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arXiv:0807.4389 · Physical Review B

Quantum phase transition in a Heisenberg antiferromagnet on a square lattice with strong plaquette interactions

A. Fabricio Albuquerque, Matthias Troyer, J. Oitmaa

Published Oct 3, 2008• 32 citations
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Abstract

We present numerical results for an $S=1/2$ 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.

DOI
10.1103/physrevb.78.132402

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Progress of Theoretical Physics Supplement•Jan 1, 2000•2 citations
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 \ensuremathν\ensuremath{\nu}\ensuremathν 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.
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.
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.
The S=1/2S=1/2S=1/2 and S=1S=1S=1 two-dimensional quantum Heisenberg antiferromagnets on the anisotropic dimerized square lattice are investigated by the quantum Monte Carlo method. By finite-size-scaling analyses on the correlation lengths, the ground-state phase diagram parametrized by strengths of the dimerization and of the spatial anisotropy is determined much more accurately than the previous works. It is confirmed that the quantum critical phenomena on the phase boundaries belong to the same universality class as that of the classical three-dimensional Heisenberg model. Furthermore, for S=1,S=1,S=1, we show that all the spin-gapped phases, such as the Haldane and dimer phases, are adiabatically connected in the extended-parameter space, though they are classified into different classes in terms of the string order parameter in the one-dimensional, i.e., the zero-interchain-coupling, case.
We have developed a general scheme for carrying out systematic perturbation expansions for groundstate properties of quantum lattice models. As an application, we study the onset of spontaneous N\'eel order in S=12S=\frac{1}{2}S=21​ Heisenberg antiferromagnets by expanding around dimerized Hamiltonians. In one dimension (1D) we recover accurately the known exact results. On the square lattice we find novel critical points separating N\'eel ordered and disordered phases; the estimated critical exponents are consistent with those of the 3D classical Heisenberg model.
We study the spin-1/2 Heisenberg model on the square lattice with first- and second-neighbor antiferromagnetic interactions J(1) and J(2), which possesses a nonmagnetic region that has been debated for many years and might realize the interesting Z(2) spin liquid. We use the density matrix renormalization group approach with explicit implementation of SU(2) spin rotation symmetry and study the model accurately on open cylinders with different boundary conditions. With increasing J(2), we find a Néel phase and a plaquette valence-bond (PVB) phase with a finite spin gap. From the finite-size scaling of the magnetic order parameter, we estimate that the Néel order vanishes at J(2)/J(1)≃0.44. For 0.5<J(2)/J(1)<0.61, we find dimer correlations and PVB textures whose decay lengths grow strongly with increasing system width, consistent with a long-range PVB order in the two-dimensional limit. The dimer-dimer correlations reveal the s-wave character of the PVB order. For 0.44<J(2)/J(1)<0.5, spin order, dimer order, and spin gap are small on finite-size systems, which is consistent with a near-critical behavior. The critical exponents obtained from the finite-size spin and dimer correlations could be compatible with the deconfined criticality in this small region. We compare and contrast our results with earlier numerical studies.
Quantum phase transition of site-diluted and bond-diluted Heisenberg antiferromagnets on square lattices is studied. By using the novel continuous-time loop algorithm, we perform quantum Monte Carlo simulations on quite larger lattices at extremely lower temperatures than the previous numerical studies. It is found that the antiferromagnetic long-range order at T = 0 persists so long as a cluster of magnetic sites percolates, that is, the critical concentration is equal to the classical percolation threshold, in both of the site-diluted and bond-diluted cases. Furthermore, we find that some critical exponents, such as the magnetization exponent β, are non-classical and strongly depend on the spin size S. On the other hand, we show that the correlation-length exponent ν is universal and is equal to the classical value (ν=4/3).