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Journal of the Physical Society of Japan

S=1/2 Quantum Heisenberg Ladder andS=1 Haldane Phase

Hiroshi K. WATANABE, K. Nomura, Satoshi Takada

Published Aug 15, 1993• 60 citations
Publisher

Abstract

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.

DOI
10.1143/jpsj.62.2845

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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.
It is shown that the ground state of a spin-1/2 antiferromagnetic ladder model is characterized by a string order parameter which is similar to but differs from that introduced by den Nijs and Rommelse. This reveals the presence of a hidden antiferromagnetic order and symmetry breaking even in the ground state which relates to high-Tc{\mathit{T}}_{\mathit{c}}Tc​ superconductors. The inverse correlation length is further calculated for any ratio of the interchain and intrachain couplings under shifted boundary conditions.
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.
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.
Ground-state properties of the spin-1 two-leg antiferromagnetic ladder are investigated precisely by means of the quantum Monte Carlo method. It is found that the correlation length along the chains and the spin gap both remain finite regardless of the strength of interchain coupling, i.e., the Haldane state and the spin-1 dimer state are connected smoothly without any quantum phase transitions between them. We propose a plaquette-singlet solid state, which qualitatively describes the ground state of the spin-1 ladder quite well, and also a corresponding topological hidden order parameter. It is shown numerically that the proposed hidden order parameter remains finite up to the dimer limit, though the conventional string order defined on each chain vanishes immediately when infinitesimal interchain coupling is introduced.
The ground-state properties of the spin-1/2 Heisenberg chain, which has two exchange couplings (J>J') alternatingly, are studied. This model interpolates the spin-1 antiferromagnetic Heisenberg chain (AFHC) (J'\ensuremath{\rightarrow}-\ensuremath{\infty}, J>0), the dimerized spin-1/2 AFHC (J'\ensuremath{\simeq}J>0), and the uniform spin-1/2 AFHC (J'=J>0) continuously. The string order introduced by den Nijs--Rommelse and Tasaki changes continuously through the Haldane phase and the dimer phase to vanish at J=J'. It is at a maximum for J'=0. This implies that the string order is related to the local singlet correlation in terms of the spin-1/2 model. The lowest excitation can be regarded as the renormalized triplet wave. The reversal of the anisotropy effect on the excitation spectrum of the spin-1 AFHC is also explained in a intuitive way based on this picture.
We obtain the phase diagram in the parameter space $({J}^{\ensuremath{'}}∕J,\ensuremath{\gamma})$ and an accurate estimate of the critical line separating the different phases. We show several measurements of the magnetization, dimerization, nearest neighbor correlation, and density of energy in the different zones of the phase diagram, as well as a measurement of the string order parameter proposed as the nonvanishing phase order parameter characterizing Haldane phases. All these results will be compared in the limit ${J}^{\ensuremath{'}}∕J⪢1$ with the behavior of the S=1\mathbf{S}=1S=1 bond alternated Heisenberg chain (BAHC). The analysis of our data supports the existence of a dimer phase separated by a critical line from a Haldane one, which has exactly the same nature as the Haldane phase in the S=1\mathbf{S}=1S=1 BAHC.