Quantum effects on criticality of an Ising model in scale-free networks: Beyond mean-field universality class

We study the effect of quantum fluctuations on the critical behavior of the Ising ferromagnetic phase transitions that do not belong to the mean-field universality class. A model system is considered, in which Ising spins are placed on the nodes of a scale-free network. Our Monte Carlo analysis shows that the critical exponents differ from those of mean-field phase transitions when degree exponent $\gamma$ is in the range $3<\gamma <5$. This confirms earlier analytic calculations based on ansatzes and approximation methods. As we apply quantum fluctuations by means of a magnetic field perpendicular to the Ising spin direction, the transition temperature $T_c$ decreases with increasing magnetic field strength. We find, however, that the quantum fluctuations do not alter the critical exponents and the universality class remains unchanged.

Figure 1

The fourth-order Binder cumulant $U_N$ as a function of temperature $T$ is drawn for $\gamma =4.5$ and $k_{\text{av}}=30$. Error bars are about the same size as the symbols. The crossing point is interpreted as the critical point. In this example, $T_c/J=34.3\pm 0.2$. This is consistent with the analytic formula [1, 2] $T_c=-2J/\text{ln}\left(1-2⟨k⟩/⟨k^2⟩\right)$, which becomes $34.35\pm 0.03$ if we use our numerical results $⟨k⟩=30.00\pm 0.01$ and $⟨k^2⟩=1060\pm 1$.

Figure 2

Scaling functions for $\gamma =4.5$ and $k_{\text{av}}=30$. (a) Specific heat, (b) magnetization, and (c) magnetic susceptibility are plotted using $T_c=34.3\pm 0.2$. Error bars are about the same as or smaller than the symbol sizes. The critical exponents are determined to be $\alpha =-0.3\pm 0.1$, $\beta =0.67\pm 0.05$, $\overline{\gamma }=1.00\pm 0.05$, and $\overline{\nu }=2.4\pm 0.1$.

Figure 3

Transition temperature $T_c$ is drawn against the perpendicular magnetic field $\Delta$. When interpolated, the curve separates the ferromagnetic and the paramagnetic phases. The parameters used in this plot are $\gamma =4.5$ and $k_{\text{av}}=30$.

Figure 4

Scaling functions for $\gamma =4.5$, $k_{\text{av}}=30$, and $\Delta =20J$. (a) Specific heat, (b) magnetization, and (c) magnetic susceptibility are plotted using $T_c=30.2\pm 0.3$. Error bars are about the same size as the symbols. We find that the critical exponents are unchanged from the classical values in Table : $\alpha =-0.3\pm 0.1$, $\beta =0.66\pm 0.07$, $\overline{\gamma }=1.00\pm 0.05$, and $\overline{\nu }=2.3\pm 0.1$.