flow in complex systems, decreasing predictably as the system evolves and exhibiting a characteristic change at critical points, signaling transitions between different phases of matter. Furthermore, the framework successfully captures the properties of anisotropic systems, providing insights into their unique behaviors and transitions. The study also suggests a connection between the timelike c-function and thermalization, how systems reach equilibrium, and establishes links to the dS/CFT correspondence.

Timelike C-function Probes Renormalization Group Flows

Scientists have developed a methodology to investigate holographic c-theorems, focusing on timelike entanglement entropy to capture irreversible renormalization group (RG) flows. The study pioneered the use of a timelike c-function, demonstrating its applicability to both relativistic and non-relativistic matter, specifically within nematic phases exhibiting broken rotational symmetry. Researchers rigorously tested the function’s behavior under anisotropic RG flows, confirming its monotonicity, a crucial consistency check. The research team established that the monotonicity of the timelike c-function depends jointly on the null energy condition, thermodynamic stability, and a constraint on effective spatial dimensionality along the RG flow.

To quantify the rate of change, scientists identified a geometric upper bound on how rapidly effective degrees of freedom can be coarse-grained, providing a precise measure of the flow’s dynamics whenever the timelike c-theorem applies. This work extends the applicability of holographic c-theorems to highly nontrivial RG flows, revealing a new theoretical diagnostic for holographic RG dynamics. Leveraging entanglement entropy formulations generalized to higher dimensions, scientists built upon the established link between entanglement entropy and the central charge at fixed points. They imposed Lorentz symmetry and unitarity to extend the proof of the c-theorem beyond two dimensions, proposing a generalized c-function based on entanglement entropy for anisotropic QFTs, which smoothly reduces to known isotropic expressions. Recognizing that broken Lorentz invariance can violate the monotonicity of standard entanglement entropy, the research team focused on timelike entanglement entropy evaluated on a fixed time slice to accurately capture the full RG monotonicity.

Timelike Entropy Reveals Stable Renormalization Group Flows

Scientists have demonstrated a new approach to understanding renormalization group (RG) flows using timelike entanglement entropy, establishing a timelike c-function that accurately captures irreversible RG flow dynamics. The research confirms that this timelike c-function applies consistently to both relativistic and non-relativistic matter, even in nematic phases exhibiting broken rotational symmetry. Crucially, the function remains monotonic, a key indicator of stability, even under anisotropic RG flows, successfully passing stringent consistency tests. The team discovered that the combined conditions of null energy condition (NEC) compliance, thermodynamic stability, and a constraint on effective spatial dimensionality are jointly sufficient to guarantee the monotonicity of the timelike c-function throughout the RG flow.

Moreover, the study identifies a geometric upper bound on the rate of change of the timelike c-function, which limits how quickly degrees of freedom can be removed during the RG flow. Measurements confirm this bound constrains how rapidly effective degrees of freedom can be coarse-grained during the RG flow, suggesting a fundamental limit on the speed of holographic RG flows. Further analysis reveals that the timelike c-function exhibits correct monotonic behavior even in theories with broken Lorentz, scale, or rotational symmetry, provided the NEC, thermodynamic stability, and effective dimension conditions are met. In Poincaré-invariant holographic flows, the entanglement entropy-based c-theorem is guaranteed as long as the NEC is satisfied, independently reproducing established results and serving as a robust consistency check. The research establishes a universal strict upper bound on the rate at which degrees of freedom are decimated along the RG flow, a value determined by the bulk metric and suggesting a deeper relationship between RG dynamics, bulk causal structure, and information-theoretic measures.

Timelike C-function Monotonicity Under Anisotropic Flow

This research establishes a refined understanding of holographic c-theorems, demonstrating their applicability to a broader range of physical systems undergoing renormalization group (RG) flow. Scientists have shown that a timelike c-function accurately captures irreversible RG flow, even in scenarios with broken rotational symmetry, such as nematic phases. Importantly, this c-function maintains its monotonic behavior under anisotropic RG flows, successfully passing stringent consistency tests. The team discovered that the combined requirements of the null energy condition, thermodynamic stability, and a constraint on effective spatial dimensionality are jointly sufficient to guarantee the monotonicity of the timelike c-function throughout the RG flow.