The numerical value of a universal quantity associated with the quantum critical regime, namely χuc2/T, for a two-dimensional (2D) dimerized spin-1/2 antiferromagnet is calculated using the quantum Monte Carlo simulations (QMC). Here χu, c, and T are the uniform susceptibility, the spin-wave velocity, and the temperature, respectively. By simulating large lattices at moderately low temperatures, we find χuc2/T∼0.32. Our estimation of χuc2/T deviates from the related analytic prediction but agrees with recent numerical calculations of other 2D dimerized spin-1/2 antiferromagnets.
Abstract By simulating a two-dimensional dimerized spin-1/2 antiferromagnet with the quantum Monte Carlo method, the numerical values of two universal quantities associated with the quantum critical regime (QCR), namely
S(π,π)/(χsT) S ( π , π ) / χ s T and
c/(Tξ) c / T ξ , are determined. Here
S(π,π) S ( π , π ) ,
χs χ s , c ,
ξ, ξ , and T are the staggered structure factor, the staggered susceptibility, the spin-wave velocity, the correlation length, and the temperature, respectively. Our calculations lead to
S(π,π)/(χsT)∼1.073 S ( π , π ) / χ s T ∼ 1.073 and
c/(Tξ)∼0.963 c / T ξ ∼ 0.963 , which are in better consistency with the leading theoretical predictions than those including the next-to-leading-order terms. This scenario was also observed for the Wilson ratio W and
χuc2/T χ u c 2 / T (
χu χ u is the uniform susceptibility) of QCR. The presented outcomes here and those in the literature suggest that it is desirable to conduct a refined analytic calculation to resolve the puzzle of why the inclusion of higher-order terms leads to less accurate predictions for these universal quantities. Graphical abstract
S(π,π)/(χsT) S ( π , π ) /
The quantum critical regimes (QCR) of both a two-dimensional (2D) disordered spin-
21 antiferromagnet and a 2D clean dimerized quantum Heisenberg model are studied using the first-principles nonperturbative quantum Monte Carlo simulations. In particular, the three well-known universal coefficients associated with QCR are investigated in detail. While in our investigation we find the obtained results are consistent with the related analytic predictions, non-negligible finite-temperature (
T) effects are observed. Moreover, a striking finding in our study is that the numerical value for one of the universal coefficients we determine is likely to be different significantly from the corresponding (theoretical) result(s) established in the literature. To better understand the sources for the discrepancy observed here, apart from carrying out the associated analytic calculations not considered previously, it will be desirable as well to conduct a comprehensive examination of the exotic features of QCR for other disordered and clean spin systems than those investigated in this study.
We study scaling behavior in two-dimensional, square lattice, S=1/2 and S=1 Heisenberg antiferromagnets using the data on full q dependences of the equal-time structure factor and the static susceptibility, calculated through high-temperature series expansions. We also carry out comparisons with a model of two coupled S=1/2 Heisenberg planes with the interlayer exchange coupling tuned to the T=0 critical point (two-plane model hereafter). For both S=1/2 and S=1 models, we separately determine the spin-wave velocity c and mass m=c/\ensuremath{\xi}, in addition to the correlation length, \ensuremath{\xi}, and find that c is temperature dependent; only for temperatures below T\ensuremath{\lesssim}JS, where J is the exchange coupling, c approaches its known T=0 values,
c0. This nonuniversal lattice effect is caused by the quantum nature of spin, and is therefore not captured by the quantum nonlinear \ensuremath{\sigma} model.Despite this temperature dependence of the spin-wave velocity, full q and \ensuremath{\omega} dependences of the dynamical susceptibility \ensuremath{\chi}(q,\ensuremath{\omega}) agree with those of the universal scaling function, computable for the \ensuremath{\sigma} model, for temperatures up to
T0\ensuremath{\sim}0.6
c0/a, and their further analysis leads us to the inference that below
T0 the S=1 model is in the renormalized classical (RC) regime, the two-plane model is in the quantum critical (QC) regime in agreement with earlier work, and the S=1/2 model exhibits a RC-QC crossover, centered around crossover and for the two-plane model at all T=0.55J. In particular, for the S=1/2 model above the RC-QC temperatures where calculated, the obtained spin-wave mass m=c/\ensuremath{\xi} is in excellent agreement with the known universal QC prediction, m\ensuremath{\simeq}1.04T. In contrast, for the S=1/2 model below the RC-QC crossover, and for the S=1 model at all temperatures, the behavior agrees with the exact RC expression. for all three models, nonuniversal behavior occurs above T\ensuremath{\sim}0.6
c0/a. Our results strongly support the conjecture by Chubukov and Sachdev that the S=1/2 model is close enough to the T=0 critical point to exhibit QC behavior at intermediate temperatures.
The scaling of the transition temperature into an ordered phase close to a quantum critical point as well as the order parameter fluctuations inside the quantum critical region provide valuable information about universal properties of the underlying quantum critical point. Here, we employ quantum Monte Carlo simulations to examine these relations in detail for two-dimensional quantum systems that exhibit a finite-temperature Ising-transition line in the vicinity of a quantum critical point that belongs to the universality class of either (i) the three-dimensional Ising model for the case of the quantum Ising model in a transverse magnetic field on the square lattice or (ii) the chiral Ising transition for the case of a half-filled system of spinless fermions on the honeycomb lattice with nearest-neighbor repulsion. While the first case allows large-scale simulations to assess the scaling predictions to a high precision in terms of the known values for the critical exponents at the quantum critical point, for the later case, we extract values of the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$, related to the order parameter fluctuations, which we discuss in relation to other recent estimates from ground-state quantum Monte Carlo calculations as well as analytical approaches.
We show that the zero-temperature susceptibility of the one-dimensional, dimerized Hubbard model at quarter-filling can be accurately determined on the basis of exact diagonalization of small clusters. The best procedure is to perform a finite-size scaling of the spin velocity ${\mathit{u}}_{\mathrm{\ensuremath{\sigma}}}$, and to calculate the susceptibility from the Luttinger-liquid relation \ensuremath{\chi}=2/\ensuremath{\pi}${\mathit{u}}_{\mathrm{\ensuremath{\sigma}}}$. We show that these results are reliable by comparing them with the analytical results that can be obtained in the weak- and strong-coupling limits. We have also used quantum Monte Carlo simulations to calculate the temperature dependence of the susceptibility for parameters that should be relevant to the Bechgaard salts. This shows that, used together, these numerical techniques are able to give precise estimates of the low-temperature susceptibility of realistic one-dimensional models of correlated electrons.
In certain Mott-insulating dimerized antiferromagnets, triplet excitations of the paramagnetic phase display both three-particle and four-particle interactions. When such a magnet undergoes a quantum phase transition into a magnetically ordered state, the three-particle interaction becomes part of the critical theory provided that the lattice ordering wave vector is zero. One microscopic example is the staggered-dimer antiferromagnet on the square lattice, for which deviations from O(3) universality have been reported in numerical studies. Using both symmetry arguments and microscopic calculations, we show that a nontrivial cubic term arises in the relevant order-parameter quantum field theory, and we assess its consequences using a combination of analytical and numerical methods. We also present finite-temperature quantum Monte Carlo data for the staggered-dimer antiferromagnet which complement recently published results. The data can be consistently interpreted in terms of critical exponents identical to that of the standard O(3) universality class, but with anomalously large corrections to scaling. We argue that the cubic interaction of critical triplons, although irrelevant in two spatial dimensions, is responsible for the leading corrections to scaling due to its small scaling dimension.