High-precision finite-size scaling analysis of the quantum-critical point of $S=1∕2$ Heisenberg antiferromagnetic bilayers

We use quantum Monte Carlo (stochastic series expansion) and finite-size scaling to study the quantum critical points of two $S=1∕2$ Heisenberg antiferromagnets in two dimensions: a bilayer and a Kondo-lattice-like system (incomplete bilayer), each with intraplane and interplane couplings $J$ and $J_{\perp }$. We discuss the ground-state finite-size scaling properties of three different quantities—the Binder moment ratio, the spin stiffness, and the long-wavelength magnetic susceptibility—which we use to extract the critical value of the coupling ratio $g=J_{\perp }∕J$. The individual estimates of $g_c$ are consistent provided that subleading finite-size corrections are properly taken into account. For both models, we find that the spin stiffness has the smallest subleading finite-size corrections; in the case of the incomplete bilayer we find that the first subleading correction vanishes or is extremely small. In agreement with predictions, we find that at the critical point the Binder ratio has a universal value and the product of the spin stiffness and the long-wavelength susceptibility scales as $1∕L^2$ with a universal prefactor. Our results for the critical coupling ratios are $g_c=2.5220\left(1\right)$ (full bilayer) and $g_c=1.3888\left(1\right)$ (incomplete bilayer), which represent improvements of more than an order of magnitude over the previous best estimates. For the correlation length exponent we obtain $\nu =0.7106\left(9\right)$, consistent with the expected three-dimensional Heisenberg universality class.

Figure 1

The arrangement of spin interactions in the (a) full and (b) incomplete bilayers. There are two different couplings: $J$ (intraplane) and $J_{\perp }$ (interplane), as indicated.

Figure 2

(Color online) Convergence as a function of inverse temperature $\beta =2^n$ of the squared sublattice magnetization to its ground state value for three different lattice sizes. The inset shows the behavior for $L=16$ on a more detailed scale.

Figure 3

The Binder ratio $Q_2$ is plotted as a function of the coupling ratio for the (a) full and (b) incomplete bilayers. Results for even $L$ from 8 to 42 are shown (all except for $L=22,26,34,38$). The slope of the curves increases with $L$.

Figure 4

The spin stiffness multiplied by the system length $L$ versus coupling ratio for the (a) full and (b) incomplete bilayers. The system sizes are the same as in Fig. 3. The slope of the curves increases with $L$.

Figure 5

The long-wavelength susceptibility multiplied by the system length versus the coupling ratio for the (a) full bilayer and (b) incomplete bilayer. The system sizes are those listed in Fig. 3. The slope of the curves again increases with $L$.

Figure 6

(Color online) Convergence versus the inverse lattice size of intersection points of curves for $L$ and $2L$, for the (a) full and (b) incomplete bilayers. The error bars are are much smaller than the symbols. The circles at $1∕L=0$ indicate the critical couplings from the careful finite-size scaling analysis carried out in Sec. 3B.

Figure 7

(Color online) The blue crosses (×) show the input points of the optimization procedure. The red plusses (+) show the output. The dashed black line indicates the magnified region correspondig to the bottom-right plot in Fig. 8.

Figure 8

(Color online) The number density of bootstrapped $(\nu ,g_c)$ solutions is plotted for $Q_2$ and ${\rho }_s$ in greyscale. The left panel shows the incomplete bilayer and the right panel the complete bilayer. The solid red lines show the contour at $1∕3$ the maximum density (this would correspond to $2∕3$ of the weight under a gaussian distribution). In the top right panel, note a two-peak structure which sometimes appears in the nonlinear fits. The dotted lines are guides to the eye. They show the $Q_2$ contour on the ${\rho }_s$ plot and vice versa.

Figure 9

(Color online) Data collapse of the Binder ratio for the (a) full and (b) incomplete bilayer. The values of $g_c$ and $\nu$ obtained are listed in Table .

Figure 10

(Color online) Data collapse of the spin stiffness ratio for the (a) full and (b) incomplete bilayer. The values of $g_c$ and $\nu$ obtained are listed in Table .

Figure 11

(Color online) The Binder ratio (top panel) and the product of the spin stiffness and the long-wavelength susceptibility (bottom panel) are plotted as a function of inverse system size for various possible values of the critical couplings. The data points represent interpolations to the hypothetical $g_c$ values. The solid lines and corresponding cones of graduated shading illustrate fits of the data for the most precise overall estimates $g_c=1.38882\left(2\right)$, $g_c=2.52181\left(3\right)$, and the fits at deviations of 2, 4, 8, and 16 error bars. Note that the scaling is not expected to be strictly $1∕L$, and all lines should be regarded as suggestive.