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Journal of the Physical Society of Japan

Twist Method for Quantum Ground-State Problems and its Application to Antiferromagnetic Heisenberg Chains with Competing Interactions

Kiyomi Okamoto, Yohtaro Ueno

Published Jan 15, 1995• 1 citation
Publisher

Abstract

We present a quantum version of the twist method which has been developed for classical systems. A finite size scaling form for the excess ground state energy is derived and tested for an S =1/2 antiferromagnetic Heisenberg chain with next nearest neighbor interactions: H = J ∑( S j · S j +1 + α S j · S j +2 ) with J >0 and α>0. Using an exact diagonalization method, we obtain the critical value \(\alpha_{c} \cong 0.246\) of the fluid-dimer transition, which is very close to the reliable value recently obtained by Okamoto and Nomura (Phys. Lett. A 169 (1992) 433).

DOI
10.1143/jpsj.64.86

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Progress of Theoretical Physics Supplement•Jan 1, 1990•22 citations
We present a quantum version of the twist method which has been developed for classical systems. A finite size scaling form for the excess ground state energy is derived and tested for an S =1/2 antiferromagnetic Heisenberg chain with next nearest neighbor interactions: H = J ∑( S j · S j +1 + α S j · S j +2 ) with J >0 and α>0. Using an exact diagonalization method, we obtain the critical value (\alpha_{c} \cong 0.246) of the fluid-dimer transition, which is very close to the reliable value recently obtained by Okamoto and Nomura (Phys. Lett. A 169 (1992) 433).
Starting from the exact result of Majumdar and Ghosh for the dimerized ground state of a Heisenberg linear-chain antiferromagnet with competing next-nearest-neighbor interaction J2{\mathit{J}}_{2}J2​=1/2J1{\mathit{J}}_{1}J1​, it is shown that this state is an exact eigenstate of the same Hamiltonian with couplings extended up to fourth-nearest neighbors, if the relations J1{\mathit{J}}_{1}J1​-2J2{\mathit{J}}_{2}J2​+J3{\mathit{J}}_{3}J3​=0 and J3{\mathit{J}}_{3}J3​=2J4{\mathit{J}}_{4}J4​ hold. Furthermore, it is proved to all orders in perturbation theory that this state is the ground state of this Hamiltonian in a finite interval of values of the parameter \ensuremath{\lambda}==1/2-j2{\mathit{j}}_{2}j2​ around \ensuremath{\lambda}=0. The actual extension of this interval is computed numerically for finite chains and is shown to be significant.
Starting from the exact result of Majumdar and Ghosh (1969) for the dimerized ground state of a Heisenberg S=1/2 antiferromagnet with competing next-nearest-neighbour interaction J2+1/2J1 on a linear chain, it is shown that this state is the ground state of the same kind of Hamiltonian, if couplings are extended up to 2n-nearest-neighbours, and the relation J1=2n, J2=2n-1, ..., J2n=1 holds. Furthermore, it is proved rigorously that this state is the ground state when n=2 and J1-2J2+J3=0, J3=2J4, in a finite interval of values of the parameter J4. A rigorous lower limit for the extension of this interval is found to be 0<or=J4<or=1/4J1. A comparison with ground-state configurations of the equivalent classical Heisenberg and Ising model is discussed.
We study the ground-state properties of the one-dimensional anisotropic spin-1/2 antiferromagnetic Heisenberg model with competing antiferromagnetic second-neighbor Ising interaction. Using the modified Lanczos method, we obtain the phase diagram in parameter space and different correlation functions, which we use to characterize the different phases. For small next-nearest-neighbor parameters we evaluate the critical exponent \ensuremath{\eta} of the spin-spin correlation function. We find good agreement between the numerical results and those obtained using the Luther and Peschel approach.
We investigate the ground-state properties of the spin-half ${J}_{1}\ensuremath{-}{J}_{2}$ Heisenberg chain with a next-nearest-neighbor spin-Peierls dimerization using conformal field theory and Lanczos exact diagonalizations. In agreement with the results of a recent bosonization analysis by Sarkar and Sen [Phys. Rev. B 65,65,65, 172408 (2002)], we find that for small frustration (J2/J1){(J}_{2}{/J}_{1})(J2​/J1​) the system is in a Luttinger spin-fluid phase, with gapless excitations, and a finite spin-wave velocity. In the regime of strong frustration the ground state is spontaneously dimerized and the bond alternation reduces the triplet gap, leading to a slight enhancement of the critical point separating the Luttinger phase from the gapped one. An accurate determination of the phase boundary is obtained numerically from the study of the excitation spectrum.
Ground-state properties are studied for the one-dimensional anisotropic spin-1/2 Heisen­ berg 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-fluid 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.