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arXiv:cond-mat/9801288 · Physical review. B, Condensed matter

Critical exponent in the magnetization curve of quantum spin chains

Tôru Sakai, Minoru Takahashi

Published Apr 1, 1998• 37 citations
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Abstract

The ground-state magnetization curve around the critical magnetic field ${H}_{c}$ of quantum spin chains with a spin gap is investigated. We propose a size scaling method to estimate the critical exponent \ensuremath{\delta} defined as $m\ensuremath{\sim}|H\ensuremath{-}{H}_{c}{|}^{1/\ensuremath{\delta}}$ from finite cluster calculation. The applications of the method to the $S=1$ antiferromagnetic chain and $S=1/2$ bond-alternating chain lead to a common conclusion that $\ensuremath{\delta}=2.$ The same result is derived for both edges of the magnetization plateau of the $S=3/2$ antiferromagnetic chain with single-ion anisotropy.

DOI
10.1103/physrevb.57.r8091

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The ground-state magnetization curve around the critical magnetic field Hc{H}_{c}Hc​ of quantum spin chains with a spin gap is investigated. We propose a size scaling method to estimate the critical exponent \ensuremath{\delta} defined as m\ensuremath∼∣H\ensuremath−Hc∣1/\ensuremathδm\ensuremath{\sim}|H\ensuremath{-}{H}_{c}{|}^{1/\ensuremath{\delta}}m\ensuremath∼∣H\ensuremath−Hc​∣1/\ensuremathδ from finite cluster calculation. The applications of the method to the S=1S=1S=1 antiferromagnetic chain and S=1/2S=1/2S=1/2 bond-alternating chain lead to a common conclusion that \ensuremathδ=2.\ensuremath{\delta}=2.\ensuremathδ=2. The same result is derived for both edges of the magnetization plateau of the S=3/2S=3/2S=3/2 antiferromagnetic chain with single-ion anisotropy.
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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 study the S =1/2 and S =1 alternating spin Heisenberg chain with the next-nearest neighbor (NNN) interaction J 2 between neighboring S =1/2 spins, by using the numerical diagonalization method of a finite size system. It is found that a fractional plateau appears at magnetization m =1/2 in the magnetization curve for J 2 > J 2 c =0.106±0.001. The phase transition at J 2 c is characterized by the exponent σ=1, which controls the critical behavior of the spin gap Δ as Δ∼ exp(- a /( J 2 - J 2 c ) 1+σ ). For J 2 < J 2 c , the model at m =1/2 is described by the conformal field theory with central charge c =1 and exponent η for the spin correlation function of the transverse component to a magnetic field is calculated.
The energy gap of the one-dimensional (1D) spin-[Formula: see text] AF (antiferromagnetic) XXZ model in a transverse magnetic field (h) is studied by using the exact diagonalization technique. We have computed the energy gap in the region of anisotropy parameter -1<Δ<1 and in the vicinity of the critical line Δ = -1. By introducing a practical finite-size scaling approach, we have computed the correct critical exponent of the energy gap in good agreement with the field theoretical and spin-wave approaches. To check this approach, we have also studied the critical exponent of the energy gap of the 1D AF-Heisenberg chain in a staggered magnetic field (h(s)).
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One dimensional S = 1 antiferromagnetic Heisenberg chains with bond alternation and uniaxial single-ion-type anisotropy are studied by numerical exact diagonalization of finite size systems. We calculate the ground state phase diagram using the twisted boundary condition method for D≥0. The ground states of this model contain the Haldane state, large-D state and the dimer state. Using conformal field theory and the level spectroscopy method, we calculate the D and δ-dependence of the Luttinger liquid paraments K and the critical exponent ν of the energy gap.