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Physical review. B, Condensed matter

Bond-operator analysis of the magnetization of spin chains

Hanting Wang, Hua Lin, Jue-Lian Shen

Published Feb 1, 2000• 25 citations
Publisher

Abstract

A bond operator representation of quantum $S=1$ spins is developed along the same line to the bond operator representation of quantum $S=\frac{1}{2}$ spins. The two representations are then applied to investigate the magnetization of the isotropic spin-$1$ antiferromagnetic chains and the spin-$\frac{1}{2}$ and spin-$1$ spin ladders. With the appropriate quantum states projected out, the spin-$1$ chains and the spin-$\frac{1}{2}$ and spin-$1$ spin ladders can be mapped to effective Hamiltonians of the $\mathcal{S}=\frac{1}{2}$ pseudospins in the effective external magnetic fields. The magnetization features of the spin-$1$ chains and the spin-$\frac{1}{2}$ and spin-$1$ spin ladders are then analyzed with the use of the existing rigorous results of the spin-$\frac{1}{2}$ $\mathrm{XXZ}$ spin chains. For spin-$1$ chains, the dependences of the magnetization on the applied magnetic fields are obtained. The $m=\frac{1}{4}$ plateau appearing in the magnetization of the spin-$\frac{1}{2}$ spin ladders and the $m=\frac{1}{4},$ $\frac{1}{2},$ and $\frac{3}{4}$ plateaus in the magnetization of the spin-$1$ spin ladders are studied. The effects of the single-ion anisotropy and the related experiments on spin-$1$ spin ladders are discussed also.

DOI
10.1103/physrevb.61.4019

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A bond operator representation of quantum S=1S=1S=1 spins is developed along the same line to the bond operator representation of quantum S=12S=\frac{1}{2}S=21​ spins. The two representations are then applied to investigate the magnetization of the isotropic spin-111 antiferromagnetic chains and the spin-12\frac{1}{2}21​ and spin-111 spin ladders. With the appropriate quantum states projected out, the spin-111 chains and the spin-12\frac{1}{2}21​ and spin-111 spin ladders can be mapped to effective Hamiltonians of the S=12\mathcal{S}=\frac{1}{2}S=21​ pseudospins in the effective external magnetic fields. The magnetization features of the spin-111 chains and the spin-12\frac{1}{2}21​ and spin-111 spin ladders are then analyzed with the use of the existing rigorous results of the spin-12\frac{1}{2}21​ XXZ\mathrm{XXZ}XXZ spin chains. For spin-111 chains, the dependences of the magnetization on the applied magnetic fields are obtained. The m=14m=\frac{1}{4}m=41​ plateau appearing in the magnetization of the spin-12\frac{1}{2}21​ spin ladders and the m=14,m=\frac{1}{4},m=41​, 12,\frac{1}{2},21​, and 34\frac{3}{4}43​ plateaus in the magnetization of the spin-111 spin ladders are studied. The effects of the single-ion anisotropy and the related experiments on spin-111 spin ladders are discussed also.
The bond operator representation of quantum S=12S=\frac{1}{2}S=21​ spins with the singlet state being projected out is applied to investigate the isotropic spin-1 Heisenberg antiferromagnetic chain $H={\ensuremath{\sum}}_{i}[{\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}\ensuremath{-}\ensuremath{\beta}({\mathbf{S}}_{i}\ensuremath{\cdot}{\mathbf{S}}_{i+1}{)}^{2}]$. In the approximation of Bose condensation, the ground-state properties of the system and their $\ensuremath{\beta}$ dependences are studied. The results at Heisenberg point $\ensuremath{\beta}=0$ and at valence-bond-solid (VBS) point $\ensuremath{\beta}=\ensuremath{-}\frac{1}{3}$ agree quite well with the analytical, numerical, and experimental studies. The excitation spectrum is nonsymmetric with respect to $k=\ensuremath{\pi}/2$ and shows a minimum at $k=\ensuremath{\pi}$. The $\ensuremath{\beta}$ dependences of the ground-state energy and the excitation gap show the similarities between the Heisenberg point and VBS point. The related properties of the spin ladders are also discussed.
The experimental realization of various spin ladder systems has prompted their detailed theoretical investigations. Here we study the evolution of ground-state magnetization with an external magnetic field for two different antiferromagnetic systems: a three-legged spin-1/21/21/2 ladder, and a two-legged spin-1/21/21/2 ladder with an additional diagonal interaction. The finite system density-matrix renormalization-group method is employed for numerical studies of the three-chain system, and an effective low-energy Hamiltonian is used in the limit of strong interchain coupling to study the two- and three-chain systems. The three-chain system has a magnetization plateau at one-third of the saturation magnetization. The two-chain system has a plateau at zero magnetization due to a gap above the singlet ground state. It also has a plateau at half of the saturation magnetization for a certain range of values of the couplings. We study the regions of transitions between plateaus numerically and analytically, and find that they are described, at first order in a strong-coupling expansion, by an XXZ\mathrm{XXZ}XXZ spin-1/21/21/2 chain in a magnetic field; the second-order terms give corrections to the XXZ\mathrm{XXZ}XXZ model. We also study numerically some low-temperature properties of the three-chain system, such as the magnetization, magnetic susceptibility and specific heat.
Ground-state properties of the spin-1 antiferromagnetic Heisenberg chain with bond alternation and uniaxial single-ion-type anisotropy, which is described by the Hamiltonian \({\cal H}_0 = \sum_{\ell=1}^N\{1-(-1)^\ell\alpha\}{\mib S}_\ell\cdot{\mib S}_{\ell+1} + D\sum_{\ell=1}^N\;(S_\ell^z)^2\) (0.0 ≤α≤1.0, -∞< D < ∞), are studied. An exact diagonalization method by means of the Lanczös technique is used. The ground-state phase diagram on the α versus D plane at the thermodynamic limit is determined by analyzing the Binder parameters associated with Néel order and z -direction string order for finite-size ( N = 6, 8, …, 16) chains. It is found that there exists a multicritical point at which the Haldane, Néel, and singlet-dimer phases meet and that the singlet-dimer phase and the large- D phase belong to the same unique phase. The ground-state magnetization curves at the thermodynamic limit for various values of α and D are obtained for the system described by the Hamiltonian, H 0 plus the Zeeman term - H ∑ ℓ=1 N S ℓ z . The results show that the 1/2-plateau appears in the magnetization curve at least when D ≥0.0 and 0.0≲α≤1.00.0\lesssim\alpha \leq 1.00.0≲α≤1.0.
We reexamine the numerical study of the magnetized state of the S=3/2S=3/2S=3/2 spin chain with single-ion anisotropy D(>0)D(>0)D(>0) for the magnetization M=MS/3,{M=M}_{S}/3,M=MS​/3, where MS{M}_{S}MS​ is the saturation magnetization. At this magnetization, we find that for D<Dc1=0.387D<{D}_{c1}=0.387D<Dc1​=0.387 the system is critical and the magnetization plateau does not appear. For D>Dc1,D>{D}_{c1},D>Dc1​, the parameter region is divided into two parts Dc1<D<Dc2=0.943{D}_{c1}<D<{D}_{c2}=0.943Dc1​<D<Dc2​=0.943 and Dc2<D.{D}_{c2}<D.Dc2​<D. In each region, the system is gapful and the M=MS/3{M=M}_{S}/3M=MS​/3 magnetization plateau appears in the magnetization process. From our numerical calculation, the intermediate region Dc1<D<Dc2{D}_{c1}<D<{D}_{c2}Dc1​<D<Dc2​ should be characterized by a magnetized valence-bond-solid state.
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In this thesis, I use a quantum Monte Carlo (QMC) method based on the stochastic series expansion (SSE) to study transport, quantum phase diagrams, and dynamics of several one- and quasi-one-dimensional spin systems, including spin-1/2 and spin-1 chains, a frustrated four-leg spin tube, and N-leg spin-1/2 ladders. I establish a diffusive channel for the finite temperature spin transport of the spin-1/2 antiferromagnetic Heisenberg chain. Extensive finite size studies are detailed, allowing to extract the diffusion kernel in the thermodynamic limit. The diffusion is found to exhibit no anomalous exponent, moreover it is shown to survive for finite values of exchange anisotropy and magnetic fields, however with a strong suppression away from the isotropic point and for increasing magnetic fields. In the case of the spin-1 chain, I obtain the quantum phase diagram and spin dynamics in the combined presence of finite single-ion anisotropies and external magnetic fields. I also prove the existence of magnon modes and multi-particle continua and determine their evolutions as a function of anisotropies and magnetic fields. Moreover, I calculate the magnetic field dependence of the nuclear magnetic relaxation as a function of the single ion anisotropy. For the quasi-one-dimensional spin system of a frustrated four-leg tube, I investigate the regime of large rung coupling, where QMC is applicable. I evaluate the uniform spin susceptibility and the dynamic structure factor and show that in this regime the spin-tube exhibts a crossover from a gapped effective antiferromagnetic spin-2 chain behavior with massive magnons to a gapless Luttinger liquid phase with spinon excitations. Finally, I investigate the spin dynamics of N-leg spin-1/2 ladders with spacially isotropic interactions for N=1,..,5. I obtain the low-lying excitations of the spin ladders as a function of the number of legs and show that they exhibt an even-odd oscillation which is consistent with Haldane's conjecture and other known results for the thermodynamics of spin ladders.