Anomalous quantum-critical scaling corrections in two-dimensional antiferromagnets
We study the Néel–paramagnetic quantum phase transition in two-dimensional dimerized 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 non-monotonic (monotonic) size dependence in the staggered (columnar) dimerized model, where cubic interactions are (are not) present. We conclude that there is an irrelevant field in the staggered model that is not present in the columnar case, but, at variance with previous claims, it is not the leading irrelevant field. The new exponent is and the prefactor of the correction is large and comes with a different sign from that of the formally leading conventional correction with exponent . Our study highlights the possibility of competing scaling corrections at quantum critical points.
One of the best understood quantum phase transitions is that between Néel antiferromagnetic (AFM) and quantum paramagnetic ground states in bipartite two- and three-dimensional (2D and 3D) dimerized Heisenberg models with inter- and intra-dimer couplings and sigma (); 3dmap (); millis (); chubukov94 (); always (); thebook (). The ground state of such a system hosts AFM order when the coupling ratio is close to , and there is a critical point at some model dependent . The 3D version of this transition for the most important case of spins has an experimental realization in TlCuCl under high pressure merchant14 (); qin15 (). While no 2D realization exists as of yet (though the magnetic field driven transition out of the QPM does have realizations bec ()), this case has been very important for developing a generic framework for 2D quantum phase transitions of the Néel AFM state gc (). The field theory of the AFM–paramagnetic transition is now well developed, and efficient quantum Monte Carlo (QMC) methods can be used to study ground states of microscopic models with tens of thousands of spins thebook (). Many non-trivial predictions for scaling in temperature, frequency, system size, etc., have been tested this way sandvik94 (); troyer1996 (); ladder (); bilayer (); sen15 (); lohofer15 ().
Despite many successes, there are still remaining questions surrounding the 2D AFM–paramagnetic transition. A long-standing unresolved issue is differences observed in QMC calculations between two classes of dimer patterns in systems different (); jiang09 (); unusual (); cubic (); accurate (), exemplified by the often studied columnar dimer model (CDM) and the initially less studied staggered dimer model (SDM), both illustrated in Fig. 1. Indications from finite-size scaling of a universality class different from the expected 3D O(3) class in the SDM different () led to several follow-up studies unusual (); cubic (); accurate (). The consensus now is that there is no new universality class, as defined by the standard critical exponents. However, because of the lack of a certain local reflection symmetry of the dimer pattern, cubic interactions arise in the bond-operator description of the SDM, which in the renormalization group corresponds to an irrelevant field that is present neither in the CDM nor in the classical O(3) model cubic (). Thus, the SDM contains an interesting quantum effect worthy of further investigations.
In this Letter we report large-scale detailed comparisons of the finite size () scaling corrections of type in the CDM and SDM. While previous works on judiciously chosen observables unusual () and lattices with optimized aspect ratios accurate () have convincingly demonstrated that there is no new universality class, the reasons for the unusual scaling behaviors of the SDM have never been adequately explained. In Ref. cubic, , QMC calculations indicated that the exponent of the leading correction is smaller than in the CDM, but the values, the observed in the SDM cubic (); accurate () versus the conventional value guida98 (); besteta () in the standard O(3) model and the CDM, are not very different. The only slightly smaller value for the SDM does not fully explain all the observed anomalous finite-size scaling properties, and, as we will show here, this scenario is actually incorrect.
We study CDM and SDM systems of size up to . Focusing on the scaling corrections, we fix the leading critical exponents at their known O(3) values in our finite-size analysis. This enables us to go to higher order in the irrelevant fields and investigate also subleading corrections. In contrast to the previous studies, we demonstrate that the SDM actually does not have a smaller than the CDM. Instead, the cubic interaction induces the next correction, which has and a large prefactor of sign different from that of the first correction. This causes non-monotonic finite-size behaviors that were previously either not observed unusual (); cubic () or were not analyzed properly accurate ().
QMC and fitting procedures.—We use the standard stochastic series expansion QMC method sandvik99 (); thebook () for spins and set the inverse temperature at ; thus the ratio is close to the value of the spinwave velocity accurate () and the effective imaginary time dimension is approximately equal to the spatial dimension. At a quantum phase transition with dynamic exponent (as is the case here), as long as the temperature does not appear as an independent argument in the scaling function obtained from renormalization group theory. In the case of a dimensionless quantity we have barber (); compost ()
if is sufficiently close to . Here denotes the irrelevant fields, which we order such that . Useful dimensionless quantities to study in QMC calculations include the Binder ratio , where is the component of the staggered magnetization along the quantization axis, the -normalized spin stiffness constants and (with and referring to the lattice directions), and the uniform susceptibility . We refer to Ref. thebook, for technical details.
Denoting the deviation from the critical point by , the standard approach to analyzing the leading critical behavior with a single correction is to expand Eq. (1) to linear order in the first irrelevant field,
where and are scaling functions related to the original . Thus, in the absence of corrections, a dimensionless quantity is completely size independent at , and by expanding we see that for different cross each other at . With the scaling correction, the crossing points only drift toward as , and for two different sizes and (where we will use ) one can derive simple expressions for the crossing value and the observable at this point luck ();
with constants and .
We extract the crossing points using polynomial fits (typically of third order) to several points (of the order 10) in the neighborhood of . The window of points used in these fits is reduced as the system size is increased, so that we are always in the regime where a low-order expansion around is expected to be valid. Since we interpolate, as opposed to extrapolate, this is a very reliable way of extracting the crossing points and their statistical errors (using bootstrapping for the latter). Examples of raw data along with fits are shown in Fig. 2 in the case of the Binder ratio of the SDM.
In the following we analyze crossing points between curves for system sizes and . When fitting the so defined and to appropriate forms from finite-size scaling theory, it should be noted that the same system size can appear in two pairs, as well as . There are therefore some covariance effects, which we take into account by using the full covariance matrix (computed using bootstrap analysis) in the definition of the goodness of the fit (normalized per degree of freedom henceforth). When jointly fitting to two different but correlated quantities, we also account for the associated covariance. For the functional forms, we will go beyond the first-order expansion leading to Eqs. (3), and this will be the key to our findings and conclusions.
Finite-size scaling.—The size dependence of crossing points is shown in Fig. 3 for both models. A striking feature is the non-monotonic behaviors apparent for the SDM but not present for the CDM. Note here that on the horizontal axis refers to the smaller of the two system sizes used for the crossing points, and the maximums are located at . In the original discovery of the anomalous behaviors for the SDM different (), all the systems were smaller, and no non-monotonic behaviors were therefore observed. It is clear that extrapolations only based on the smaller system sizes cannot reproduce the correct asymptotic behaviors.
We will first assume that only one irrelevant field is important but treat the corrections beyond the first-order expansion, Eq. (2), in . Later we will argue that one has to include also the next exponent in the case of the SDM, while for the CDM this exponent is much larger and does not have to be considered. Even with only one irrelevant field, if the associated exponent is small, the higher order terms such as will clearly also be important. As a guide to how far to go, we here compare the previous estimates cubic (); accurate () in the SDM with the second correction exponent of the O(3) model, newman84 (), and also note that several additional corrections with exponents close to are expected hasen01 (). It would then be pointless to go to higher order than in the first irrelevant field, and with we also do not include mixed corrections with and . Thus, for the SDM we use
and exclude small systems until good fits are obtained. For the CDM, with , by the above arguments we do not include the corrections.
The fitting coefficients and in Eq. (4) are not fully independent of each other but are related because they originate from the same scaling function, Eq. (1). We do not write down the rather complicated relationships here but fully take them into account in joint fits of the and data. These nonlinear fits are quite demanding and we make use of a slow but reliable stochastic approach similar to the one discussed in Ref. optim, . The stability of the fits is greatly aided by fixing to its known 3D O(3) value besteta (). The resulting curves are shown in Fig. 3. Here, as in all cases below, all data points shown in the figure were included in the fits (with smaller sizes excluded until the fits become acceptable).
For the CDM, our result for the critical coupling is , where the number within parathesis here and henceforth denotes the statistical error (one standard deviation of the mean) in the preceding digit. This is consistent with the best previous result, thebook () and accurate (), but with reduced statistical error. For the correction, we obtain , which agrees with the O(3) value besteta ().
For the SDM we obtain , which is consistent with obtained previously using lattices different (), but with a much smaller error bar. It should be noted that the previous analysis was different from our approach here. Using rectangular lattices with optimized aspect ratio, the critical point was estimated at in Ref. accurate (), which agrees with our result within error bars. For the correction we obtain , which is clearly smaller than the known O(3) value cited above but in good agreement with the values presented in both Refs. cubic () and accurate ().
Although is universal in the sense that it does not depend on the micro structure of lattice and details of the interactions, its value does depend on boundary conditions kamienartz (); selke (), including aspect ratios. The CDM and SDM have different critical spin wave velocities and, therefore, effectively different time-space aspect ratios even though is the same. This explains the different values in Fig. 3; see also Supplemental Material smcite ().
By analyzing also the spin stiffness and the uniform susceptibility in the manner described above, we obtain the results summarized in Tab. 1. The results for the CDM consistently reproduce the known O(3) value of , while in the case of the SDM the different quantities produce a wide range of results. This behavior makes us suspect that in this case the extracted may not be the true smallest correction exponent, but, as also pointed out in Ref. cubic (), should be regarded as an “effective exponent”, i.e., one influenced by neglected further corrections. The inability of a single irrelevant field to describe the data is actually not unexpected within the scenario of irrelevant cubic interactions cubic (), because the standard leading correction with should still be present and may produce various “effective” scaling behaviors over a limited range of system sizes when combined with the cubic perturbation. Thus, a reliable analysis of the SDM data should require at least and .
We can generalize Eqs. (4) to two correction exponents, and , but in that case it is very difficult to determine both of them with sufficient precision. However, since the standard leading correction should still be present cubic (), we now also can fix and only treat as a free parameter. We find that it is then sufficient to go only to linear order in the corrections and yet obtain fully acceptable fits with . We obtain and for the SDM. The new fitted curve is shown in the inset of Fig. 3(a). The estimate of is now a bit higher than the previous value from the fit (though not much outside one error bar of the difference).
The key result here is clearly that comes out larger than the leading O(3) exponent. It is, however, significantly smaller than the expected second irrelevant exponent with value newman84 (); hasen01 (), and it is also less than . The new correction should therefore be due to the cubic interactions cubic () in the low-energy theory of the SDM. To test the stability of across different quantities, we also used a slightly different procedure of fitting only to (instead of the joint fit with ) and requiring the same value of for all the quantities considered. We still also fix and but keep free for all individual quantities. The SDM data with fits are displayed in Fig. 4(a), with the resulting and estimates listed in the caption. All four estimates are statistically consistent with the value obtained above. In the case of the CDM, shown Fig. 4(b), we follow the same procedures but replace by and there is no free exponent. This fit is of marginally good statistical quality even when starting the fits from , indicating some effects still of the higher-order terms that were included in Fig. 3(b). We therefore keep the value from in Tab. 1 as our best estimate for this model.
To further ascertain our conclusions about the SDM, we also consider the squared order parameter itself. Having determined a precise estimate of , we study the scaling of at this point, where we expect
We can then define a size-dependent exponent as
which should scale as
To test this form and extract , we use the known value besteta () and fix . As shown in Fig. 5, the form fits the data very well and gives . Here one can again see how access to only system sizes less than would lead to the wrong conclusion. A fit with two adjustable exponents give and , perfectly consistent with the fit with fixed. In the case of the CDM, also shown in Fig. 5, we find that the data are well described with a single correction with the known value of the exponent.
Conclusions.—We have analyzed the SDM under the scenario cubic () of an O(3) quantum phase transition with an additional irrelevant perturbation that is absent in the CDM. Our results are consistent with this picture and demand a new scaling correction with exponent that is larger than the also present conventional 3D O(3) exponent but smaller than the next known O(3) exponent. Thus, the cubic interactions in the low-energy theory are formally more irrelevant than previously believed cubic (); accurate (), but their effects are important in finite-size scaling of many quantities because of their different signs and larger prefactors of the correction terms (four times larger than the factor of the leading correction in the case of the order parameter), thus giving rise to non-monotonic behaviors.
In addition to resolving the role of the cubic interactions in the class of models represented by the SDM, our study also serves as an example of finite-size behaviors that may at first sight appear puzzling but can be understood once the possibility of competing scaling corrections is recognized. Nonmonotonic scaling has also been observed at the deconfined quantum phase transitions, which has complicated efforts to extract the critical point and exponents shao16 ().
Acknowledgements.Acknowledgments.—We would like to thank Ning Su, Stefan Wessel, and Matthias Vojta for useful discussions. The work of N.M. and D.X.Y. was supported by Grants No. NKRDPC-2017YFA0206203, No. NSFC-11574404, No. NSFC-11275279, No. NSFG-2015A030313176, Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund, and the Leading Talent Program of Guangdong Special Projects. H.S. was supported by the China Postdoctoral Science Foundation under Grants No. 2016M600034 and No. 2017T100031. W.G. was supported by NSFC under Grants No. 11775021 and No. 11734002. A.W.S was supported by the NSF under Grant No. DMR-1710170 and by a Simons Investigator Award. Some of the calculations were carried out on Boston University’s Shared Computing Cluster.
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Appendix A Supplemental Material
Anomalous quantum-critical scaling corrections in two-dimensional antiferromagnets
N. Ma, P. Weinberg, H. Shao, W. Guo, D.-X. Yao, and A. W. Sandvik
Here we discuss the dependence of the critical Binder ratio on the time-space aspect ratio of the system in the QMC simulations, to explain the fact that the results for the SDM and the CDM in Fig. 3 of the main text do not extrapolate to the same value when . We also comment more broadly on the role of aspect ratios when analyzing quantum phase transitions.
The dependence of the Binder ratio on the spatial aspect ratio in classical systems is well understood kamienartz (); selke (), and in quantum systems also acts as an aspect ratio. In addition, the CDM and SDM lack 90 lattice rotational invariance and therefore have different velocities of excitations in the two lattice directions. In order to obtain the universal value of , one has to find both the correct spatial aspect ratio , corresponding to the ratio of the two velocities, and the temporal ratio . This was done in Ref. accurate (), and the crossing values of the CDM and SDM were shown to indeed be universal, agreeing with the value obtained for the 3D classical Heisenberg model at its critical temperature.
Here we just illustrate the dependence on the temporal ratio in the case of the CDM, keeping the spatial geometry. The results shown in Figure S1 demonstrate that the critical point consistently flows to the same value, while the asymptotic crossing value depends on . We do not extrapolate these results to infinite size, as the purpose is just to illustrate the very clear flows toward incompatible infinite-size values for different ratios. Although the CDM and SDM have the same ratio in the QMC simulations leading to the results in Fig. 3, the effective aspect ratio is still different because of the different spinwave velocities. The two models also have effectively different spatial aspect ratios.
While we agree with Ref. accurate () on the point of the common universality of the CDM, SDM, and O(3) models, and the importance of tuning aspect ratios if one desires to observe the universal Binder cumulant, we are not convinced of the practical utility of finding the special aspect ratios and make the system effectively perfectly space-time isotropic. Optimizing the aspect ratios is an additional complication in the simulations, though potentially the symmetry between the directions also could have advantageous effects on the scaling, thoough this is not clear from the results presented so far. In Ref. accurate () some non-monotonic behaviors were also seen, i.e., the corrections arising from the cubic interactions do not vanish at the special aspect ratios, which one should also not expect. As we have shown in the main text, one can reach the correct conclusions on the universality class also with lattices and with any fixed reasonable ratio (where one should also keep in mind that the QMC simulation time scales linearly with and with ). The key to understand fully the role of the cubic interactions in the SDM is to realize the importance of two irrelevant fields in the finite-size analysis.