Holographic Timelike c-function

The integration of high-energy degrees of freedom along the renormalization group (RG) flow in Poincar\'e-invariant theories can be captured by a monotonic c-function. For such theories, holographic monotonic c-functions have been constructed using entanglement entropy. However, in theories with broken Lorentz invariance, such constructions generally fail, reflecting both the violation of the entanglement RG monotonicity and its limitations in capturing certain properties of non-relativistic RG flows. Since many quantum many-body systems lack Lorentz invariance, it is of significant importance to identify a quantity that reflects the decrease in degrees of freedom along non-relativistic RG flows. We show that the recently introduced holographic timelike entanglement entropy naturally gives rise to a new c-function applicable to all such theories. We further demonstrate the existence of this c-function in theories with Lifshitz and hyperscaling-violating fixed points, showing that, provided the null energy conditions and thermodynamic stability are satisfied, the proposed c-function exhibits the expected monotonic behavior along the RG flow.