Quantum magnetism on small-world networks

While classical spin systems in random networks have been intensively studied, much less is known about quantum magnets in random graphs. Here, we investigate interacting quantum spins on small-world networks, building on mean-field theory and extensive quantum Monte Carlo simulations. Starting from one-dimensional (1D) rings, we consider two situations: all-to-all interacting and long-range interactions randomly added. The effective infinite dimension of the lattice leads to a magnetic ordering at finite temperature $T_\mathrm{c}$ with mean-field criticality. Nevertheless, in contrast to the classical case, we find two distinct power-law behaviors for $T_\mathrm{c}$ versus the average strength of the extra couplings. This is controlled by a competition between a characteristic length scale of the random graph and the thermal correlation length of the underlying 1D system, thus challenging mean-field theories. We also investigate the fate of a gapped 1D spin chain against the small-world effect.