Criticality and Mott glass phase in a disordered two-dimensional quantum spin system

We use quantum Monte Carlo simulations to study a disordered $S=1/2$ Heisenberg quantum spin model with three different nearest-neighbor interactions, $J_1\le J_2\le J_3$, on the square lattice. We consider the regime in which $J_1$ represents weak bonds, and $J_2$ and $J_3$ correspond to two kinds of stronger bonds (dimers) which are randomly distributed on columns forming coupled two-leg ladders. When increasing the average intradimer coupling $(J_2+J_3)/2$, the system undergoes a Néel to quantum glass transition of the ground state and later a second transition into a quantum paramagnet. The quantum glass phase is of the gapless Mott glass type (i.e., in boson language it is incompressible at temperature $T=0$), and we find that the temperature dependence of the uniform magnetic susceptibility follows the stretched exponential form $\chi \sim exp(-b/T^{\alpha })$, with $0<\alpha <1$. At the Néel-glass transition, we observe the standard O(3) critical exponents, which implies that the Harris criterion for the relevance of the disorder is violated in this system.

Figure 1

2D square-lattice columnar-dimerized Heisenberg model with two types of strong bonds. Blue and red thick lines represent the strong bonds, which reside on vertical columns separated by one lattice spacing. The corresponding couplings, $J_2$ or $J_3$, are chosen at random. The thin lines stand for the weaker interdimer coupling $J_1=1$.

Figure 2

The spin stiffness (upper panel) and the staggered structure factor (lower panel) vs the inverse temperature for $L=32$ systems at coupling ratio $g=1.98$, $p=1/2$ (close to the quantum critical point). Both of the quantities exhibit sufficient convergence within statistical errors when $\beta \gtrsim 60$.

Figure 3

The staggered (lower panel) and uniform (upper panel) susceptibilities vs the inverse temperature $\beta$ for $L=32$ systems at $g=1.98$, $p=1/2$. The uniform susceptibility tends to 0 when $\beta \to \infty$ and is not useful for $T=0$ studies, while the staggered susceptibility converges to a finite nonzero value.

Figure 4

The size-scaled spin stiffness, assuming a dynamic exponent $z=1$, vs the average dimer coupling $g$ for different system sizes. The inverse temperature is scaled with the size as $\beta =2L$, and the results were averaged over 1000 disorder realizations. Curves for different $L$ are expected to cross each other at the critical coupling ratio.

Figure 5

Size-scaled uniform susceptibility (upper panel) and spin stiffness (lower panel) with dynamic exponent $z=1$ assumed. The data for each $L$ were averaged over 1000 configurations. The curves show fitted polynomials, using which crossing points are extracted. The horizontal and vertical drifts of the crossing points are analyzed in Figs. 6 and 7, respectively.

Figure 6

Size-dependent critical couplings $g_c\left(L\right)$ defined using $(L/2,L)$ crossing points of the spin stiffness and the uniform susceptibility extracted from the data in Fig. 5. The two data sets were fitted to the form (11), with different subleading exponents ($\omega \approx$ 3 and $\approx 2$ for the $\chi$ and ${\rho }_s$ fits, respectively) but constrained to a common critical point in the thermodynamic limit. This fit resulted in the infinite-size critical point $g_c=1.974\left(4\right)$.

Figure 7

Size dependence of the vertical crossing points in Fig. 5 for system-size pairs ($L/2,L$). The values based on $L{\rho }_s$ and $L{\chi }_u$ are shown in the left and right panels, respectively, along with line fits.

Figure 8

Uniform magnetic susceptibility vs temperature at $g=1.98$, for convenience graphed on a log-log scale. The size dependence is nonmonotonic, and for $L=128$ only small finite-size effects should remain for the temperatures $T\ge 0.02$ considered here.

Figure 9

Temperature dependence of the uniform susceptibility at the estimated critical coupling, $g=1.974$, for $L=128$. The main panel and the inset show the data on log-log and lin-lin scales, respectively. The data follow very closely the expected quantum-critical linear-$T$ form (12) with $a=0$, $z=1$, and $b=0.093$, as shown with the straight lines.

Figure 10

Dependence of the staggered susceptibility on the system size $L$ for $g=1.974$ (the estimated critical point). The linear fit to the data (red line) is unsatisfactory when all system sizes are included, with ${\chi }^2/\mathrm{dof}\approx 4$. The slope is $2-\eta =2.015$. The black line shows the slope $\eta =1.9625$ expected asymptotically in the standard O(3) universality class applicable to the clean system [45].

Figure 11

Finite-size scaling of the staggered susceptibility at the critical point. The data are the same as in Fig. 10, but here the fitted curve is of the form (16), including a correction to scaling. The leading power is now $2-\eta =1.971\left(6\right)$, which is consistent with the O(3) universality class, and the fit has ${\xi }^2/\mathrm{dof}\approx 1$. Going away from the estimated critical point by one error bar, the exponents change to $2-\eta =2.046\left(9\right)$ for $g=1.970$ and $2-\eta =1.966\left(4\right)$ for $g=1.978$ (${\chi }^2/\mathrm{dof}$ remains close to 1 in these fits).

Figure 12

Scaling of the staggered structure factor at $g=1.970$, $1.974$, and $1.978$. Pure power-law fits (lines on the log-log scale) give $1-\eta =0.927\left(5\right),0.948\left(1\right)$, and $0.970\left(1\right)$ with ${\chi }^2/\mathrm{dof}<1$ in all cases.

Figure 13

Tests of scaling collapse of the spin stiffness, using the finite-size scaling form (9) with SSE data at $g=1.974$. Upper panel: with the correlation-length exponent at its O(3) value [45] $\nu =0.7115$. The data collapse is good, with ${\chi }^2/\mathrm{dof}=1.02$ computed relative to a fitted polynomial. Lower panel: using $\nu =1$, corresponding to the lower edge of the range of values satisfying the Harris inequality $\nu \ge 2/d$. Here the data collapse is visibly much less satisfactory, and quantitatively the goodness is ${\chi }^2/\mathrm{dof}\approx 6.5$.

Figure 14

Uniform susceptibility vs inverse temperature for $L=32$ systems. When $g<1.974$ (in the Néel phase), the curves flatten out and converge to a nonzero constant (flat line). For $g=g_c\approx 1.974$, the $T$-linear behavior applies, as shown with the fitted curve. When $g=3$, a standard exponential decay $e^{-\Delta /T}$ is found (fitted curve shown), indicating a gapped quantum paramagnet, while for smaller $g>g_c$ there are significant deviations from the exponential form (the lines are drawn here to go through the data for the higher temperatures), suggesting the system is a Mott glass.

Figure 15

Uniform susceptibility of systems in the MG phase. The fitted curves are of the form (18) with $\alpha =0.15\left(1\right)$, 0.47(1), and 0.66(1) for $g=2.0$, 2.1, and 2.2, respectively.