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Physical review. B, Condensed matter

Ground-state alternation and excitation energy of S=(1/2) linear Heisenberg antiferromagnets

Z. G. Soos, Satoru Kuwajima, Jennifer E. Mihalick

Published Sep 1, 1985• 63 citations
Publisher

Abstract

The ground-state energy per site, \ensuremath{\epsilon}(\ensuremath{\delta}), and the singlet-triplet gap, \ensuremath{\Delta}(\ensuremath{\delta}), of the linear spin-(1/2) alternating Heisenberg antiferromagnet are found by extrapolation of direct solutions to N\ensuremath{\le}26 sites for even rings and N\ensuremath{\le}21 sites for odd rings, whose orbital degeneracy leads to qualitatively different critical behavior. The \ensuremath{\epsilon}(\ensuremath{\delta}) accuracy of ${<10}^{\mathrm{\ensuremath{-}}4}$ (in units of J) decisively rules out power-law behavior for any alternation \ensuremath{\delta}>0.001 and is consistent with ${\ensuremath{\delta}}^{4/3}$/\ensuremath{\Vert}ln\ensuremath{\delta}\ensuremath{\Vert} for \ensuremath{\delta}<0.05. The gap \ensuremath{\Delta}(\ensuremath{\delta}) is slightly sublinear in \ensuremath{\delta}. Direct ground-state solutions are found via diagrammatic valence-bond methods and provide sensitive tests for theoretical approaches to linear spin systems.

DOI
10.1103/physrevb.32.3124

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The ground-state energy per site, \ensuremath{\epsilon}(\ensuremath{\delta}), and the singlet-triplet gap, \ensuremath{\Delta}(\ensuremath{\delta}), of the linear spin-(1/2) alternating Heisenberg antiferromagnet are found by extrapolation of direct solutions to N\ensuremath{\le}26 sites for even rings and N\ensuremath{\le}21 sites for odd rings, whose orbital degeneracy leads to qualitatively different critical behavior. The \ensuremath{\epsilon}(\ensuremath{\delta}) accuracy of <10\ensuremath−4{<10}^{\mathrm{\ensuremath{-}}4}<10\ensuremath−4 (in units of J) decisively rules out power-law behavior for any alternation \ensuremath{\delta}>0.001 and is consistent with \ensuremathδ4/3{\ensuremath{\delta}}^{4/3}\ensuremathδ4/3/\ensuremath{\Vert}ln\ensuremath{\delta}\ensuremath{\Vert} for \ensuremath{\delta}<0.05. The gap \ensuremath{\Delta}(\ensuremath{\delta}) is slightly sublinear in \ensuremath{\delta}. Direct ground-state solutions are found via diagrammatic valence-bond methods and provide sensitive tests for theoretical approaches to linear spin systems.
Alternation rarameter δ dependences of the ground-state energy (GE) ε (δ) and the gap Δ (δ) in the excitation spectrum of the spin Hamiltonian H = J ∑ l =1 N [1+(-1) l δ] [ S l x S l +1 x + S l y S l +1 y +σ S l z S l +1 z ] are calculated in two numerical methods, with interest in the exponents β of the GE and α of the gap which are defined as ε (δ) - ε (0)∝δ β and Δ (δ)∝δ α for small δ. First by means of the 2-level renormalization group calculation the exponents are obtained as β∼1.78 and α∼0.96 for the isotropic Heisenberg model (σ=1). Second by means of the finite chain extrapolation, β for σ=1 is obtained as 1.68 -0.36 +0.13 from the data of the extrapolated GE between δ=0.05 and 0.1.
The linear, spin-12\frac{1}{2}21​, alternating Heisenberg chain has attracted theoretical and experimental attention from physical chemists for about two decades, particularly in relation to spin exciton theory and the properties of linear, exchange-coupled free radicals. The model is somewhat unfamiliar to physicists but has become of increasing interest recently, primarily because of its relation to spin-Peierls transition systems. A striking feature of this model is that it has so far proved resistant to any form of analytic attack. Existing theories are therefore all approximate, and are not in agreement with one another. In particular, there is disagreement about the existence of an energy gap in the excitation spectrum for nonzero alternation, such a gap being crucial to spin-Peierls theory and spin exciton theory. In this paper we employ the method which has so far proved more reliable than any other approximate technique, namely the method of extrapolating exact finite-chain calculations to the thermodynamic limit. Our study is an extension of earlier work in this direction, and focuses on the nature of the ground state and on low-lying excitations in general, and the existence and properties of the gap in particular. We introduce the features of the linear alternating antiferromagnet through an initial description of the spin-Peierls transition and with brief reference to organic free radicals and spin exciton theory. This is followed by a survey of existing approximate theories. Features of the excitation spectrum are discussed and finite-chain extrapolations for the ground-state energy and energy gap as a function of alternation are presented. Comparisons are made with similar procedures performed on exactly solvable models, as a test of the expected accuracy of the extrapolations. Excitation spectra for a variety of other alternating models, classical and quantum, are calculated and surveyed comparatively. An unusual variety of behavior is observed, with striking differences between quantum and classical systems. Finally, a detailed comparison is made between our results and those of other approximate methods, including the new quantum renormalization-group approach. Particular attention is paid to values for the T=0T=0T=0 spin-Peierls critical exponents.
An even-site block renormalization-group method is applied to the spin-1 and spin-(3/2 anisotropic or alternating Heisenberg antiferromagnets. In the spin-1 case it shows an energy gap \ensuremath{\Delta} \ensuremath{\simeq}0.368 166 and ground-state energy E0{E}_{0}E0​/NJ\ensuremath{\simeq}1.449 724, which are in good agreement with the recent Monte Carlo results. In the spin-(3/2 case it shows an energy excitation behavior similar to that in the spin-(1/2 case. Therefore, our study tends to support Haldane's conjecture. In addition, a possible novel property of the spin-1 Heisenberg antiferromagnet has been predicted.
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.
We investigate the critical behavior of the S=1/2S=1/2S=1/2 alternating Heisenberg chain using the density matrix renormalization group. The ground-state energy per spin, e0,{e}_{0},e0​, and singlet-triplet energy gap $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{\ensuremath{\Delta}}$ are determined for a range of alternations $\ensuremath{\delta}.$ Our results for the approach of e0{e}_{0}e0​ to the uniform chain limit are well described by $c{\ensuremath{\delta}}^{p},$ with $p\ensuremath{\approx}1.45.$ The singlet-triplet gap is also well described by a power law, with $p\ensuremath{\approx}0.73,$ half of the e0{e}_{0}e0​ power. The renormalization group predictions of power laws with logarithmic corrections can also accurately describe our data provided that a surprisingly large-scale parameter ${\ensuremath{\delta}}_{0}$ is present in the logarithms.
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