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Physical review. B, Condensed matter

Numerical diagonalization study of anS=1/2 ladder model with open boundary conditions

Hiroshi K. WATANABE

Published Nov 1, 1994• 32 citations
Publisher

Abstract

Low-lying energy states of an S=1/2 ladder model are investigated by applying the numerical diagonalization method to finite clusters. This model has the antiferromagnetic intrachain coupling J (J>0) and the ferromagnetic interchain one -\ensuremath{\lambda}J (\ensuremath{\lambda}>0). Both the inverse correlation length ${\ensuremath{\xi}}^{\mathrm{\ensuremath{-}}1}$(\ensuremath{\lambda}) and an energy gap are shown to be finite at least for \ensuremath{\lambda}\ensuremath{\ge}0.05, the former of which is shown to approach the same value as that of the S=1 antiferromagnetic Heisenberg chain with increasing \ensuremath{\lambda}. A generation mechanism of the gap is also discussed in terms of a simple model by using the Lieb-Mattis theorem.

DOI
10.1103/physrevb.50.13442

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Low-lying energy states of an S=1/2 ladder model are investigated by applying the numerical diagonalization method to finite clusters. This model has the antiferromagnetic intrachain coupling J (J>0) and the ferromagnetic interchain one -\ensuremath{\lambda}J (\ensuremath{\lambda}>0). Both the inverse correlation length \ensuremathξ\ensuremath−1{\ensuremath{\xi}}^{\mathrm{\ensuremath{-}}1}\ensuremathξ\ensuremath−1(\ensuremath{\lambda}) and an energy gap are shown to be finite at least for \ensuremath{\lambda}\ensuremath{\ge}0.05, the former of which is shown to approach the same value as that of the S=1 antiferromagnetic Heisenberg chain with increasing \ensuremath{\lambda}. A generation mechanism of the gap is also discussed in terms of a simple model by using the Lieb-Mattis theorem.
We investigate an S =(1/2) ladder model with a ferromagnetic interchain coupling (-λ) and an antiferromagnetic intrachain coupling based on the bosonization method, which reduces to the S =1 antiferromagnetic Heisenberg model in the limit λ→∞. We argue with the aid of Wilson's renormalization group method that the ground state is the Haldane state with an excitation gap for λ>0, that the gap opens at λ=0 exponentially and that the critical index of the coherence length is unity (ν=1) on the critical line between the antiferromagnetic and the Haldane phases. Furthermore, it is found that the string order parameter proposed by den Nijs and Rommelse is expressed by a local order parameter in the bosonization method.
Using the density matrix renormalization group method, we have obtained the ground-state phase diagram of a spin-1/2 two-leg antiferromagnetic Heisenberg ladder with a ferromagnetic diagonal interaction ${J}_{\ifmmode\times\else\texttimes\fi{}}.$ The boundary between the two phases found in the system, the Haldane phase and the singlet phase, lies in the frustrated area of the parameter space, i.e., both the rung coupling ${J}_{\ensuremath{\perp}}$ and the diagonal coupling ${J}_{\ifmmode\times\else\texttimes\fi{}}$ are ferromagnetic. The spin gap is found to be insensitive to the change of the ferromagnetic diagonal coupling strength when the ratio of the rung coupling constant ${J}_{\ensuremath{\perp}}$ to the leg coupling constant ${J}_{\ensuremath{\Vert}}$ is 0.5, in agreement with the analysis of recent experimental spin susceptibility data.
Heisenberg antiferromagnetic spin ``ladders'' (two coupled spin chains) are low-dimensional magnetic systems which for S=1/2 interpolate between half-integer-spin chains, when the chains are decoupled, and effective integer-spin one-dimensional chains in the strong-coupling limit. The spin-1/2 ladder may be realized in nature by vanadyl pyrophosphate, (VO)2{)}_{2})2​P2{\mathrm{P}}_{2}P2​O7{\mathrm{O}}_{7}O7​. In this paper we apply strong-coupling perturbation theory, spin-wave theory, Lanczos techniques, and a Monte Carlo method to determine the ground-state energy and the low-lying excitation spectrum of the ladder. We find evidence of a nonzero spin gap for all interchain couplings ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$>0. A band of spin-triplet excitations above the gap is also analyzed. These excitations are unusual for an antiferromagnet, since their long-wavelength dispersion relation behaves as (k-k0{\mathit{k}}_{0}k0​)2{)}^{2})2 (in the strong-coupling limit ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$\ensuremath{\gg}J, where J is the in-chain antiferromagnetic coupling). Their band is folded, with a minimum energy at k0{\mathit{k}}_{0}k0​=\ensuremath{\pi}, and a maximum between k1{\mathit{k}}_{1}k1​=\ensuremath{\pi}/2 (for ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=0) and 0 (for ${\mathit{J}}_{\mathrm{\ensuremath{\perp}}}$=\ensuremath{\infty}). We also give numerical results for the dynamical structure factor S(q,\ensuremath{\omega}), which can be determined in neutron scattering experiments. Finally, possible experimental techniques for studying the excitation spectrum are discussed.
We study two Heisenberg spin-1/2 chains coupled by a frustrating ``zigzag'' interaction. We are particularly interested in the regime of weak interchain coupling, which is difficult to analyze by either numerical or analytical methods. Previous density matrix renormalization-group studies of the isotropic model with open boundary conditions and sizable interchain coupling have established the presence of incommensurate correlations and of a spectral gap. By using twisted boundary conditions with an arbitrary twist angle, we are able to determine the incommensurabilities both in the isotropic case and in the presence of an exchange anisotropy by means of exact diagonalization of relatively short finite chains of up to 242424 sites. Using twisted boundary conditions results in a very smooth dependence of the incommensurabilities on system size, which makes the extrapolation to infinite systems significantly easier than for open or periodic chains.
We study two Heisenberg spin-1/2 chains coupled by a frustrating ''zigzag'' interaction. We are particularly interested in the regime of weak interchain coupling, which is difficult to analyze by either numerical or analytical methods. Previous density matrix renormalization-group studies of the isotropic model with open boundary conditions and sizable interchain coupling have established the presence of incommensurate correlations and of a spectral gap. By using twisted boundary conditions with an arbitrary twist angle, we are able to determine the incommensurabilities both in the isotropic case and in the presence of an exchange anisotropy by means of exact diagonalization of relatively short finite chains of up to 24 sites. Using twisted boundary conditions results in a very smooth dependence of the incommensurabilities on system size, which makes the extrapolation to infinite systems significantly easier than for open or periodic chains.
The ground-state phase diagram of the S =1/2 X X Z ladder model is investigated by using the density-matrix renormalization-group method. This model is composed of two S =1/2 X X Z chains with ferromagnetic rung couplings (λ) and intra-chain anisotropy (Δ), and continuously changes from the two independent S =1/2 X X Z chains at λ=0 to the S =1 X X Z chain in the limit of λ→∞ . It is shown that there exist edge states at Δ=1 and λ=1 in relation to the four-fold degeneracy of the ground state and that an excitation gap opens nearly at λ=0 for 0.8≤Δ≤1. In addition, it is clarified that the transition between the Haldane phase and the antiferromagnetic phase occurs at λ=1.15 ±0.01 for Δ=1. A possible phase diagram in the λ-Δ plane is proposed, which is consistent with the scaling analysis based on the bosonization method.