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Physical review. B, Condensed matter

Experimental and theoretical studies of finite-size effects in an S= 12 antiferromagnetic Heisenberg chain

Hitoshi Asakawa, Masaaki Matsuda, Kazuhiko Minami, Hiroki Yamazaki, K. Katsumata

Published Apr 1, 1998• 18 citations
Publisher

Abstract

Finite-size effects in a spin-$1/2$ one-dimensional (1D) Heisenberg antiferromagnet (HAF) are studied experimentally and theoretically. The calculated susceptibility is compared with that observed in the quasi-1D HAF ${\mathrm{SrCu}}_{1\ensuremath{-}x}{\mathrm{Mg}}_{x}{\mathrm{O}}_{2}$. Excellent agreement between theory and experiment is obtained without any adjustable parameters above about $0.02J$, where $J$ is the dominant exchange interaction constant.

DOI
10.1103/physrevb.57.8285

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Physical review. B, Condensed matter•Aug 1, 1993•30 citations
Finite-size effects in a spin-1/21/21/2 one-dimensional (1D) Heisenberg antiferromagnet (HAF) are studied experimentally and theoretically. The calculated susceptibility is compared with that observed in the quasi-1D HAF SrCu1\ensuremath−xMgxO2{\mathrm{SrCu}}_{1\ensuremath{-}x}{\mathrm{Mg}}_{x}{\mathrm{O}}_{2}SrCu1\ensuremath−x​Mgx​O2​. Excellent agreement between theory and experiment is obtained without any adjustable parameters above about 0.02J0.02J0.02J, where JJJ is the dominant exchange interaction constant.
Haldane conjectures the fundamental difference in the energy spectrum of the Heisenberg antiferromagnetic (HAF) of the spin S chain is that the half-integer and the integer S chain have gapless and gapped energy spectrums, respectively. The ground state (gs) of the HAF spin-1/2 and spin-1 chains have a quasi-long-range and short-range correlation, respectively. We study the effect of the exchange interaction between an HAF spin-1/2 and an HAF spin-1 chain forming a normal ladder system and its gs properties. The inter-chain exchange interaction J⊥ can be either ferromagnetic (FM) or antiferromagnetic (AFM). Using the density matrix renormalization group method, we show that in the weak AFM/FM coupling limit of J⊥, the system behaves like two decoupled chains. However, in the large AFM J⊥ limit, the whole system can be visualized as weakly coupled spin-1/2 and spin-1 pairs which behave like an effective spin-1/2 HAF chain. In the large FM J⊥ limit, coupled spin-1/2 and spin-1 pairs can form pseudo spin-3/2 and the whole system behaves like an effective spin-3/2 HAF chain. We also derive the effective model Hamiltonian in both strong FM and AFM rung exchange coupling limits.
Magnetic susceptibility is measured on the one-dimensional spin 1/2 antiferromagnets Sr2CuO3{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}Sr2​CuO3​ and SrCuO2\mathrm{Sr}{\mathrm{CuO}}_{2}SrCuO2​ using single crystals. Large, nearly contamination-free single crystals enable us to measure the intrinsic spin susceptibility over a wide temperature range between 5 and 800 K. The results are in excellent agreement with the recent calculation by Eggert, Affleck, and Takahashi, and the exchange interaction energy JJJ is estimated to be $2200\ifmmode\pm\else\textpm\fi{}200$ and $2100\ifmmode\pm\else\textpm\fi{}200\mathrm{K}$, respectively. In Sr2CuO3{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}Sr2​CuO3​, an isotropic susceptibility drop is observed below about 20 K, which is also consistent with the result of this rigorous calculation.
The specific heat and magnetic susceptibility of the antiferromagnetic Heisenberg chain have been calculated using a numerical finite-size technique. An agreement with experiment for the quasi-one-dimensional compound CsNiCl3{\mathrm{CsNiCl}}_{3}CsNiCl3​, which displays a Haldane gap, is revealed.
Data from susceptibility measurements [Ami et al., Phys. Rev. B 51, 5994 (1995)] on Sr2{\mathrm{Sr}}_{2}Sr2​CuO3{\mathrm{CuO}}_{3}CuO3​ are compared with recent theoretical predictions [Eggert et al., Phys. Rev. Lett. 73, 332 (1994)] for the magnetic susceptibility of the antiferromagnetic spin-1/2 Heisenberg chain. The experimental data fully confirms the theoretical predictions and in turn we establish that Sr2{\mathrm{Sr}}_{2}Sr2​CuO3{\mathrm{CuO}}_{3}CuO3​ behaves almost perfectly like a one-dimensional antiferromagnet with an exchange coupling of J=${1700}_{\mathrm{\ensuremath{-}}100}^{+150}$ K. \textcopyright{} 1996 The American Physical Society.
Effects of an uniaxial single-ion anisotropy tsumi{\mathit{tsum}}_{\mathit{i}}tsumi​Dz{\mathit{D}}_{\mathit{z}}Dz​(Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​)2{)}^{2})2 on the magnetic properties of an S=1 antiferromagnetic Heisenberg chain with open boundaries are studied by a quantum Monte Carlo method. In the ground stae, how the anisotropy affects the staggered magnetization 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 and the spin correlation functions 〈S1z{\mathit{S}}_{1}^{\mathit{z}}S1z​Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 and 〈S1x{\mathit{S}}_{1}^{\mathit{x}}S1x​Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​〉 is investigated. Amplitudes of 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 decrease with with increase of Dz{\mathit{D}}_{\mathit{z}}Dz​. The correlation length ${\ensuremath{\xi}}_{\mathit{S}\mathit{S}}^{\mathit{z}}$ of 〈S1z{\mathit{S}}_{1}^{\mathit{z}}S1z​Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 monotonously decreases with increase of Dz{\mathit{D}}_{\mathit{z}}Dz​, while the correlation length ${\ensuremath{\xi}}_{\mathit{S}\mathit{S}}^{\mathit{x}}$ of 〈S1x{\mathit{S}}_{1}^{\mathit{x}}S1x​Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​〉 has the opposite but weak Dz{\mathit{D}}_{\mathit{z}}Dz​ dependence. The fourfold degeneracy of the ground state also holds with nonzero Dz{\mathit{D}}_{\mathit{z}}Dz​'s in the thermodynamic limit although it is lifted up in finite chains. At finite temperatures, the magnetic susceptibilities along the z axis (${\mathrm{\ensuremath{\chi}}}^{\mathit{z}}$) and the x axis (${\mathrm{\ensuremath{\chi}}}^{\mathit{x}}$) are calculated. Temperature dependences of the bulk susceptibilites, ${\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\infty}}}^{\mathit{z}}$ and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\infty}}}^{\mathit{x}}$, are consistent with the previous experimental result on Ni(C2{\mathrm{C}}_{2}C2​H8{\mathrm{H}}_{8}H8​N2{\mathrm{N}}_{2}N2​)2{)}_{2})2​NO2{\mathrm{NO}}_{2}NO2​ClO4{\mathrm{ClO}}_{4}ClO4​ (NENP). Finite-size corrections, ${\mathrm{\ensuremath{\chi}}}_{0}^{\mathit{z}}$ and ${\mathrm{\ensuremath{\chi}}}_{0}^{\mathit{x}}$, and the Curie susceptibilities resulting from chain ends, ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}^{\mathit{z}}$ and ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}^{\mathit{x}}$, are also investigated in detail. Both ${\mathrm{\ensuremath{\chi}}}_{0}$ and ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}$ diverge as temperature goes to zero and they are combined through a relation ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}$=${\mathrm{\ensuremath{\chi}}}_{0}$+${\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\infty}}}$. Temperature dependences of ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}$ are dominated by the edge effect at low temperatures, while at high temperatures they are attributed to an S=1 free spin.
The dynamical properties of spin-1 and spin-1/2 antiferromagnetic Heisenberg chains (AHC's) are studied by diagonalizing exactly clusters of up to 18 and 26 sites, respectively. It is shown that the spin-1 AHC has a quasi-single-mode spectrum for momentum k\ensuremath{\ge}0.3\ensuremath{\pi}, while the low-energy edge of the spin-1/2 AHC is dominated by a spin-wave continuum. The dispersion curve obtained for the spin-1 chain is in excellent agreement with recent experiments on Ni(C2{\mathrm{C}}_{2}C2​H8{\mathrm{H}}_{8}H8​N2{\mathrm{N}}_{2}N2​)2{)}_{2})2​NO2{\mathrm{NO}}_{2}NO2​ClO4{\mathrm{ClO}}_{4}ClO4​. The size dependence of the low-energy spectral weights is also analyzed.