Griffiths-McCoy singularities in the dilute transverse-field Ising model: A numerical linked cluster expansion study

We use numerical linked cluster expansions (NLCEs) to study the site-diluted transverse-field Ising model on the square lattice at $T=0$. NLCE with a self-consistent mean field on the boundary of the clusters is used to obtain the ground-state magnetization, susceptibility, and structure factor as a function of transverse field $h$ and exchange constant $J$. Adding site dilution to the model turns NLCE into a series expansion in the dilution parameter $p$. Studying the divergence of the structure factor allows us to establish the phase diagram in the $h/J$ and $p$ plane. By studying the magnetization of the system in a longitudinal field, we investigate the Griffiths-McCoy singularities. We find that the magnetization develops nonlinearities in the Griffiths phase with exponents that vary continuously with $h$. Additionally, the probability distribution of the local susceptibility develops long tails in the Griffiths phase, which is studied in terms of its moments.

Figure 1

Panel (a) shows the structure factor $S$ and panel (b) shows the susceptibility $\chi$. For both quantities, the NLCE produces a series expansion in the number of bonds in clusters. Our analysis includes clusters up to order 10. Curves of order 1–10 in the number of bonds are shown simultaneously with increasing redness starting with blue at first order. The NLCE converge well for $h>h_c$ and increase sharply close to $h_c$ before saturating for small values of $h$. The point at which the steep rise begins occurs closer and closer to $h_c$ the higher the order.

Figure 2

In all three plots, orders 4–10 in the number of bonds are shown, with higher orders shown in redder colors. Panel (a) shows the magnetization computed by using the self-consistency constraint. Its shape is qualitatively typical for the magnetization of a ferromagnet, falling to zero near the critical point. Panels (b) and (c) show curves of the structure factor $S$ and susceptibility $\chi$ plotted by using the magnetization self-consistent mean field. Both quantities now converge very well in both the ordered and disordered phases, peaking near the critical point.

Figure 3

(a) Phase diagram, showing distinct ferromagnetic (FM), paramagnetic (PM), and Griffiths (GP) phases. Using linear regression on data like that shown in Fig. 4 for many values of $h$, we established the boundary shown by the full line. This line begins at the approximate $h_c$ of the pure system, which we find to be 3.03, close to the known value of 3.04. The flat continuation of the phase boundary we expect after convergence of the NLCE breaks down for $h<h_M$ is shown by the horizontal dashed line. This point begins at $h_M\simeq 1.65$. This line intersects the $p$ axis at the percolation probability $p_c$, which we find to be 0.58, also close to the known value of 0.59. The dotted line marked with $h_c^{d-1}$ shows the value of the one-dimensional critical point, a lower bound on $h_M$. The vertical dashed line indicates a separation between the ordinary paramagnetic phase and the disordered Griffiths phase where we observe Griffiths-McCoy singularities to be present. Panel (b) shows the value of $\gamma$ computed from the slope of the same linear regression. This gives the rough bounds $0.47\lesssim \gamma \lesssim 0.61$.

Figure 4

Plots of ratios $a_n/a_{n-1}$ as a function of $1/n$ for a representative range of $h$ values with $n$ ranging from 4 to 10. For large values of $h$, these plots are all relatively flat. For values of $h\lesssim h_c$, the intercept of a linear regression of the values shown here will yield a value of $1/p_c$ that is $<1$. This implies an absence of any phase transition in the physically relevant range of $p$ values in the system. Subsequently, the $h$ at which the computed $p_c$ becomes unity gives an approximation of the pure system critical point $h_c$. Additionally, for sufficiently small values of $h$, the plots become very nonlinear and irregular, indicating that the convergence of NLCE breaks down below some $h_M$. This is to be expected as at the multicritical point $h_M$ critical behavior switches from being governed by the value of $h$ to being controlled by the geometry of the lattice, and the singularity is no longer a power-law for $h$ below $h_M$.

Figure 5

Both plots show computations for $p=0.5$, well below the percolation threshold $p_c=0.59$ and in the Griffiths phase for small $h$. Computations were done to 10th order in the number of sites. Panel (a) shows magnetization $M$ plotted as a function of longitudinal field $h_L$ for a representative range of $h$ values. Below the critical point $h_c=3.04$, curvature begins to develop in the small-$h_L$ limit. This can be better visualized in panel (b), showing a log-log plot of the same quantities. For $h>h_c$, the slopes of the log-log plot are about 1, while for values $h<h_c$, they are noticeably less than 1.

Figure 6

Plot of the exponent $a$ as a function of $h$ for $p=0.5$. For $h>h_c$, $a\simeq 1$ as is typical of the pure system. For $h<h_c$, $a$ begins decreasing continuously with $h$, indicating the influence of GM singularities.

Figure 7

Panel (a) shows plots of the moments of the local susceptibility ${\chi }_{\text{loc}}^n$ shown for $p=0.5$ for $n$ ranging from 1 to 10, with smaller $n$ values appearing in blue and larger $n$ values in red. All values were computed to 10th order in the number of sites. Panel (b) shows a series of points where each moment crosses a selection of values $v$ [indicated by horizontal lines in panel (a)], indicating roughly where each moment begins to diverge, plotted as a function of $1/n$. As $n$ is increased, these plots begin to curve up pushing them towards $h_c$.