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Physical review. B, Condensed matter

Crossover effects in a spin-½ antiferromagnetic Heisenberg-Ising ring

Hiroshi Betsuyaku

Published Oct 1, 1987• 3 citations
Publisher

Abstract

Numerical results are presented for the crossover effects existing in the $S=\frac{1}{2}$ one-dimensional antiferromagnetic Heisenberg-Ising model. A novel Heisenberg singlet phase appears unexpectedly between the gapless $\mathrm{XY}$ phase and the doublet ground-state N\'eel phase. It extends over the range of anisotropy ($0.5\ensuremath{\leqq}\ensuremath{\Delta}\ensuremath{\leqq}{\ensuremath{\Delta}}_{c}$) above the isotropic Heisenberg point $\ensuremath{\Delta}=0.5$. It is ascertained that at ${\ensuremath{\Delta}}_{c}\ensuremath{\simeq}0.67$ (1) the second derivative of the ground-state energy exhibits a rounded peak, (2) the order parameter of the N\'eel state behaves as $\ensuremath{\Delta}\ensuremath{-}{\ensuremath{\Delta}}_{c}$, and (3) the mobility of soliton-antisoliton pairs changes drastically.

DOI
10.1103/physrevb.36.5613

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Comment on "Ground-state properties of a spin-1 antiferromagnetic chain"

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Physical review. B, Condensed matter•Oct 1, 1983•250 citations
Numerical results are presented for the crossover effects existing in the S=12S=\frac{1}{2}S=21​ one-dimensional antiferromagnetic Heisenberg-Ising model. A novel Heisenberg singlet phase appears unexpectedly between the gapless XY\mathrm{XY}XY phase and the doublet ground-state N'eel phase. It extends over the range of anisotropy (0.5\ensuremath≦\ensuremathΔ\ensuremath≦\ensuremathΔc0.5\ensuremath{\leqq}\ensuremath{\Delta}\ensuremath{\leqq}{\ensuremath{\Delta}}_{c}0.5\ensuremath≦\ensuremathΔ\ensuremath≦\ensuremathΔc​) above the isotropic Heisenberg point \ensuremathΔ=0.5\ensuremath{\Delta}=0.5\ensuremathΔ=0.5. It is ascertained that at \ensuremathΔc\ensuremath≃0.67{\ensuremath{\Delta}}_{c}\ensuremath{\simeq}0.67\ensuremathΔc​\ensuremath≃0.67 (1) the second derivative of the ground-state energy exhibits a rounded peak, (2) the order parameter of the N'eel state behaves as \ensuremathΔ\ensuremath−\ensuremathΔc\ensuremath{\Delta}\ensuremath{-}{\ensuremath{\Delta}}_{c}\ensuremathΔ\ensuremath−\ensuremathΔc​, and (3) the mobility of soliton-antisoliton pairs changes drastically.
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.
A finite-cell-scaling analysis of the spin-1 antiferromagnetic Heisenberg-Ising chain, shows that the ground-state properties are completely different than in the spin-\textonehalf{} case. Between the XY\mathrm{XY}XY gapless phase and the doublet ground-state N\'eel phase, it appears, in an extended range of the anisotropy ($0\ensuremath{\lesssim}\ensuremath{\lambda}\ensuremath{\lesssim}1.18$) a characteristic Heisenberg-phase with a nonmagnetic singlet ground state, nonzero gap, and exponential decay of the correlation functions.
Botet and Jullien [Phys. Rev. B 27, 613 (1983)] have conducted a finite-chain scaling analysis on the spin-1 antiferromagnetic Heisenberg-Ising chain. They claim that the ground-state properties are completely different from the case of spin 12\frac{1}{2}21​. Using finite-size scaling techniques, they conclude that between the XY\mathrm{XY}XY gapless phase and the doublet-ground-state N\'eel phase, a new phase appears in an extended anisotropy range. This phase, which encompasses the isotropic Heisenberg model, is characterized by a nonmagnetic singlet ground state and a nonzero excitation-energy gap. We perform a similar finite-chain scaling analysis on the spin-12\frac{1}{2}21​ antiferromagnetic Heisenberg-Ising chain and obtain strikingly similar results. Since the spin-14\frac{1}{4}41​ system is known rigorously not to exhibit the type of behavior proposed by Botet and Jullien, their analysis is open to doubt. The possibility arises that a strong singularity at the Heisenberg point for spin 1 (similar to the essential singularity for spin 12\frac{1}{2}21​) may be giving rise to misleading results. The T=0T=0T=0 phase behavior of the spin-1 Heisenberg-Ising antiferromagnetic chain, therefore, remains in question and specific additional numerical studies are proposed to clarify the matter.
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.
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.
Finite-cell calculations (up to N=12N=12N=12 spins) have been performed on the spin-1 Heisenberg-Ising chain with an uniaxial anisotropy, $\mathcal{H}=\ensuremath{\Sigma}{i}^{}[{S}_{i}^{x}{S}_{i+1}^{x}+{S}_{i}^{y}{S}_{i+1}^{y}+\ensuremath{\lambda}{S}_{i}^{z}{S}_{i+1}^{z}+D{({S}_{i}^{z})}^{2}]$. From a scaling analysis of the gap between the ground state and the first excited state, a phase diagram has been drawn in the ($\ensuremath{\lambda},D$) plane and the transition lines between the "ferromagnetic," $X\ensuremath{-}Y$," "singlet-ground-state," and "antiferromagnetic" phases have been estimated for the infinite-NNN system. One of the most important results is that a singlet-ground-state phase with a nonzero gap exists in an extended range of $\ensuremath{\lambda}$ and DDD values including the Heisenberg point $\ensuremath{\lambda}=1$, D=0D=0D=0, in contrast with the spin-12\frac{1}{2}21​ case. Moreover, for $\ensuremath{\lambda}\ensuremath{\simeq}1$, the gap decreases with increasing positive anisotropy DDD, goes through a minimum, estimated to be zero, and then increases with DDD.