Finite Entanglement Scaling of a Kibble-Zurek sweep in the Transverse Field Ising Model

Quantum criticality harbors universal physics with observables near a quantum critical point (QCP). Such points in 1D are usually constrained by conformal symmetry, thus vastly reducing the number of parameters needed to describe the system. Although conformal symmetry provides immense simplifications analytically, the presence of logarithmic corrections to the entanglement entropy makes QCPs challenging to probe numerically. As conjectured in the finite entanglement scaling hypothesis (FES), using an MPS with a finite bond-dimension introduces a length scale into the system, spoiling the conformal symmetry. However, a careful scaling analysis in the bond-dimension can extract insights about the nature of quantum critical points. Furthermore, performing time evolution with a time dependent Hamiltonian that continuously travels through a QCP leads to excitation production, via the Kibble-Zurek (KZ) mechanism, and introduces a length scale into the problem determined by the rate v at which the Hamiltonian is changed. Deviations from the quantum critical properties are fully determined by v, and the critical exponents of the QCP. In this work, we find that observables reproduce the KZ predictions, but are modulated by a scaling function of the ratio of the two length scales. Moreover, we find that the length scale introduced by finite bond-dimension in KZ dynamics is the same length scale conjectured by the FES.