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arXiv:0808.1418 · Physical Review B

Comprehensive quantum Monte Carlo study of the quantum critical points in planar dimerized/quadrumerized Heisenberg models

S. C. Wenzel, Wolfhard Janke E

Published Jan 8, 2009• 70 citations
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Abstract

We study two planar square lattice Heisenberg models with explicit dimerization or quadrumerization of the couplings in the form of ladder and plaquette arrangements. We investigate the quantum critical points of those models 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 ladder model is determined as ${\ensuremath{\alpha}}_{c}=1.9096(2)$ improving on previous studies. For the plaquette model, we obtain ${\ensuremath{\alpha}}_{c}=1.8230(2)$ establishing a first benchmark for this model from quantum Monte Carlo simulations. Based on those values, we give further convincing evidence that the models are in the three-dimensional classical Heisenberg universality class. The results of this contribution shall be useful as references for future investigations on planar Heisenberg models such as concerning the influence of nonmagnetic impurities at the quantum critical point.

DOI
10.1103/physrevb.79.014410

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We study two planar square lattice Heisenberg models with explicit dimerization or quadrumerization of the couplings in the form of ladder and plaquette arrangements. We investigate the quantum critical points of those models by means of (stochastic series expansion) quantum Monte Carlo simulations as a function of the coupling ratio \ensuremathα=J\ensuremath′/J\ensuremath{\alpha}={J}^{\ensuremath{'}}/J\ensuremathα=J\ensuremath′/J. The critical point of the order-disorder quantum phase transition in the ladder model is determined as \ensuremathαc=1.9096(2){\ensuremath{\alpha}}_{c}=1.9096(2)\ensuremathαc​=1.9096(2) improving on previous studies. For the plaquette model, we obtain \ensuremathαc=1.8230(2){\ensuremath{\alpha}}_{c}=1.8230(2)\ensuremathαc​=1.8230(2) establishing a first benchmark for this model from quantum Monte Carlo simulations. Based on those values, we give further convincing evidence that the models are in the three-dimensional classical Heisenberg universality class. The results of this contribution shall be useful as references for future investigations on planar Heisenberg models such as concerning the influence of nonmagnetic impurities at the quantum critical point.
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 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.
In order to gain a deeper understanding of the quantum criticality in the explicitly staggered dimerized Heisenberg models, we study a generalized staggered dimer model named the J 0 — J 1 — J 2 model, which corresponds to the staggered J — J ′ model on a square lattice and a honeycomb lattice when J 1 / J 0 equals 1 and 0, respectively. Using the quantum Monte Carlo method, we investigate all the quantum critical points of these models with J 1 / J 0 changing from 0 to 1 as a function of coupling ratio α = J 2 / J 0 . We extract all the critical values of the coupling ratio α c for these models, and we also obtain the critical exponents ν, β/ν, and η using different finite-size scaling ansätz,. All these exponents are not consistent with the three-dimensional Heisenberg universality class, indicating some unconventional quantum critical points in these models.
We present numerical results for an S=1/2S=1/2S=1/2 Heisenberg antiferromagnet on an inhomogeneous square lattice with tunable interaction between spins belonging to different plaquettes. Employing quantum Monte Carlo, we significantly improve on previous results for the critical point separating singlet-disordered and N\'eel-ordered phases and obtain an estimate for the critical exponent $\ensuremath{\nu}$ consistent with the three-dimensional classical Heisenberg universality class. Additionally, we show that a fairly accurate result for the critical point can be obtained from a contractor renormalization expansion by applying a surprisingly simple analysis to the effective Hamiltonian.
Motivated by the unexpected Monte Carlo results as well as the theoretical proposal of a large correction to scaling for the critical theory of the 2-d staggered-dimer spin-1/2 Heisenberg model on the square lattice, we study the phase transitions induced by dimerization of several dimerized quantum Heisenberg models with spatial anisotropy using first principles Monte Carlo method. Remarkably, while our Monte Carlo data for all the models considered here, including the herringbone- and ladder-dimer models on the square lattice, are compatible with the recently proposed scenario of an enhanced correction to scaling, we find it is likely that the enhanced correction to scaling manifests itself as amplification of the nonuniversal prefactors appeared in the scaling forms. In other words, our data are in consistence with the established numerical values for the critical exponents, including the confluent exponent, in the O(3) universality class. Convincing numerical evidence is provided to support this proposed scenario.
We study the S>1/2S>1/2S>1/2 antiferromagnetic Heisenberg model on the 1/5-depleted square lattice as a function of the ratio of the intraplaquette coupling to the interplaquette coupling. Using stochastic series expansion quantum Monte Carlo simulations, we numerically identify three quantum phases, including the dimer phase, N\'eel phase, and plaquette valence-bond-solid phase. We also obtain the accurate quantum critical points that belong to the OOO(3) universality class using the large-scale finite-size scaling. Most importantly, we study the dynamic spin structure factors of different phases, which can be measured by inelastic neutron scattering experiments. The low-energy excitations can be explained as triplons in the dimer phase and plaquette valence-bond-solid phase, while in the N\'eel phase, the more prominent magnon mode can be found as the spin magnitude increases. Furthermore, we find a broader continuum at smaller SSS, which may be the dynamical signature of nearly deconfined spinon excitations.