Quantum criticality in SU(3) and SU(4) antiferromagnets

We study the quantum phase transition out of the Néel state in SU(3) and SU(4) generalizations of the Heisenberg antiferromagnet with a sign-problem-free four-spin coupling (so-called $JQ$ model), by extensive quantum Monte Carlo simulations. We present evidence that the SU(3) and SU(4) order parameters and the SU(3) and SU(4) stiffness go to zero continuously without any evidence of a first-order transition. However, we find considerable deviations from simple scaling laws for the stiffness even in the largest system sizes studied. We interpret these as arising from multiplicative scaling terms in these quantities that affect the leading behavior, i.e., they will persist in the thermodynamic limit, unlike the conventional additive corrections from irrelevant operators. We conjecture that these multiplicative terms arise from dangerously irrelevant operators whose contributions to the quantities of interest are nonanalytic.

Figure 1

Binder ratio of the $JQ$ model for SU(3) and SU(4). Note particular the fact that the Binder ratio becomes negative. For the SU(3) case, it is clear that the minimum does not saturate with the system size. For the SU(4) system, the negativity appears only at very large system sizes and hence we cannot make a conclusion on the behavior in the thermodynamic limit. As explained in the text in Sec. 3, the divergence in the SU(3) case is far too slow to indicate a first-order transition.

Figure 2

$\langle W_x^2\rangle$ data for the same set of runs as Fig. 1. The $y$ axis is in a log scale. There is clear evidence for a crossing in the data. The crossing is analyzed in detail in Sec. 4.

Figure 3

A zoom-in view of the crossing of the binder ratio $B$ for SU(3)- and SU(4)-symmetric $JQ$ models. Error bars were determined by bootstrapping the data. The solid lines are polynomial fits on the data and are plotted to guide the eye. The crossing point converges well for the larger system sizes. For $L=64,96,128,\text{and}192$ the data crosses nicely within the estimated error bars, allowing very accurate brackets for the quantum critical points. ${(J/Q)}_c^3=(1.9905,1.9920)$ and ${(J/Q)}_c^4=(11.235,11.255)$.

Figure 4

A zoom-in view of the fluctuations of the spatial winding number (stiffness) for SU(3)- and SU(4)-symmetric $JQ$ models. This is the same set of simulations as for the Binder ratio data shown in Fig. 2 (this is a somewhat wider view to accommodate drift of crossing; compare $x$ axes). Clearly, there are significant deviations from the simple scaling behavior expected for a quantity of scaling dimension zero.

Figure 5

The $x$ intercept ($J/Q$) of the crossing of $L$ and $2L$ for the stiffness shown in Fig. 4, plotted versus $1/L$. The error bars are estimated by a bootstrap method detailed in the text. The error bars get smaller for large systems because these curves are steeper; the $x$ intercept of the crossing can hence be determined more precisely. At $1/L=0$, the thick red line shows the bracket for the critical point from the analysis of the Binder-ratio data (Fig. 3). Clearly, the $x$ intercept of the crossing converges to a finite value, and this value is completely consistent with the Binder-ratio crossing. The dashed line is a nonlinear fit to the form $g-A/L^a$, with $g=1.992,A=3.20,\text{and}a=1.47$ for SU(3) and $g=11.255,A=51.3,\text{and}a=1.48$ for SU(4). A motivation for this form is given in Appendix .

Figure 6

Testing linear and logarithmic divergences in the $y$ intercept of the crossing points. We have made the “fits” using only the two biggest system sizes [insets fit to $A+Bx$ and main panels to $A+B\ln \left(x\right)$]. Thus this is more of a test of the model than a fit, hence, we refer to it as a “fit”. The inset shows a linear fit (straight line on a linear-linear plot) and the main graphs a log fit (straight line on a log-linear plot). The insets clearly show that a linear divergence is completely inconsistent with our data. The main graphs show that a multiplicative log form fits the data reasonably, but not as good as the power law (see Fig. 7).

Figure 7

Same as the previous figure, but now it is a test of the possibility of a power-law divergence. Again, the straight line drawn is made by “fitting” the form $A{x}^B$ to the two largest $L$ points. This model fits the SU(3) data extremely well down to the smallest system size. For the SU(4) data, the fit is also excellent if one attributes the deviation of the smallest system size to a finite-size effect. The unfamiliar numbers on the $y$ axis are because we have chosen to tick and label the log axis with points on a base of $1.1$, which although convenient here, is not one of the familiar bases. Doing a standard linear-regression analysis on the entire data set for SU(3) and all the data except for the $L=16$ point for SU(4), we find estimates for the exponent: $B_3=0.20\left(2\right)$ and $B_4=0.16\left(2\right)$. The errors quoted here are based on roughly fitting the power-law form to different data sets and making an estimate for how much the exponent differs.

Figure 8

An identical analysis as that leading up to Fig. 7, but now for the fluctuations of the temporal winding number. The power-law fit does not work quite as well here. Again the fit is carried out by fitting the power-law form $A{x}^B$ to the two largest system sizes. The exponents we find here are $B_3=0.18\left(2\right)$ and $B_4=0.17\left(2\right)$, where the errors are determined roughly by fitting different sets of the data. Reassuringly, they are in agreement with the values found for the stiffness. We note that a fit to a multiplicative logarithm form (not shown) for $\langle W_{\tau }^2\rangle$ does just as well. See also Fig. 9 for a comparison between the susceptibility and stiffness.

Figure 9

The ratio of the fluctuations of the temporal and spatial winding numbers evaluated at the crossing points of the curves for the susceptibility. From the saturation to a finite (nonzero, noninfinite) value of the ratio at large volume shown here, it is clear that in the large-volume limit the asymptotic behavior is identical for $\langle W_x^2\rangle$ and $\langle W_{\tau }^2\rangle$. The ratio is not expected to go exactly to 1 away from the $LT/c=1$ point. The number is of order one, because of our choice of $LT$ as discussed in Appendix .