Absence of small-world effects at the quantum level and stability of the quantum critical point

TL;DR

This paper demonstrates that, unlike classical systems, quantum systems do not exhibit small-world effects at the quantum critical point, which remains stable despite the addition of random long-range couplings.

Contribution

It proves that small-world effects do not manifest at the quantum level, showing the stability of the quantum critical point against random long-range couplings.

Findings

Quantum critical point remains unchanged with added random couplings.

Small-world effects are absent at the quantum level due to quantum fluctuations.

Quantum criticality is a stable thermodynamic feature not mirrored in classical systems.

Abstract

The small-world effect is a universal feature used to explain many different phenomena like percolation, diffusion, and consensus. Starting from any regular lattice of $N$ sites, the small-world effect can be attained by rewiring randomly an $\mathcal{O}(N)$ number of links or by superimposing an equivalent number of new links onto the system. In a classical system this procedure is known to change radically its critical point and behavior, the new system being always effectively mean-field. Here, we prove that at the quantum level the above scenario does not apply: when an $\mathcal{O}(N)$ number of new couplings are randomly superimposed onto a quantum Ising chain, its quantum critical point and behavior both remain unchanged. In other words, at zero temperature quantum fluctuations destroy any small-world effect. This exact result sheds new light on the significance of the quantum critical point as a thermodynamically stable feature of nature that has no analogous at the classical level and essentially prevents a naive application of network theory to quantum systems. The derivation is obtained by combining the quantum-classical mapping with a simple topological argument.