# Quantum critical response: from conformal perturbation theory to holography

## TL;DR

This paper explores the behavior of dynamical response functions near quantum critical points using conformal perturbation theory and holography, deriving sum rules and comparing Lifshitz and conformal theories.

## Contribution

It bridges conformal field theory and holography to analyze quantum critical responses, including high-frequency behavior and sum rules, in a unified framework.

## Key findings

- Holography reproduces CFT results and sum rules for UV-complete theories.
- Numerical response functions reveal breakdown of high-frequency asymptotics.
- Lifshitz theories show similar high-frequency responses to conformal cases.

## Abstract

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.