Quantum Ising chains with small-world couplings: No small-world effect at the quantum level; Stability of the quantum critical point

Massimo Ostilli

Published 2019-12-03Version 1

Due to the small-world effect, the critical behavior of finite dimensional classical systems of N spins is known to change radically when an O(N) number of couplings are randomly rewired or superimposed onto the original system. In particular, one-dimensional systems acquire a finite critical temperature while two-dimensional systems get higher critical temperatures and, in both cases, the critical behavior turns out to be mean-field like. Here, we prove that at the quantum level the above scenario does not apply: when an O(N) number of extra ferromagnetic couplings are randomly superimposed onto a quantum Ising chain, its quantum critical point and behavior remain both unchanged. In other words, at zero temperature quantum fluctuations destroy any small-world effect (quantum networks cannot be small-world). This exact result sheds new light on the significance of the quantum critical point as a thermodynamically stable feature of nature and might be crucial for quantum annealing.