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arXiv:cond-mat/9712298

One-dimensional Heisenberg antiferromagnet in which two S=1/2 and two S=1 spins are arranged alternatively

Takashi Tonegawa, Takashi Hikihara, Tomotoshi Nishino, Makoto Kaburagi, Seiji Miyashita, H.-J. Mikeska

Published Dec 26, 1997• 0 citations
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Abstract

Ground-state and thermodynamic properties of the one-dimensional Heisenberg antiferromagnet in which two S=1/2 and two S=1 spins are arranged alternatively are studied by a quantum Monte Carlo method and by analytical estimates. It is found that the ground state of the system undergoes a second-order phase transition which accompanies the vanishing of the energy gap.

DOI
10.48550/arxiv.cond-mat/9712298

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Ground-state and thermodynamic properties of the one-dimensional Heisenberg antiferromagnet in which two S=1/2 and two S=1 spins are arranged alternatively are studied by a quantum Monte Carlo method and by analytical estimates. It is found that the ground state of the system undergoes a second-order phase transition which accompanies the vanishing of the energy gap.
The ground-state properties of the one-dimensional spin-1/2 alternating Heisenberg-Ising antiferromagnet are studied both analytically and numerically. The phase diagram is qualitatively predicted by the use of the phase Hamiltonian: The ground state is either the Néel or the effective singlet state according to the values of the alternation and anisotropy parameters. The phase transition between these states is of the first order. The ground state energy, its derivative with respect to the alternation parameter, excitation gap, and the various long-range order parameters are exactly calculated by the numerical method for finite systems ( N ≦20). The numerical calculations support the analytically predicted phase diagram. The correlation length is also numerically calculated at finite temperatures by the use of the quantum transfer-matrix method.
Thermodynamic properties of S=1 antiferromagnetic Heisenberg chains with free and periodic boundaries are investigated by a quantum Monte Carlo method. In particular, temperature dependences of the specific heat, the magnetic susceptibility, and the hidden order parameter, which are inherent to the Haldane phase, are investigated. The specific heat turns out to have a peak at a temperature Tpeak{\mathit{T}}_{\mathrm{peak}}Tpeak​\ensuremath{\sim}2\ensuremath{\Delta}, where \ensuremath{\Delta} is the energy gap of the present model, although the temperature dependence of the specific heat is very similar to the Schottky type. In open chains, due to the fourfold degeneracy of the ground state, the magnetic susceptibility shows a Curie-like divergence regardless of the number of spins in the chain. The amplitude of the Curie-like divergence is consistent with the edge moments (S=1/2). On the other hand, it is confirmed that the long-range hidden order exists only in the ground state. How the hidden order grows as the temperature goes to zero (T\ensuremath{\rightarrow}0) is investigated by introducing an intrinsic correlation length ${\ensuremath{\xi}}_{\mathit{D}}$ of the ordering, which diverges as T\ensuremath{\rightarrow}0. A qualitative difference in fluctuations between the ground state and the finite-temperature states is discussed making use of snapshots of the transformed two-dimensional Ising system on which a Monte Carlo simulation is being performed.
A real space renormalization group method is applied to study the properties of the anisotropic and alternating Heisenberg antiferromagnets with spin 1/2, 1 and 3/2. To improve the accuracy of the numerical estimates larger spin cluster is used. The ground state energy and the energy gap are obtained. For the case of S = 1 at Heisenberg point, the ground state energy per site is E (0)/N = -1.4021 ± 0.0005 and the energy gap is 0.4097 ± 0.0005, in excellent agreement with a recent Monte Carlo calculation. For the cases of spin 1/2 and 3/2 the energy gap is absent. Therefore our results are shown to support Haldane's conjecture. We briefly discuss the properties of the novel phase.
Ground-state properties of S =1 antiferromagnetic Heisenberg chains with bond alternation, H = J ∑ i {1-(-1) i δ} S i · S i +1 , are studied by a quantum Monte Carlo method. Spin correlation between chain ends on the open chain appears for δ<0.25, while it vanishes for δ≥0.25. At δ≃0.25, the spin correlation length has a peak and the S =1/2 edge states for the open chain disappear. The fourfold degeneracy of the ground state of the open chain holds for δ<0.25, while it is removed for δ≥0.25. All the results support the continuous phase transition at δ≃0.25 demonstrated in the previous works.
We describe a numerical simulation of the isotropic spin-\textonehalf{} Heisenberg antiferromagnet on square lattices of size 4\ifmmode\times\else\texttimes\fi{}4, 6\ifmmode\times\else\texttimes\fi{}6, and 8\ifmmode\times\else\texttimes\fi{}8. A Hamiltonian Monte Carlo method is used to measure the energies of the lowest-lying singlet and triplet states, and we extrapolate these results to give estimates of the properties of large antiferromagnets. Our result for the ground-state energy per spin is $\frac{{E}_{0}}{{N}^{2}}=\ensuremath{-}0.672\ifmmode\pm\else\textpm\fi{}0.001$ for an infinite system. Within our statistical accuracy, the singlet-triplet gap behaves as ${E}_{1}\ensuremath{-}{E}_{0}\ensuremath{\approx}\frac{2.1}{N}$, which implies that the infinite system is gapless.
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.