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Physical review. B, Condensed matter

Gap ofS=1/2alternating antiferromagnetic Heisenberg chains atT=0K

C. Tannous

Published Oct 1, 1985• 6 citations
Publisher

Abstract

Results for the gap of Heisenberg alternating antiferromagnetic finite chains of N spins S=(1/2) are reported. For N=2,4,...,18, the calculations are done with the modified boundary conditions and the results are compared with those obtained with periodic boundary conditions. For the largest considered chain it is shown that in both cases one is still far from the asymptotic regime.

DOI
10.1103/physrevb.32.4770

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Results for the gap of Heisenberg alternating antiferromagnetic finite chains of N spins S=(1/2) are reported. For N=2,4,...,18, the calculations are done with the modified boundary conditions and the results are compared with those obtained with periodic boundary conditions. For the largest considered chain it is shown that in both cases one is still far from the asymptotic regime.
The one-dimensional Heisenberg antiferromagnets of large-integer-SSS spins are studied; their Haldane gaps are estimated by the numerical diagonalization method for S=5S=5S=5 and 666. We successfully obtain a monotonically increasing sequence of finite-size energy difference data corresponding to the Haldane gaps from the huge-scale parallel calculations of diagonalization under the twisted boundary condition and create a monotonically decreasing sequence within the range of system sizes treated in this study from the monotonically increasing sequence. Consequently, the gaps for S=5S=5S=5 and 666 are estimated to be 0.000050±0.0000050.000050 \pm 0.0000050.000050±0.000005 and 0.0000030±0.00000050.0000030 \pm 0.00000050.0000030±0.0000005, respectively. The asymptotic formula of the Haldane gap for S→∞S\rightarrow\inftyS→∞ is examined from the new estimates to determine the coefficient in the formula more precisely.
Full magnetization curves of S=1 antiferromagnetic Heisenberg chains with periodic and open boundaries are investigated by a quantum Monte Carlo method. The boundary condition has a significant effect on the magnetization process especially in the field range where the Zeeman energy is less than the Haldane gap. The magnetization of the periodic chain in the range is strongly suppressed at low temperatures because the ground state is nonmagnetic. On the other hand, the magnetic ground of the open chain bring about finite magnetization. The difference between both the magnetization curves is also investigated in detail by subtracting the magnetization of the periodic chain with L spins from one of the open chains with L+1 spins. It agrees well with the double of the Brillouin function for S=1/2, rather than the Brillouun function for S=1.
A zero-temperature renormalization group (RG) approach is applied to the one-dimensional, spin-1/2 antiferromagnetic Heisenberg dimerized (alternating) chain. Specifically, the ground state energy and lowest-lying spectral excitations are examined. The calculation indicates the existence of a gap in the spectrum of the dimerized chain which vanishes only in the limit of a uniform spin chain, in contrast to a recent Green’s function approach. The RG results are in reasonable agreement with numerical extrapolations on the exact eigenvalue spectrum of finite chains of up to 12 spins. Both methods are compared with several other approximate treatments of the Heisenberg system, and tested by comparison with exact results for the spin-1/2 XY dimerized chain.
The one-dimensional Heisenberg antiferromagnet of S = 1 spins is studied by numericaldiagonalization method based on the Lanczos algorithm. The Haldane gap of this system isnumerically estimated from finite-size clusters under the twisted boundary condition up tothe 30-site system. The finite-size results are analyzed by a convergence-acceleration technique; the multistep procedure of the convergence acceleration is successfully carried out.The present estimate for the gap magnitude is more precisely than the previously reportedvalue from a Lanczos-diagonalization study. It also agrees well with the result reported ina study based on the density matrix renormalization group calculation. The present studyclarifies that Lanczos-diagonalization calculations can sufficiently contribute for preciselyestimating an energy gap although this method can treat only small clusters.
We have calculated the two-point correlation functions ${\ensuremath{\omega}}_{\mathrm{il}}$(N)=(4/3)〈S${\ensuremath{\rightarrow}}_{i}$\ensuremath{\cdot}S${\ensuremath{\rightarrow}}_{i+l}$ 〉 and their averages over i,${\ensuremath{\omega}}_{l}$ (N), in the ground state of the one-dimensional antiferromagnetic Heisenberg model for N=4,6,8,...,16 spins. Both periodic (rings) and free-end (chains) boundary conditions are considered. Surprisingly tight lower and upper bounds have been obtained for ${\ensuremath{\omega}}_{l}$(\ensuremath{\infty}) under reasonable assumptions. In addition to showing the rather strong even-l--odd-l alternation in \ensuremath{\Vert}${\ensuremath{\omega}}_{l}$(N)\ensuremath{\Vert}, known from earlier results of Bonner and Fisher for rings with N up to 10, our bounds indicate a smooth behavior in l\ensuremath{\Vert}${\ensuremath{\omega}}_{l}$(\ensuremath{\infty})\ensuremath{\Vert} for l odd and l even, with, surprisingly, a broad maximum attained within the odd-l values. The bounds obtained from the chain results were essential to seeing this maximum (because of the larger l values available for given N). The quantity l\ensuremath{\Vert}${\ensuremath{\omega}}_{l}$(N)\ensuremath{\Vert} for chains with fixed N also shows such a maximum, and in addition shows a similar maximum for even l's. If the indicated trends for large l and N continue in ${\ensuremath{\omega}}_{l}$(\ensuremath{\infty}) and in SN{S}_{N}SN​, the structure factor at wave vector \ensuremath{\pi}, then finite-size contributions to ${\ensuremath{\omega}}_{l}$(N) will have to contribute to the (seemingly) logarithmic divergence of SN{S}_{N}SN​ as N\ensuremath{\rightarrow}\ensuremath{\infty}. We are not aware of any models where a similarly weak divergence shows such a finite-size contribution.
Thermodynamic properties of S =1 antiferromagnetic Heisenberg chains with free and periodic boundaries are investigated by means of a quantum Monte Carlo method. The specific heat is shown to have a peak at a temperature T peak ∼2Δ, where Δ is the energy gap of the present model, although global shape of the temperature dependence is very similar to the Schottky type specific heat. In open chains, due to the four-fold degeneracy of the ground state, the magnetic susceptibility shows a Curie-like divergence regardless of the number of spins. The extracted diverging part is consistent with the edge moment S =1/2.