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arXiv:cond-mat/0509747 · Physical Review B

High-precision finite-size scaling analysis of the quantum-critical point of S=1∕2 Heisenberg antiferromagnetic bilayers

Ling Wang, K. S. D. Beach, Anders W. Sandvik

Published Jan 27, 2006• 175 citations
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Abstract

We use quantum Monte Carlo (stochastic series expansion) and finite-size scaling to study the quantum critical points of two $S=1∕2$ Heisenberg antiferromagnets in two dimensions: a bilayer and a Kondo-lattice-like system (incomplete bilayer), each with intraplane and interplane couplings $J$ and ${J}_{\ensuremath{\perp}}$. We discuss the ground-state finite-size scaling properties of three different quantities---the Binder moment ratio, the spin stiffness, and the long-wavelength magnetic susceptibility---which we use to extract the critical value of the coupling ratio $g={J}_{\ensuremath{\perp}}∕J$. The individual estimates of ${g}_{c}$ are consistent provided that subleading finite-size corrections are properly taken into account. For both models, we find that the spin stiffness has the smallest subleading finite-size corrections; in the case of the incomplete bilayer we find that the first subleading correction vanishes or is extremely small. In agreement with predictions, we find that at the critical point the Binder ratio has a universal value and the product of the spin stiffness and the long-wavelength susceptibility scales as $1∕{L}^{2}$ with a universal prefactor. Our results for the critical coupling ratios are ${g}_{c}=2.5220(1)$ (full bilayer) and ${g}_{c}=1.3888(1)$ (incomplete bilayer), which represent improvements of more than an order of magnitude over the previous best estimates. For the correlation length exponent we obtain $\ensuremath{\nu}=0.7106(9)$, consistent with the expected three-dimensional Heisenberg universality class.

DOI
10.1103/physrevb.73.014431

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The ground-state parameters of the two-dimensional S=1/2S=1/2S=1/2 antiferromagnetic Heisenberg model are calculated using the stochastic series expansion quantum Monte Carlo method for $L\ifmmode\times\else\texttimes\fi{}L$ lattices with LLL up to 161616. The finite-size results for the energy EEE, the sublattice magnetization MMM, the long-wavelength susceptibility ${\ensuremath{\chi}}_{\ensuremath{\perp}}(q=2\ensuremath{\pi}/L)$, and the spin stiffness ${\ensuremath{\rho}}_{s},$ are extrapolated to the thermodynamic limit using fits to polynomials in 1/L1/L1/L, constrained by scaling forms previously obtained from renormalization-group calculations for the nonlinear $\ensuremath{\sigma}$ model and chiral perturbation theory. The results are fully consistent with the predicted leading finite-size corrections, and are of sufficient accuracy for extracting also subleading terms. The subleading energy correction $(\ensuremath{\sim}{1/L}^{4})$ agrees with chiral perturbation theory to within a statistical error of a few percent, thus providing numerical confirmation of the finite-size scaling forms to this order. The extrapolated ground- state energy per spin is $E=\ensuremath{-}0.669437(5)$. The result from previous Green's function Monte Carlo (GFMC) calculations is slightly higher than this value, most likely due to a small systematic error originating from ``population control'' bias in GFMC. The other extrapolated parameters are M=0.3070(3)M=0.3070(3)M=0.3070(3), ${\ensuremath{\rho}}_{s}=0.175(2)$, ${\ensuremath{\chi}}_{\ensuremath{\perp}}=0.0625(9)$, and the spin-wave velocity c=1.673(7)c=1.673(7)c=1.673(7). The statistical errors are comparable with those of previous estimates obtained by fitting loop algorithm quantum Monte Carlo data to finite-temperature scaling forms. Both MMM and ${\ensuremath{\rho}}_{s}$ obtained from the finite-TTT data are, however, a few error bars higher than the present estimates. It is argued that the T=0T=0T=0 extrapolations performed here are less sensitive to effects of neglected higher-order corrections, and therefore should be more reliable.
Abstract Motivated by the need for high-precision benchmark results for basic quantum many-body models, several ground state properties of the square lattice S = 1 / 2 Heisenberg antiferromagnet are computed (the energy, order parameter, spin stiffness, spinwave velocity, long-wavelength susceptibility and staggered susceptibility) using extensive quantum Monte Carlo simulations with the stochastic series expansion method. Moderately sized lattices are studied at temperatures T sufficiently low to realize the T → 0 limit, with significantly reduced statistical errors relative to previous works. Results for periodic L × L lattices with L ∈ [ 6 , 96 ] are tabulated versus L and extrapolations to infinite system size are carried out. The relative statistical errors of finite-size energies are below 10 −7 . The extrapolated ground state energy density is e 0 = − 0.669 441 857 ( 7 ) , where the number within parentheses is the statistical error (one standard deviation) of the preceding digit. This value represents an improvement in precision of three orders of magnitude over the previous best result. The leading and subleading finite-size corrections to e 0 are in full quantitative agreement with predictions from chiral perturbation theory, thus further supporting the soundness of both the extrapolations and the theory. The extrapolated sublattice magnetization is m s = 0.307 447 ( 2 ) , which agrees well with previous estimates but with a much smaller statistical error. The coefficient of the linear in L −1 correction to m s 2 agrees with the value from chiral perturbation theory and the presence of a factor ln γ ⁡ ( L ) in the second-order correction is also confirmed, with the previously unknown value of the exponent being γ = 0.82 ( 4 ) . The finite-size corrections to the staggered susceptibility point to logarithmic corrections in this quantity as well. To facilitate benchmarking of methods for which periodic boundary conditions are challenging, results for systems with open and cylindrical boundaries are also listed and their spatially inhomogeneous order parameters are analyzed.
We use the stochastic series expansion quantum Monte Carlo method to study the Heisenberg models on the square lattice with strong and weak couplings in the form of three different plaquette arrangements known as checkerboard models $\mathrm{C}2\ifmmode\times\else\texttimes\fi{}2$, $\mathrm{C}2\ifmmode\times\else\texttimes\fi{}4$, and $\mathrm{C}4\ifmmode\times\else\texttimes\fi{}4$. The $a\ifmmode\times\else\texttimes\fi{}b$ here stands for the shape of the plaquette consisting of spins connected by strong couplings. Through detailed analysis of a finite-size scaling study, the critical point of the $\mathrm{C}2\ifmmode\times\else\texttimes\fi{}2$ model is improved as gc=0.548524(3){g}_{c}=0.548\phantom{\rule{0.16em}{0ex}}524(3)gc​=0.548524(3) compared with previous studies where ggg is the ratio of weak and strong couplings in the models. For $\mathrm{C}2\ifmmode\times\else\texttimes\fi{}4$ and $\mathrm{C}4\ifmmode\times\else\texttimes\fi{}4$ we give gc=0.456978(2){g}_{c}=0.456\phantom{\rule{0.16em}{0ex}}978(2)gc​=0.456978(2) and 0.314451(3)0.314\phantom{\rule{0.16em}{0ex}}451(3)0.314451(3). We also study the critical exponents $\ensuremath{\nu}$ and $\ensuremath{\eta}$ and the universal property of the Binder ratio to give further evidence that all quantum phase transitions in these three models are in the three-dimensional O(3) universality class. Furthermore, our fitting results show the importance of effective corrections in the scaling study of these models.
The three-dimensional quenched random bond diluted $({J}_{1}\ensuremath{-}{J}_{2})$ quantum Heisenberg antiferromagnet is studied on a simple-cubic lattice. Using extensive stochastic series expansion quantum Monte Carlo simulations, we perform very long runs for an $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ lattice up to L=48L=48L=48. By employing a standard finite-size scaling method, the numerical values of the N\'eel temperature are determined with high precision as a function of the coupling ratio r=J2/J1r={J}_{2}/{J}_{1}r=J2​/J1​. Based on the estimated critical exponents, we find that the critical behavior of the considered model belongs to the pure classical three-dimensional O(3)O(3)O(3) Heisenberg universality class.
The two-dimensional $J\mathrm{\text{\ensuremath{-}}}{J}^{\ensuremath{'}}$ dimerized quantum Heisenberg model is studied on the square lattice by means of (stochastic series expansion) quantum Monte Carlo simulations as a function of the coupling ratio $\ensuremath{\alpha}={J}^{\ensuremath{'}}/J$. The critical point of the order-disorder quantum phase transition in the $J\mathrm{\text{\ensuremath{-}}}{J}^{\ensuremath{'}}$ model is determined as ${\ensuremath{\alpha}}_{c}=2.5196(2)$ by finite-size scaling for up to approximately 10 000 quantum spins. By comparing six dimerized models we show, contrary to the current belief, that the critical exponents of the $J\mathrm{\text{\ensuremath{-}}}{J}^{\ensuremath{'}}$ model are not in agreement with the three-dimensional classical Heisenberg universality class. This lends support to the notion of nontrivial critical excitations at the quantum critical point.
We analyze the critical properties and the entanglement scaling at the quantum critical points of the spin-half XY model on a two-dimensional square-lattice bilayer and necklace lattice, based on quantum Monte Carlo simulations on finite tori and for different subregion shapes. For both models, the finite-size scaling of the transverse staggered spin structure factor is found in accord with a quantum critical point described by the two-component, three-dimensional ${\ensuremath{\phi}}^{4}$ theory. The second R\'enyi entanglement entropy in the absence of corners along the subsystem boundary exhibits area-law scaling in both models, with an area-law prefactor of 0.0674(7)0.0674(7)0.0674(7) [0.0664(4)][0.0664(4)][0.0664(4)] for the bilayer (necklace) model, respectively. Furthermore, the presence of ${90}^{\ensuremath{\circ}}$ corners leads to an additive logarithmic term in both models. We estimate a contribution of $\ensuremath{-}0.010(2)$ $[\ensuremath{-}0.009(2)]$ due to each ${90}^{\ensuremath{\circ}}$ corner to the logarithmic correction for the bilayer (necklace) model, and compare our findings to recent numerical linked-cluster calculations and series expansion results on related models.