Weyl consistency conditions have been used in unitary relativistic quantum\nfield theory to impose constraints on the renormalization group flow of certain\nquantities. We classify the Weyl anomalies and their renormalization scheme\nambiguities for generic non-relativistic theories in 2+1 dimensions with\nanisotropic scaling exponent z=2; the extension to other values of z are\ndiscussed as well. We give the consistency conditions among these anomalies. As\nan application we find several candidates for a C-theorem. We comment on\npossible candidates for a
C-theorem in higher dimensions.
Weyl consistency conditions have been used in unitary relativistic quantum field theory to impose constraints on the renormalization group flow of certain quantities. We classify the Weyl anomalies and their renormalization scheme ambiguities for generic non-relativistic theories in 2 + 1 dimensions with anisotropic scaling exponent z = 2; the extension to other values of z are discussed as well. We give the consistency conditions among these anomalies. As an application we find several candidates for a C-theorem. We comment on possible candidates for a C-theorem in higher dimensions.
We extend the cohomological analysis in arXiv:1410.5831 of anisotropic Lifshitz scale anomalies. We consider non-relativistic theories with a dynamical critical exponent z = 2 with or without non-relativistic boosts and a particle number symmetry. We distinguish between cases depending on whether the time direction does or does not induce a foliation structure. We analyse both 1 + 1 and 2 + 1 spacetime dimensions. In 1 + 1 dimensions we find no scale anomalies with Galilean boost symmetries. The anomalies in 2 + 1 dimensions with Galilean boosts and a foliation structure are all B-type and are identical to the Lifshitz case in the purely spatial sector. With Galilean boosts and without a foliation structure we find also an A-type scale anomaly. There is an infinite ladder of B-type anomalies in the absence of a foliation structure with or without Galilean boosts. We discuss the relation between the existence of a foliation structure and the causality of the field theory.
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.