Hyperscaling of Fidelity and Operator Estimations in the Critical Manifold

Abstract

By formulating the renormalization group as a quantum channel acting on density matrices in Quantum Field Theories (QFTs), we show that ground-state expectation values of observables supported on slow momentum modes can be approximated by their averages on the fixed-point theories to which the QFTs flow. This is done by studying the fidelity between ground states of different QFTs and arriving at certain hyperscaling relations satisfied at criticality. Our results allow for a clear identification of cases in which one can replace a QFT by its scale-invariant limit in the calculation of expectation values, opening the way for a range of applications, including the improvement of numerical and analytical methods used to tackle the costly computer simulation of critical models.

v2: Main discussions and calculations in End Matter expanded, example of hyperscaling relation in the (2+1)d Ising CFT and proposal of application to models of deconfined criticality included, references added and other minor changes, 8 pages