Quantum critical response: from conformal perturbation theory to holography

We discuss dynamical response functions near quantum critical points, allowing for both a finite temperature and detuning by a relevant operator. When the quantum critical point is described by a conformal field theory (CFT), conformal perturbation theory and the operator product expansion can be used to fix the first few leading terms at high frequencies. Knowledge of the high frequency response allows us then to derive non-perturbative sum rules. We show, via explicit computations, how holography recovers the general results of CFT, and the associated sum rules, for any holographic field theory with a conformal UV completion -- regardless of any possible new ordering and/or scaling physics in the IR. We numerically obtain holographic response functions at all frequencies, allowing us to probe the breakdown of the asymptotic high-frequency regime. Finally, we show that high frequency response functions in holographic Lifshitz theories are quite similar to their conformal counterparts, even though they are not strongly constrained by symmetry.Comment: 45+14 pages, 9 figures. v2: small clarifications, added reference

Quantum critical responses via holographic models and conformal perturbation theory

We investigate response functions near quantum critical points, allowing for finite temperature and a mild deformation by a relevant scalar. When the quantum critical point is described by a conformal field theory, we use conformal perturbation theory and holography to determine the two leading corrections to the scalar two-point function and to the conductivity. We build a bridge between the couplings fixed by conformal symmetry with the interaction couplings in the gravity theory. Knowledge of the high-frequency response allows us to derive non-perturbative sum rules. We construct a minimal holographic model that allows us to numerically obtain the response functions at all frequencies, independently confirming the corrections to the high-frequency response functions. In addition to probing the physics of the ultraviolet, the holographic model probes the physics of the infrared giving us qualitative insight into new physics scalings. We briefly investigate the hydrodynamic modes that occur in the field theory by observing the diffusive nature of the gauge field deep in the bulk using the membrane paradigm, allowing us to calculate the diffusion constant and DC conductivity

The implicit function theorem in a non-Archimedean setting

In this paper, the inverse function theorem and the implicit function theorem in a non-Archimedean setting will be discussed. We denote by N any non-Archimedean field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order; and we study the properties of locally uniformly differentiable functions from N n to N m. Then we use that concept of local uniform differentiability to formulate and prove the inverse function theorem for functions from N n to N n and the implicit function theorem for functions from N n to N m with m<n