Long-range random transverse-field Ising model in three dimensions

We consider the random transverse-field Ising model in $d=3$ dimensions with long-range ferromagnetic interactions which decay as a power $\alpha >d$ with the distance. Using a variant of the strong-disorder renormalization group method we study numerically the phase-transition point from the paramagnetic side. We find that the fixed point controlling the transition is of the strong-disorder type, and based on experience with other similar systems, we expect the results to be qualitatively correct, but probably not asymptotically exact. The distribution of the (sample dependent) pseudocritical points is found to scale with $1/lnL, L$ being the linear size of the sample. Similarly, the critical magnetization scales with ${(lnL)}^{\chi }/L^d$ and the excitation energy behaves as $L^{-\alpha }$. Using extreme-value statistics we argue that extrapolating from the ferromagnetic side the magnetization approaches a finite limiting value and thus the transition is of mixed order.

Figure 1

Illustration of the spin clusters formed during the SDRG process at the critical point of the 1D short-range random transverse-field Ising model for two samples at $L=256$. The fate of a spin cluster is either to be decimated out (indicated by vertical spikes) or to be fused together with an other spin cluster (horizontal lines). Higher trees indicate clusters being present at later stages of the SDRG procedure, corresponding to the low-energy modes of the system. The magnetic moment is related to the size of the largest cluster, scaling as $\mu \left(L\right)\propto L^{d_f}$, where the fractal dimension is $d_f=\frac{\sqrt{5}+1}{4}\approx 0.809$ [26].

Figure 2

The same dendrogram illustration as in Fig. 1 for the critical 1D long-range random transverse-field Ising model at $L=256$. As opposed to the short-range model, mostly transverse fields are decimated resulting in spikes and smaller spin clusters, following a scaling of the form $\mu \left(L\right)\propto {ln}^{2}L$.

Figure 3

The large-scale spin clusters of the critical long-range random transverse-field Ising model appear to be sparse and they can be embedded in a quasi-one-dimensional object, as illustrated in 2D ($L=64$) and 3D ($L=24$). The size of these clusters provides the magnetic moment following a similar scaling as in 1D as shown in Eq. (4).

Figure 4

Distribution of the pseudocritical points, which are estimated to cross each other for different $L$ at ${\theta }_c\approx 3.25$, is indicated by a dotted line. The ratio of the accumulated distributions on two sides of ${\theta }_c$ is given by $r_{{\theta }_c}\left(L\right)$ (see the text and Fig. 5). The inset shows the rescaled distributions with $L_0=2$.

Figure 5

The scaled decimation ratio, $r_{{\theta }_c}\left(L\right)$, as a function of the $L$ size at $L_0=2$, indicating a similar logarithmic scaling as in 1D.

Figure 6

The average mass of the last decimated cluster plotted against ${(lnL/L_0)}^2$ with $L_0=3.2$. The data has been obtained by numerical renormalization of the three-dimensional model with decay exponent $\alpha =4$, for different values of the control parameter $\theta$.

Figure 7

Effective dynamical exponents obtained by two-point fits using Eq. (5) as a function of the system size $L$. The straight line, which connects the data obtained for the critical value $\theta =3.25$ is a guide to the eye.

Figure 8

(a) Distributions of the last decimated transverse fields at the critical point for different sizes. The straight line indicates the asymptotic shape of the tail according to EVS, with $z=\alpha$ (see text). (b) As the dynamical exponent, $z$ decreases, the shape of the distribution changes in the paramagnetic phase as illustrated at $L=16$.

Figure 9

Schematic SDRG phase diagram obtained through the maximum rule: the arrows indicate the direction the parameters evolve as the energy scale is reduced. Fixed points (blue circles) are at $r=0$: the attractive fixed points of the paramagnetic phase ($\alpha /z>1$) and the repulsive ones ($\alpha /z<1$) are separated by the critical fixed point (red circle); see the text.