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Physical review. B, Condensed matter

Ground-state properties of axially anisotropic quantum Heisenberg chains

Jenő Sólyom, Timothy A. L. Ziman

Published Oct 1, 1984• 103 citations
Publisher

Abstract

The ground state and low-lying excited states of a one-dimensional Heisenberg model with axial anisotropy are studied by extrapolation from finite lattices. The ground-state properties and the phase diagram are determined for quantum spins, especially for $S=\frac{1}{2}, 1, \mathrm{and} \frac{3}{2}$. Higher spin values do not give new features in the phase diagram. the limitations of finite-lattice extrapolation are discussed in connection with the possibility of a singlet phase for the $S=1$ isotropic Heisenberg antiferromagnet.

DOI
10.1103/physrevb.30.3980

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

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Journal of the Physical Society of Japan•May 15, 1993•24 citations
The ground state and low-lying excited states of a one-dimensional Heisenberg model with axial anisotropy are studied by extrapolation from finite lattices. The ground-state properties and the phase diagram are determined for quantum spins, especially for S=12,1,and32S=\frac{1}{2}, 1, \mathrm{and} \frac{3}{2}S=21​,1,and23​. Higher spin values do not give new features in the phase diagram. the limitations of finite-lattice extrapolation are discussed in connection with the possibility of a singlet phase for the S=1S=1S=1 isotropic Heisenberg antiferromagnet.
The S=1/2S=1/2S=1/2 and S=1S=1S=1 two-dimensional quantum Heisenberg antiferromagnets on the anisotropic dimerized square lattice are investigated by the quantum Monte Carlo method. By finite-size-scaling analyses on the correlation lengths, the ground-state phase diagram parametrized by strengths of the dimerization and of the spatial anisotropy is determined much more accurately than the previous works. It is confirmed that the quantum critical phenomena on the phase boundaries belong to the same universality class as that of the classical three-dimensional Heisenberg model. Furthermore, for S=1,S=1,S=1, we show that all the spin-gapped phases, such as the Haldane and dimer phases, are adiabatically connected in the extended-parameter space, though they are classified into different classes in terms of the string order parameter in the one-dimensional, i.e., the zero-interchain-coupling, case.
Using density matrix renormalization group calculations, ground state properties of the spin–1 Heisenberg chain with exchange and quadratic single-ion anisotropies in an external field are studied, for special choices of the two kinds of anisotropies. In particular, the phase diagram includes antiferromagnetic, spin-liquid (or spin-flop), IS2, and supersolid (or biconical) phases. Especially, new features of the spin-liquid and supersolid phases are discussed. Properties of the quantum chains are compared to those of corresponding classical spin chains.
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.
Ground-state properties are studied for the one-dimensional anisotropic spin-1/2 Heisenberg magnet with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions, which are assumed to have the same form as regards the anisotropy. The singlet-triplet energy gap, the magnetization, the energy, and the two-spin correlation function in the ground states of finite-size systems of up to 20 spins are calculated exactly. By extrapolating these results, the values of the above quantities at the infinite-size limit are estimated. Special attention is paid to the limiting ground-state phase diagram on the plane of the ratio of the next-nearest-neighbor interaction constant to the nearest-neighbor one versus the anisotropy parameter. It is found that, when the interactions have the XY-type anisotropy, the limiting ground state is in the spin-fiuid phase or in the dimer phase depending upon whether the ratio between the interaction constants is smaller or larger than a critical value. It is also found that, when the interactions have the Ising-type anisotropy, a novel phase with unsaturated magnetization appears between the ferromagnetic and (2, 2) antiphase phases in the limiting ground-state phase diagram.
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.