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Chinese Physics B

Quantum Monte Carlo study on the phase transition for a generalized two-dimensional staggered dimerized Heisenberg model

Rui Zheng, Bang‐Gui Liu

Published Nov 1, 2012• 2 citations
Publisher

Abstract

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.

DOI
10.1088/1674-1056/21/11/116401

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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.
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.
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 Wenzel (2008 Phys. Rev. Lett. 101 127202), we study a similar anisotropic spin-1/2 Heisenberg model on the honeycomb lattice. The critical point where the phase transition occurs due to the dimerization as well as the critical exponent ν are analyzed in great detail. Remarkably, using most of the available data points in conjunction with the expected finite-size scaling ansatz with a sub-leading correction indeed leads to a consistent ν = 0.691(2) with that calculated in Wenzel (2008 Phys. Rev. Lett. 101 127202). However, by using the data with a large number of spins N , we obtain ν = 0.707(6) which agrees with the most accurate Monte Carlo O(3) value ν = 0.7112(5) as well.
Motivated by the indication of a new critical theory for the spin-1/2 Heisenberg model with a spatially staggered anisotropy on the square lattice, we reinvestigate the phase transition of this model induced by dimerization using first-principle Monte Carlo simulations. We focus on studying the finite-size scaling of ${\ensuremath{\rho}}_{s1}2L$ and ${\ensuremath{\rho}}_{s2}2L$, where LLL stands for the spatial box size used in the simulations and ${\ensuremath{\rho}}_{si}$, with $i\ensuremath{\in}{1,2}$, is the spin-stiffness in the iii-direction. Remarkably, while we observe a large correction to scaling for the observable ${\ensuremath{\rho}}_{s1}2L$, the data for ${\ensuremath{\rho}}_{s2}2L$ exhibit a good scaling behavior without any indication of a large correction. As a consequence, we are able to obtain a numerical value for the critical exponent $\ensuremath{\nu}$, which is consistent with the known O(3)O(3)O(3) result with moderate computational effort. Further, we additionally carry out an unconventional finite-size scaling analysis with which we assume that the ratio of the spatial winding numbers squared is fixed through all simulations. The theoretical correctness of our idea is argued and its validity is confirmed. Using this unconventional finite-size scaling method, even from ${\ensuremath{\rho}}_{s1}L$, which receives the most serious correction among the observables considered in this study, we are able to arrive at a value for $\ensuremath{\nu}$ consistent with the expected O(3)O(3)O(3) value. A detailed investigation to compare these two finite-size scaling methods should be performed.
The quantum phase transition, scaling behaviors, and thermodynamics in the spin-1/2 quantum Heisenberg model with antiferromagnetic coupling J>0 in the armchair direction and ferromagnetic interaction J^{'}<0 in the zigzag direction on a honeycomb lattice are systematically studied using the continuous-time quantum Monte Carlo method. By calculating the Binder ratio Q_{2} and spin stiffness ρ in two directions for various coupling ratios α=J^{'}/J under different lattice sizes, we found that a quantum phase transition from the dimerized phase to the stripe phase occurs at the quantum critical point α_{c}=-0.93. Through the finite-size scaling analysis on Q_{2}, ρ_{x}, and ρ_{y}, we determined the critical exponent related to the correlation length ν to be 0.7212(8), implying that this transition falls into a classical Heisenberg O(3) universality. A zero magnetization plateau is observed in the dimerized phase, whose width decreases with increasing α. A phase diagram in the coupling ratio α-magnetic field h plane is obtained, where four phases, including dimerized, stripe, canted stripe, and polarized, are identified. It is also unveiled that the temperature dependence of the specific heat C(T) for different α's intersects precisely at one point, similar to that of liquid ^{3}He under different pressures and several magnetic compounds under various magnetic fields. The scaling behaviors of Q_{2}, ρ, and C(T) are carefully analyzed. The susceptibility is compared with the experimental data to give the magnetic parameters of both compounds.
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 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.