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Physical review. B, Condensed matter

Variational calculations for the square-lattice quantum antiferromagnet

Zhiping Liu, Efstratios Manousakis

Published Dec 1, 1989• 35 citations
Publisher

Abstract

We study the spin-1/2 Heisenberg antiferromagnet on a square lattice using variational Monte Carlo calculations. We construct a variational wave function which gives accurate ground-state energy and is consistent with sum rules for the dynamical structure function. The spin-wave velocity and the spin-stiffness constant can be calculated using sum rules. We obtain ${Z}_{c}$==c/${c}_{0}$, where ${c}_{0}$ is the classical spin-wave velocity. This and other ground-state quantities are in good agreement with spin-wave theory and Monte Carlo calculations.

DOI
10.1103/physrevb.40.11437

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We study the spin-1/2 Heisenberg antiferromagnet on a square lattice using variational Monte Carlo calculations. We construct a variational wave function which gives accurate ground-state energy and is consistent with sum rules for the dynamical structure function. The spin-wave velocity and the spin-stiffness constant can be calculated using sum rules. We obtain Zc{Z}_{c}Zc​==c/c0{c}_{0}c0​, where c0{c}_{0}c0​ is the classical spin-wave velocity. This and other ground-state quantities are in good agreement with spin-wave theory and Monte Carlo calculations.
The Green's-function Monte Carlo (GFMC) method is used to calculate very accurate ground-state energies of the two-dimensional, spin-1/2 Heisenberg antiferromagnet. The computations are performed on L\ifmmode\times\else\texttimes\fi{}L square lattices up to L=16 with varying uniform magnetization, which allows the extraction of the perpendicular susceptibility (\ensuremath{\chi}) and spin-wave velocity (c). These two quantities are the lattice- or cutoff-dependent parameters that allow one to map the long-wavelength properties of the antiferromagnet onto the nonlinear \ensuremath{\sigma} model and so are of general interest. Systematic errors present in previous GFMC calculations are addressed and corrected to yield results in excellent agreement with other numerical methods. I find, for the ground-state energy per site, -0.669 34(3); the susceptibility renormalization factor, ${\mathit{Z}}_{\mathrm{\ensuremath{\chi}}}$=0.535(5); and the spin-wave velocity renormalization factor, Zc{\mathit{Z}}_{\mathit{c}}Zc​=1.10(3). Finite-size effects in the extraction of Zc{\mathit{Z}}_{\mathit{c}}Zc​ and ${\mathit{Z}}_{\mathrm{\ensuremath{\chi}}}$ are discussed. The value of ${\mathit{Z}}_{\mathrm{\ensuremath{\chi}}}$ computed here is in agreement with the series-expansion results of Singh and of Zheng, Oitmaa, and Hamer, thereby clearing up a previous inconsistency between the series-expansion and quantum Monte Carlo predictions.
In this work, a long-range resonating valence bond state is proposed as a variational wave function for the ground state of the S=1/2S=1/2S=1/2 antiferromagnetic Heisenberg model on the honeycomb lattice. Employing Variational Monte Carlo (VMC) method, we show that the ground state energy obtained from such RVB wave function, lies well below the energy of the Néel state and compares very well to the energies evaluated from spin-wave theory and series expansion method. We also obtain the spin-spin correlation function along zig-zag and armchair directions and find that the two correlations are different, which indicates the anisotropic nature of the system. We compare our results with the square lattice and we show that although the quantum fluctuations on honeycomb lattice are much stronger, but do not completely destroy the magnetic order.
The ground-state properties of the spin-(1/2 Heisenberg antiferromagnet on a square lattice are studied by using a simple variational wave function that interpolates continuously between the N\'eel state and short-range resonating-valence-bond states. Exact calculations of the variational energy for small systems show that the state with the lowest energy has long-range antiferromagnetic order. The staggered magnetization in this state is approximately 70% of its maximum possible value. The variational estimate of the ground-state energy is substantially lower than the value obtained for the nearest-neighbor resonating-valence-bond wave function.
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We present numerical exact results for the ground state and the low-lying excitations for the spin-1/2 J1-J2 Heisenberg antiferromagnet on finite square lattices of up to N = 40 sites. Using finite-size extrapolation we determine the ground-state energy, the magnetic order parameters, the spin gap, the uniform susceptibility, as well as the spin-wave velocity and the spin stiffness as functions of the frustration parameter J2/J1. In agreement with the generally excepted scenario we find semiclassical magnetically ordered phases for J2 < J c1 2 and J2> J c2 2 separated by a gapful quantum paramagnetic phase. We estimate Jc1 2 ≈ 0.35J1 and J c2 2 ≈ 0.66J1. PACS codes: 75.10.Jm Quantized spin models 75.45.+j Macroscopic quantum phenomena in magnetic systems 1