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arXiv:2501.15234 · The European Physical Journal B

A Monte Carlo examination of the numerical values of universal quantities in spatial dimension two

Fu-Jiun Jiang

Published Aug 1, 2026• 0 citations
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Abstract

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 $$S(\pi ,\pi )/\left( \chi _s T\right) $$ S ( π , π ) / χ s T and $$c/\left( T\xi \right) $$ c / T ξ , are determined. Here $$S(\pi ,\pi )$$ S ( π , π ) , $$\chi _s$$ χ s , c , $$\xi ,$$ 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 $$S(\pi ,\pi )/\left( \chi _s T\right) \sim 1.073$$ S ( π , π ) / χ s T ∼ 1.073 and $$c/\left( T\xi \right) \sim 0.963$$ 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 $$\chi _u c^2/T$$ χ u c 2 / T ( $$\chi _u$$ 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 $$S(\pi ,\pi ){/}\left( \chi _s T\right) $$ S ( π , π ) /

DOI
10.1140/epjb/s10051-026-01227-x

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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(\pi ,\pi )/\left( \chi _s T\right) S(π,π)/(χs​T) S ( π , π ) / χ s T and c/(Tξ)c/\left( T\xi \right) c/(Tξ) c / T ξ , are determined. Here S(π,π)S(\pi ,\pi )S(π,π) S ( π , π ) , χs\chi _sχs​ χ s , c , ξ,\xi ,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)∼1.073S(\pi ,\pi )/\left( \chi _s T\right) \sim 1.073S(π,π)/(χs​T)∼1.073 S ( π , π ) / χ s T ∼ 1.073 and c/(Tξ)∼0.963c/\left( T\xi \right) \sim 0.963c/(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\chi _u c^2/Tχu​c2/T χ u c 2 / T ( χu\chi _uχu​ 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(\pi ,\pi ){/}\left( \chi _s T\right) S(π,π)/(χs​T) S ( π , π ) /
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.
The quantum critical regimes (QCR) of both a two-dimensional (2D) disordered spin-12\frac{1}{2}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 (TTT) 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.
A quantum Monte Carlo simulation scheme for spin systems is presented. The method is a generalization of Handscomb's method but applicable to any length of the spin, i.e., when the spin traces cannot be evaluated analytically. The Monte Carlo sampling is extended to the space of spin vectors in addition to the usual operator-index sequences. An important technical point is that the index sequences are augmented with the aid of unit operators to a constant, self-consistently determined length. The scheme is applied to the one-dimensional antiferromagnetic spin-S Heisenberg model. Results at low temperatures are reported for S=1 and S=3/2 and system sizes up to N=64. The computed magnetic structure factor in the S=1 chain is in agreement with earlier ground-state calculations. For S=3/2 we find the exponent \ensuremath{\gamma}\ifmmode\bar\else\textasciimacron\fi{}=0.49\ifmmode\pm\else\textpm\fi{}0.04 for the divergence of the antiferromagnetic structure factor. Further, the susceptibility as a function of the wave number is computed. For S=1 the staggered susceptibility \ensuremath{\chi}(\ensuremath{\pi}) at T=0 is found to take the value 20.0\ifmmode\pm\else\textpm\fi{}1.5 in units such that \ensuremath{\chi}(q)\ensuremath{\rightarrow}${\mathit{T}}^{\mathrm{\ensuremath{-}}1}$ at high temperatures (with the temperature scale defined by kB{\mathit{k}}_{\mathit{B}}kB​=1). For S=3/2 we obtain the exponent \ensuremath{\gamma}=1.45\ifmmode\pm\else\textpm\fi{}0.05 for the divergence of the staggered susceptibility.
We simulate the spin-1/2 Heisenberg model with a spatially staggered anisotropy using first principles Monte Carlo method. In particular, the critical exponents β/νβ/νβ/ν and ωωω associated with the quantum phase transition induced by dimerization are determined with high precision. Here βββ and ννν are the exponents related to the magnetization and the correlation length, respectively. In addition, ωωω is the confluent exponent. With very accurate data of the relevant observables, we first obtain a value of ωωω compatible with the known result in the O(3) universality class. Further, using either the value of ωωω determined here or the established one in the literature, the exponent β/νβ/νβ/ν calculated from our data is in quantitative agreement with the known result β/ν=0.519(1)β/ν= 0.519(1)β/ν=0.519(1) as well. Our investigation suggests that the quantum phase transition studied here is fully consistent with the O(3) universality class.
Puzzled by the indication of a new critical theory for the spin-1/2 Heisenberg model with a spatially staggered anisotropy on the square lattice as suggested in \cite{Wenzel08}, we re-investigate the phase transition of this model induced by dimerization using first principle Monte Carlo simulations. We focus on studying the finite-size scaling of ρs1Lρ_{s1} Lρs1​L and ρs2Lρ_{s2} Lρs2​L, where LLL stands for the spatial box size used in the simulations and ρsiρ_{si}ρsi​ with i∈{1,2}i \in \{1,2\}i∈{1,2} is the spin-stiffness in iii-direction. From our Monte Carlo data, we find that ρs2Lρ_{s2} Lρs2​L suffers a much less severe correction compared to that of ρs1Lρ_{s1} Lρs1​L. Therefore ρs2Lρ_{s2} Lρs2​L is a better quantity than ρs1Lρ_{s1} Lρs1​L for finite-size scaling analysis concerning the limitation of the availability of large volumes data in our study. Further, motivated by the so-called cubical regime in magnon chiral perturbation theory, we additionally perform a finite-size scaling analysis on our Monte Carlo data with the assumption that the ratio of spatial winding numbers squared is fixed through all simulations. As a result, the physical shape of the system remains fixed in our calculations. The validity of this new idea is confirmed by studying the phase transition driven by spatial anisotropy for the ladder anisotropic Heisenberg model. With this new strategy, even from ρs1Lρ_{s1} Lρs1​L which receives the most serious correction among the observables considered in this study, we arrive at a value for the critical exponent ννν which is consistent with the expected O(3)O(3)O(3) value by using only up to L=64L = 64L=64 data points.
We present detailed Monte Carlo results for the susceptibility \ensuremath{\chi}, correlation length \ensuremath{\xi}, and specific heat Cv{\mathit{C}}_{\mathit{v}}Cv​ for the XY model. The simulations are done on 642{64}^{2}642, 1282{128}^{2}1282, 2562{256}^{2}2562, and 5122{512}^{2}5122 lattices over the temperature range 0.98\ensuremath{\le}T\ensuremath{\le}1.43 corresponding to 270. Fits to \ensuremath{\chi} and \ensuremath{\xi} data favor a Kosterlitz-Thouless (KT) singularity over a second-order transition; however, unconstrained four-parameter KT fits do not confirm the predicted values \ensuremath{\nu}=0.5 and \ensuremath{\eta}=0.25. Our best estimate, Tc{\mathit{T}}_{\mathit{c}}Tc​=0.894(5), is obtained using KT fits with \ensuremath{\nu} fixed at 0.5. The exponent \ensuremath{\eta} is calculated as a function of temperature in the spin-wave phase using Monte Carlo renormalization-group and finite-size-scaling methods. Both methods give consistent results and we find \ensuremath{\eta}\ensuremath{\approxeq}0.235 at T=0.894. We also present results for the behavior of vortex density across the transition and exhibit how the dilute-gas approximation breaks down.