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Physical review. B, Condensed matter

Finite-lattice extrapolations for the ground-state energies of antiferromagnetic Heisenberg rings

Hiroshi Betsuyaku

Published Dec 1, 1986• 33 citations
Publisher

Abstract

The sequence-transformation method due to Vanden Broeck and Schwartz is used to accelerate the convergence of numerical finite-lattice data on ground-state energies to their infinite-lattice limit. Highly accurate estimates of the ground-state energy are obtained for the $S=\frac{1}{2}$ antiferromagnetic Heisenberg and $\mathrm{XY}$ rings. The ground-state energy of the $S=1$ antiferromagnetic Heisenberg ring is also estimated accurately.

DOI
10.1103/physrevb.34.8125

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The sequence-transformation method due to Vanden Broeck and Schwartz is used to accelerate the convergence of numerical finite-lattice data on ground-state energies to their infinite-lattice limit. Highly accurate estimates of the ground-state energy are obtained for the S=12S=\frac{1}{2}S=21​ antiferromagnetic Heisenberg and XY\mathrm{XY}XY rings. The ground-state energy of the S=1S=1S=1 antiferromagnetic Heisenberg ring is also estimated accurately.
The excitation energies of magnons and solitons as well as the ground-state energy of the spin-1 antiferromagnetic Heisenberg-Ising ring are studied by the numerical method. The method of Vanden Broeck and Schwartz is used to estimate the infinite-lattice limits accurately. The result confirms Haldane's prediction that a novel singlet phase with a nonzero excitation gap appears between the gapless XY phase and the doublet ground-state N\'eel phase. It extends over the range of anisotropy (${\ensuremath{\Delta}}_{l}$\ensuremath{\le}\ensuremath{\Delta}\ensuremath{\le}${\ensuremath{\Delta}}_{u}$) around the isotropic Heisenberg point \ensuremath{\Delta}=0.5. It is ascertained that at the upper transition point ${\ensuremath{\Delta}}_{u}$=0.540\ifmmode\pm\else\textpm\fi{}0.001 (1) the longitudinal magnon gap closes, (2) the soliton existing in the Ising side softens, and (3) the second derivative of the ground-state energy diverges. Around the lower transition point of ${\ensuremath{\Delta}}_{l}$=0.38\ifmmode\pm\else\textpm\fi{}0.02, the transverse magnon gap exhibits an exponential singularity accompanying with the slight anomaly of the second derivative of the ground-state energy.
The ground-state energy, staggered magnetization, and correlation function are computed for the two-dimensional spin-(1/2 quantum antiferromagnetic Heisenberg model (believed to be relevant to the high-temperature superconducting materials without doping) on lattices up to 24\ifmmode\times\else\texttimes\fi{}24 by Monte Carlo methods extrapolated to zero temperature. A more accurate value for the ground-state energy is determined by the method of Neumann and Ulam by extrapolating from lattices up to 12\ifmmode\times\else\texttimes\fi{}12.
We present results of exact diagonalization of the isotropic antiferromagnetic S=1 Heisenberg chain by the Lanczos method, for finite rings of up to N=22 sites. The Haldane gap G(N) and the ground-state energy per site e(N) converge, with increasing N, faster than a power law. By VBS and Shanks transformations, the extrapolated values are G(\ensuremath{\infty})=0.41049(2) and e(\ensuremath{\infty})=-1.401485(2). The spin-spin correlation function is well fit by exp(-r/\ensuremath{\xi})/ \ensuremath{\surd}r with \ensuremath{\xi}=6.2.
Systematic properties of the first excited state are presented for various ring sizes and spin quantum numbers which are only partly covered by the theorem of Lieb, Schultz, and Mattis. For odd ring sizes the first excited energy eigenvalue shows unexpected degeneracy and related shift quantum numbers. As a byproduct the ground state energy as well as the energy of the first excited state of infinite chains are calculated by extrapolating the properties of only a few, finite, antiferromagnetically coupled Heisenberg rings using the powerful Levin sequence acceleration method.