Anomalous Quantum-Critical Scaling Corrections in Two-Dimensional Antiferromagnets

We study the Néel-paramagnetic quantum phase transition in two-dimensional dimerized $S=1/2$ Heisenberg antiferromagnets using finite-size scaling of quantum Monte Carlo data. We resolve the long-standing issue of the role of cubic interactions arising in the bond-operator representation when the dimer pattern lacks a certain symmetry. We find nonmonotonic (monotonic) size dependence in the staggered (columnar) dimerized model, where cubic interactions are (are not) present. We conclude that there is a new irrelevant field in the staggered model, but, at variance with previous claims, it is not the leading irrelevant field. The new exponent is ${\omega }_2\approx 1.25$ and the prefactor of the correction $L^{-{\omega }_2}$ is large and comes with a different sign from that of the conventional correction with ${\omega }_1\approx 0.78$. Our study highlights competing scaling corrections at quantum critical points.

Figure 1

The Heisenberg SDM and CDM studied in this work. Black (thinner) and red (thicker) bonds represent intra- and interdimer exchange ${\mathbf{S}}_i·{\mathbf{S}}_j$, of strength (prefactor) $J_1$ and $J_2$, respectively, between $S=1/2$ spins.

Figure 2

Binder ratio of the SDM for several system sizes in the neighborhood of $g_c$. The curves are polynomial fits giving crossing points $(g^{*},R^{*})$ between $(L,2L)$ data.

Figure 3

Inverse system size dependence of $(L,2L)$ crossing data for the SDM (a),(b) and the CDM (c),(d) along with joint fits (green curves) of the forms in Eqs. (4). The exponent $\omega$ is adjusted for optimal fits, giving $\omega =0.60(4)$ for the SDM and $\omega =0.80(2)$ for the CDM. The insets show the large system data on more detailed scales. The red curve in the inset of (a) shows a fit with only the leading terms arising from the first and second irrelevant fields, with ${\omega }_1=0.78$ fixed and ${\omega }_2=1.22(5)$ resulting from the fit; the corresponding fitting curve in (b) barely changes and is not shown.

Figure 4

Joint fits of crossing data for several quantities where $g^{*}(\infty )=g_c$ is fixed to a common value and two corrections are used to first order, with ${\omega }_1=0.78$ and $1/\nu =1.406$. The insets zoom in on the data for the larger system sizes. For the SDM (a), the fit delivers $g_c(\infty )=2.51945(1)$ and ${\omega }_2=1.30(7)$, 1.3(1), 1.2(1), and 1.0(2) from $R$, $L{\chi }_u$, $L{\rho }_x$, and $L{\rho }_y$, correspondingly. In the CDM fits (b), $2{\omega }_1=1.56$ was used in place of ${\omega }_2$ and $g_c=1.90956(2)$.

Figure 5

Size dependence of the exponent $\eta$ as defined in Eq. (6). The known infinite-size value $\eta =0.0375$ is fixed in the fits (curves). The CDM data are fitted with only the first correction term in Eq. (7), with ${\omega }_1=0.78$ fixed. In the SDM fit, ${\omega }_1=0.78$ is also fixed and the second exponent ${\omega }_2=1.29(5)$ is the result of the fit.