A bstract We study applications of spectral positivity and the averaged null energy condition (ANEC) to renormalization group (RG) flows in two-dimensional quantum field theory. We find a succinct new proof of the Zamolodchikov c -theorem, and derive further independent constraints along the flow. In particular, we identify a natural C -function that is a completely monotonic function of scale, meaning its derivatives satisfy the alternating inequalities (–1) n C ( n ) ( μ 2 ) ≥ 0. The completely monotonic C -function is identical to the Zamolodchikov C -function at the endpoints, but differs along the RG flow. In addition, we apply Lorentzian techniques that we developed recently to study anomalies and RG flows in four dimensions, and show that the Zamolodchikov c -theorem can be restated as a Lorentzian sum rule relating the change in the central charge to the average null energy. This establishes that the ANEC implies the c -theorem in two dimensions, and provides a second, simpler example of the Lorentzian sum rule.
A bstract We study applications of spectral positivity and the averaged null energy condition (ANEC) to renormalization group (RG) flows in two-dimensional quantum field theory. We find a succinct new proof of the Zamolodchikov c -theorem, and derive further independent constraints along the flow. In particular, we identify a natural C -function that is a completely monotonic function of scale, meaning its derivatives satisfy the alternating inequalities (–1) n C ( n ) ( μ 2 ) ≥ 0. The completely monotonic C -function is identical to the Zamolodchikov C -function at the endpoints, but differs along the RG flow. In addition, we apply Lorentzian techniques that we developed recently to study anomalies and RG flows in four dimensions, and show that the Zamolodchikov c -theorem can be restated as a Lorentzian sum rule relating the change in the central charge to the average null energy. This establishes that the ANEC implies the c -theorem in two dimensions, and provides a second, simpler example of the Lorentzian sum rule.
A bstract Energy-momentum tensor in general conformal field theories have improvement ambiguity and it can affect the argument in deriving c-theorem. While the derivation of Zamolodchikov’s c-theorem is still formally valid with the improved energy-momentum tensor, the behavior of the c-function proposed by Zamolodchikov becomes different. When the theory suffers an IR divergence due to the non-compactness, the c-function can be unbounded and may even show non-monotonicity. We find that one of the three-point function sum rule proposed by Hartman and Mathys motivated by the averaged null energy condition, however, is agnostic about the improvement and gives the (effective) Virasoro central charge, when the IR theory is gapped.
In the successful framework of the Renormalization Group (RG) approach to Quantum Field Theories (QFT), establishing general constraints on the RG flow has become a central problem. This is formally addressed by the monotonicity theorems, such as Zamolodchikov’s c-theorem in d=2, the a-theorem in d=4, and the F-theorem in d=3. Essentially, these theorems state that if a RG trajectory leads from a UV fixed point to a IR one, each described by a Conformal Field Theory (CFT), then a suitable positive quantity decreases. In both even and odd dimensions, this quantity can be extracted, at fixed points, from the free energy of the CFT on a sphere, leading to the conjecture of a generalized F-theorem valid in continuous dimension d. In this thesis we investigate this framework in a multicritical multiscalar theory with a generic sextic interaction for N massless scalar fields below its upper critical dimension d=3. This model captures, among many others, the critical features of the Tricritical Ising universality class. A first step is to establish whether the RG flow admits a gradient structure, i.e. that there exists a monotonic function A whose gradient yields the beta function up to a positive-definite metric in the space of couplings. This is one of the strongest realizations of monotonicity, as it guarantees the irreversibility of the RG flow and gives an intrinsic geometric structure to the theory space. Given the known beta function up to next-to-leading order, we solve the gradient flow equation perturbatively, finding a parametric family of solutions for the monotonic A-function together with the metric at and below the upper critical dimension, so that the RG flow admits a gradient structure at this order. As this procedure leaves some parameters undetermined, to turn it into a physical statement these functions must be matched to suitable observables of the CFTs describing the fixed points. We therefore compute the CFT candidates suggested by the generalized F-theorem, namely the sphere free energy of the CFT and the Zamolodchikov metric. Specializing to the O(N)-symmetric fixed point, we consistently match the RG and CFT results, fixing all the free parameters in a unique way. This matching is the central result of this work: it establishes that the RG flow admits a gradient structure perturbatively, with F as the monotonic function and Zamolodchikov's metric as the metric on the theory space, recovered from the geometric construction for a specific and unique choice of the free parameters. Importantly, this result provides a nontrivial, perturbative confirmation of the F-theorem in its strongest form, at and below the upper critical dimension, for renormalizable three-dimensional multiscalar theories.
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 We prove a conjectured lower bound on 〈 T__ ( x )〉 ψ in any state ψ of a CFT on Minkowski space, dubbed the Quantum Null Energy Condition (QNEC). The bound is given by the second order shape deformation, in the null direction, of the geometric entanglement entropy of an entangling cut passing through x . Our proof involves a combination of the two independent methods that were used recently to prove the weaker Averaged Null Energy Condition (ANEC). In particular the properties of modular Hamiltonians under shape deformations for the state ψ play an important role, as do causality considerations. We study the two point function of a “probe” operator O O in the state ψ and use a lightcone limit to evaluate this correlator. Instead of causality in time we consider causality on modular time for the modular evolved probe operators, which we constrain using Tomita-Takesaki theory as well as certain generalizations pertaining to the theory of modular inclusions. The QNEC follows from very similar considerations to the derivation of the chaos bound and the causality sum rule. We use a kind of defect Operator Product Expansion to apply the replica trick to these modular flow computations, and the displacement operator plays an important role. We argue that the proof extends to more general relativistic QFT with an interacting UV fixed point and also prove a higher spin version of the QNEC. Our approach was inspired by the AdS/CFT proof of the QNEC which follows from properties of the Ryu-Takayanagi (RT) surface near the boundary of AdS, combined with the requirement of entanglement wedge nesting. Our methods were, as such, designed as a precise probe of the RT surface close to the boundary of a putative gravitational/stringy dual of any QFT with an interacting UV fixed point.
After showing how to prove the integrated c-theorem within the functional RG framework based on the effective average action, we derive an exact RG flow equation for Zamolodchikov’s c-function in two dimensions by relating it to the flow of the effective average action. In order to obtain a non-trivial flow for the c-function, we will need to understand the general form of the effective average action away from criticality, where nonlocal invariants, with beta functions as coefficients, must be included in the ansatz to be consistent. Then we apply our construction to several examples: exact results, local potential approximation and loop expansion. In each case we construct the relative approximate c-function and find it to be consistent with Zamolodchikov’s c-theorem. Finally, we present a relation between the c-function and the (matter induced) beta function of Newton’s constant, allowing us to use heat kernel techniques to compute the RG running of the c-function.
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.