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.

Figure 1

Two types of small-world networks with $N=12$ sites. (a), (b) Long-range bonds are added with probability $p$, without diluting the underlying 1D structure. (c) All possible long-range links are added, but with a reduced strength $\propto 1/N$ vanishing at large sizes.

Figure 2

The crossover probability $p^{★}$, defined in Eq. (3.11), is plotted versus the long-range coupling $J^{\prime }/J$ for different values of Ising anisotropy $\mathrm{\Delta }$. The inset shows the behavior of the ($J^{\prime }/J$)-independent prefactor of $p^{★}$ versus $\mathrm{\Delta }$. Note its singular behavior as $\left|\mathrm{\Delta }\right|\to 1$.

Figure 3

QMC transverse susceptibility of the XY chain ($\mathrm{\Delta }=0$ and $p=0$) as a function of the temperature $T$ for different system sizes from $N=128$ to $N=2048$. The straight black line is the parameter-free expression of Eq. (3.7). It fits asymptotically well the QMC data at low energy for $T/J\lesssim 0.1$, when the Tomonaga-Luttinger liquid description becomes valid.

Figure 4

QMC results for the staggered susceptibility of the Heisenberg chain ($p=0$) as a function of the temperature $T$. Different symbols show system sizes from $N=128$ to $N=2048$. The classical result [72] is also shown by the solid line. Inset: ${\left(T\chi \right)}^2$ is plotted against ${\left(T/J\right)}^{-1}$. The solid line is a fit to the form ${\chi }_0^2\left(ln\mathrm{\Lambda }-lnT\right)$, according to Eq. (3.14), with ${\chi }_0=0.2823\left(16\right)$ and $\mathrm{\Lambda }=22.7\left(20\right)$.

Figure 5

Hastings model. Data collapse for the staggered XXX antiferromagnet with $J^{\prime }/J=0.25$. Three estimates are considered: (a) the binder cumulant, (b) the square of the order parameter evaluated from the spin-spin correlation at long distance, and (c) the normalized structure factor. Setting $\tilde{\nu }=2$ and ${\beta }_{\mathrm{MF}}=1/2$, we find that $T_{\mathrm{c}}\simeq 0.16$. The other fitting parameters are reported in Appendix pp2. The insets show the data crossing without any correction to the scaling.

Figure 6

Critical temperature of the Hastings model, plotted as a function of the average long-range coupling strength $\overline{J^{\prime }\left(p\right)}/J$ for the ferromagnetic XY and staggered antiferromagnetic Heisenberg models. The RPA estimates (analytical and QMC) are also displayed for each model.

Figure 7

Data collapse for the XY ferromagnet on the SW network with $p=0.03125$: the square of the order parameter evaluated from (a) the normalized structure factor and (b) the spin-spin correlation at long distance. Setting $\tilde{\nu }=2$ and ${\beta }_{\mathrm{MF}}=1/2$, we find that $T_{\mathrm{c}}\simeq 0.065$. The other fitting parameters are reported in Appendix pp2. The insets show the data crossing without any correction to the scaling. (c) Critical temperature of the ferromagnetic XY and staggered antiferromagnetic Heisenberg models versus the average strength of the extra couplings. In each case, the analytical RPA estimate is also displayed. In the staggered antiferromagnetic Heisenberg model, the two estimates agree as $\overline{J^{\prime }\left(p\right)}\to 0$. We plot in the inset the renormalization parameter $\alpha$ (see text). As $\overline{J^{\prime }\left(p\right)}\to 0$, one sees that $\alpha \to 1$. In the ferromagnetic XY case, the RPA and QMC estimates deviate below $\overline{J^{\prime }\left(p\right)}/J\sim 0.1$. The dashed line is a linear fit $\propto \overline{J^{\prime }\left(p\right)}/J$.

Figure 8

Critical temperature of the dimerized antiferromagnetic model, Eq. (4.11), with $\delta =0.25$ and additional long-range couplings of the Hastings form of strength $\overline{J^{\prime }\left(p\right)}$. The RPA estimate compares perfectly to the QMC results. Inset: Temperature dependence of the staggered susceptibility of the $1\mathrm{D}$ dimerized system.