Universal and non-universal facets of quantum critical phenomena unveiled along the Schmidt decomposition theorem

TL;DR

This paper explores the universal and non-universal features of quantum critical phenomena in the 1D Ising model, focusing on the quantum Grüneisen parameter and the role of the entropic index in the Schmidt decomposition at critical points.

Contribution

It reveals how the quantum Grüneisen parameter varies with spin magnitude, the conditions for entropy extensivity, and the dependence of universality classes on system symmetry, along with proposing an experimental setup.

Findings

Higher spin increases the quantum Grüneisen parameter but keeps it finite.

The Schmidt decomposition recovers entropy extensivity only at a special entropic index q.

Universality class depends solely on the system's symmetry.

Abstract

Critical phenomena have been extensively investigated both theoretically and experimentally in many fields, such as condensed matter physics, biology, e.g., brain criticality, and cosmology. In particular, the behaviour of response functions right at critical points (CPs) is highly topical. It turns out that in the frame of Boltzmann-Gibbs-von Neumann-Shannon approach, the extensive character of entropy breaks down at CPs. The latter implies diverging susceptibilities, which is at odds with experimental observations. Here, we investigate the influence of the spin magnitude $S$ on the quantum Gr\"uneisen parameter $\Gamma^{0\text{K}}_{q}$ right at CPs for the 1D Ising model under a transverse magnetic field. Our findings are fourfold: $\textit{i}$) for higher $S$, $\Gamma^{0\text{K}}_{q}$ is increased, but remains finite, reflecting the enhancement of the Hilbert space dimensionality; $\textit{ii}$) the Schmidt decomposition theorem recovers the extensivity of the nonadditive $q$-entropy $S_q$ only for a $\textit{special}$ value of the entropic index $q$; $\textit{iii}$) the universality class in the frame of $S_q$ depends only on the symmetry of the system; $\textit{iv}$) we propose an experimental setup to explore finite-size effects in connection with the Hilbert space occupation at CPs. Our findings unveil both universal and non-universal aspects of quantum criticality in terms of $\Gamma^{0\text{K}}_{q}$ and $S_q$.