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arXiv:0911.0653

The Phase Transition of the Spin-1/2 Heisenberg Model with a Spatially Staggered Anisotropy on the Square Lattice

Fu-Jiun Jiang

Published Nov 3, 2009• 0 citations
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Abstract

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. We focus on studying the finite-size scaling of the observables $ρ_{s1} L$ and $ρ_{s2} L$, where $L$ stands for the spatial box sizes used in the simulations and $ρ_{si}$ with $i \in \{1,2\}$ is the spin-stiffness in $i$-direction. We find by performing finite-size scaling using the observable $ρ_{s2} L$, which corresponds to the spatial direction with a fixed antiferromagnetic coupling, one would suffer a much less severe correction compared to that of using $ρ_{s1} L$. Therefore $ρ_{s2} L$ is a better quantity than $ρ_{s1} L$ for finite-size scaling analysis concerning the limitation for the availability of large volumes data in our study. Remarkably, by employing the method of fixing the aspect-ratio of spatial winding numbers squared in the simulations, even from $ρ_{s1} L$ which receives the most serious correction among the observables considered in this study, we arrive at a value for the critical exponent $ν$ which is consistent with the expected O(3) value by using only up to $L = 64$ data points.

DOI
10.48550/arxiv.0911.0653

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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. We focus on studying the finite-size scaling of the observables ρs1Lρ_{s1} Lρs1​L and ρs2Lρ_{s2} Lρs2​L, where LLL stands for the spatial box sizes 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. We find by performing finite-size scaling using the observable ρs2Lρ_{s2} Lρs2​L, which corresponds to the spatial direction with a fixed antiferromagnetic coupling, one would suffer a much less severe correction compared to that of using 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 for the availability of large volumes data in our study. Remarkably, by employing the method of fixing the aspect-ratio of spatial winding numbers squared in the simulations, 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) value by using only up to L=64L = 64L=64 data points.
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.
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.
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 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.
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 ground state phase of a spin-$\frac{1}{2} {J}_{1}\text{\ensuremath{-}}{J}_{2}$ antiferromagnetic Heisenberg model on a square lattice around the maximally frustrated regime $({J}_{2}\ensuremath{\sim}0.5{J}_{1})$ has been debated for decades. Here we study this model using the cluster update algorithm for tensor-product states (TPSs). The ground state energies at finite sizes and in the thermodynamic limit (with finite size scaling) are in good agreement with exact diagonalization study. Through finite size scaling of the spin correlation function, we find the critical point J2c1=0.572(5)J1{J}_{2}^{{c}_{1}}=0.572(5){J}_{1}J2c1​​=0.572(5)J1​ and critical exponents $\ensuremath{\nu}=0.50(8),\phantom{\rule{0.16em}{0ex}}{\ensuremath{\eta}}_{s}=0.28(6)$. In the range of $0.572<{J}_{2}/{J}_{1}\ensuremath{\le}0.6$ we find a paramagnetic ground state with an exponentially decaying spin-spin correlation. Up to a $24\ifmmode\times\else\texttimes\fi{}24$ system size, we observe power law decaying dimer-dimer and plaquette-plaquette correlations with an anomalous plaquette scaling exponent ${\ensuremath{\eta}}_{p}=0.24(1)$ and an anomalous columnar scaling exponent ${\ensuremath{\eta}}_{c}=0.28(1)$ at J2/J1=0.6{J}_{2}/{J}_{1}=0.6J2​/J1​=0.6. These results are consistent with a potential gapless U(1)U(1)U(1) spin-liquid phase. However, since the U(1)U(1)U(1) spin liquid is unstable due to the instanton effect, a valence bond solid order with very small amplitude might develop in the thermodynamic limit. Thus, our numerical results strongly indicate a deconfined quantum critical point at J2c1{J}_{2}^{{c}_{1}}J2c1​​. Remarkably, all the observed critical exponents are consistent with the $J\text{\ensuremath{-}}Q$ model.