Journal of Physics A Mathematical and General

Comparison of ground state properties for odd half-integer and integer spin antiferromagnetic Heisenberg chains

M. Kolb, Robert S. Botet, Rémi Jullien

Published Dec 1, 1983• 75 citations

Abstract

Ground state properties of finite anisotropic antiferromagnetic Heisenberg chains are studied for odd half-integer spins S= 1 / 2 , 3 / 2 ... as well as integer spins S=1, 2 .... Finite size scaling analysis of the results clearly distinguishes the half-integer (S= 1 / 2 , 3 / 2 ) from the integer (S=1) spin situation. It gives strong support to a recent conjecture which postulates that the T=0 phase structure is very different in the two cases. According to this idea there exists a new phase between the planar and the antiferromagnetic region, for integer spins only. This phase, which includes the isotropic point, has a finite energy gap and no long range order.

DOI: 10.1088/0305-4470/16/17/008

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Journal of Physics A Mathematical and General

Comparison of ground state properties for odd half-integer and integer spin antiferromagnetic Heisenberg chains

M. Kolb, Robert S. Botet, Rémi Jullien

Published Dec 1, 1983• 75 citations

Publisher

Abstract

DOI: 10.1088/0305-4470/16/17/008

Related papers

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

Jill C. Bonner, Gerhard H. Muller

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 $\frac{1}{2}$ . Using finite-size scaling techniques, they conclude that between the $\mathrm{XY}$ 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- $\frac{1}{2}$ antiferromagnetic Heisenberg-Ising chain and obtain strikingly similar results. Since the spin- $\frac{1}{4}$ 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 $\frac{1}{2}$ ) may be giving rise to misleading results. The $T=0$ 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.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 $\frac{1}{2}$. Using finite-size scaling techniques, they conclude that between the $\mathrm{XY}$ 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-$\frac{1}{2}$ antiferromagnetic Heisenberg-Ising chain and obtain strikingly similar results. Since the spin-$\frac{1}{4}$ 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 $\frac{1}{2}$) may be giving rise to misleading results. The $T=0$ 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.

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Ground state properties, dispersion relations and scaling behaviour of spin gap of a bond alternating spin-1 2 anisotropic Heisenberg chain have been studied where the exchange interactions on alternate bonds are ferromagnetic (FM) and antiferromagnetic (AFM) in two separate cases.The resulting models separately represent nearest neighbour (NN) AFM-AFM and AFM-FM bond alternating chains.Ground state energy has been estimated analytically by using both bond operator and Jordan-Wigner representations and numerically by using exact diagonalization.Dispersion relations, spin gap and several ground state orders have been obtained.Dimer order and string orders are found to coexist in the ground state.Spin gap is found to develop as soon as the non-uniformity in alternating bond strength is introduced in the AFM-AFM chain which further remains non-zero for the AFM-FM chain.This spin gap along with the string orders attribute to the Haldane phase.The Haldane phase is found to exist in most of the anisotropic region similar to the isotropic point.Ground state properties, dispersion relations and scaling behaviour of spin gap of a bond alternating spin-1 2 anisotropic Heisenberg chain have been studied where the exchange interactions on alternate bonds are ferromagnetic (FM) and antiferromagnetic (AFM) in two separate cases.The resulting models separately represent nearest neighbour (NN) AFM-AFM and AFM-FM bond alternating chains.Ground state energy has been estimated analytically by using both bond operator and Jordan-Wigner representations and numerically by using exact diagonalization.Dispersion relations, spin gap and several ground state orders have been obtained.Dimer order and string orders are found to coexist in the ground state.Spin gap is found to develop as soon as the non-uniformity in alternating bond strength is introduced in the AFM-AFM chain which further remains non-zero for the AFM-FM chain.This spin gap along with the string orders attribute to the Haldane phase.The Haldane phase is found to exist in most of the anisotropic region similar to the isotropic point.

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Robert S. Botet, Rémi Jullien

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 $\mathrm{XY}$ 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.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 $\mathrm{XY}$ 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.

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\frac{1}{2}

. Using finite-size scaling techniques, they conclude that between the

\mathrm{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-

\frac{1}{2}

antiferromagnetic Heisenberg-Ising chain and obtain strikingly similar results. Since the spin-

\frac{1}{4}

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

\frac{1}{2}

) may be giving rise to misleading results. The

T=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.

We study the ground-state properties of antiferromagnetic ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ chains with half-integer spins ranging from

S=\frac{3}{2}

S=\frac{11}{2}

using the density-matrix renormalization group method. We map out the ground-state phase diagrams as a function of

\frac{{J}_{2}}{{J}_{1}}

containing topological phases with alternating $\frac{2S\ensuremath{-}1}{2}$ and

\frac{2S+1}{2}

valence bonds. We identify these topological phases and their boundaries by calculating the string order parameter, the dimer order parameter, and the spin gap for those $\mathrm{high}\text{\ensuremath{-}}S$ systems in the thermodynamic limit (finite-size scaling). We find that these topological regions narrow down inversely with

S

and converge to a single point at

\frac{{J}_{2}}{{J}_{1}}=\frac{1}{4}

in the classical limit---a critical threshold between commensurate and incommensurate orders. In addition, we extend the discussion of the Majumder-Ghosh state, previously noted only for

S=\frac{1}{2}

, and speculate about its possible presence as a ground state in half-integer high-spin systems over a substantial range of

\frac{{J}_{2}}{{J}_{1}}

values.

\mathrm{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.