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Physical Review Letters

Magnetic Susceptibility of Ideal Spin 1 / 2 Heisenberg Antiferromagnetic Chain Systems, Sr2CuO3 and SrCuO2

N. Motoyama, Hiroshi Eisaki, S. Uchida

Published Apr 22, 1996• 444 citations
Publisher

Abstract

Magnetic susceptibility is measured on the one-dimensional spin 1/2 antiferromagnets ${\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}$ and $\mathrm{Sr}{\mathrm{CuO}}_{2}$ 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 $J$ is estimated to be $2200\ifmmode\pm\else\textpm\fi{}200$ and $2100\ifmmode\pm\else\textpm\fi{}200\mathrm{K}$, respectively. In ${\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}$, an isotropic susceptibility drop is observed below about 20 K, which is also consistent with the result of this rigorous calculation.

DOI
10.1103/physrevlett.76.3212

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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±\else\textpm\fi2002200\ifmmode\pm\else\textpm\fi{}2002200\ifmmode±\else\textpm\fi200 and 2100\ifmmode±\else\textpm\fi200K2100\ifmmode\pm\else\textpm\fi{}200\mathrm{K}2100\ifmmode±\else\textpm\fi200K, 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.
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.
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 ${\mathrm{SrCu}}_{1\ensuremath{-}x}{\mathrm{Mg}}_{x}{\mathrm{O}}_{2}$. Excellent agreement between theory and experiment is obtained without any adjustable parameters above about 0.02J0.02J0.02J, where JJJ is the dominant exchange interaction constant.
The NMR relaxation data on Sr2CuO3{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}Sr2​CuO3​ [Phys. Rev. Lett. 76, 4612 (1996)] are reexamined and compared with the analytic theory of the dynamic susceptibility in the S=1/2S=1/2S=1/2 antiferromagnetic Heisenberg chain including multiplicative logarithmic corrections [Phys. Rev. Lett. 78, 539 (1997)]. Comparisons of the spin-lattice and the Gaussian spin-echo decay rates (1/T1{1/T}_{1}1/T1​ and 1/T2G{1/T}_{2G}1/T2G​) and their ratio all show good quantitative agreement. Our results demonstrate the importance of the logarithmic corrections in the analysis of experimental data for quasi-one-dimensional systems and indicate that the dynamics of Sr2CuO3{\mathrm{Sr}}_{2}{\mathrm{CuO}}_{3}Sr2​CuO3​ is well described by an S=1/2S=1/2S=1/2 one-dimensional Heisenberg model with a nearest-neighbor exchange.
The magnetic susceptibility ${\ensuremath{\chi}}^{*}(t)$ and specific heat C(t)C(t)C(t) versus temperature t of the spin S=1/2S=1/2S=1/2 antiferromagnetic (AF) alternating-exchange (J1{(J}_{1}(J1​ and J2){J}_{2})J2​) Heisenberg chain are studied for the entire range $0<~\ensuremath{\alpha}<~1$ of the alternation parameter $\ensuremath{\alpha}\ensuremath{\equiv}{J}_{2}{/J}_{1}{(J}_{1},$ J2> 0,{J}_{2}>~0,J2​> 0, J2< J1,{J}_{2}<~{J}_{1},J2​< J1​, t=kBT/J1,{t=k}_{\mathrm{B}}{T/J}_{1},t=kB​T/J1​, ${\ensuremath{\chi}}^{*}=\ensuremath{\chi}{J}_{1}{/Ng}^{2}{\ensuremath{\mu}}_{B}^{2}).$ For the uniform chain $(\ensuremath{\alpha}=1),$ the high-accuracy ${\ensuremath{\chi}}^{*}(t)$ and C(t)C(t)C(t) Bethe ansatz data of Kl\"umper and Johnston (unpublished) are shown to agree very well at low t with the respective exact theoretical low-ttt logarithmic correction predictions of Lukyanov [Nucl. Phys. B 522,522,522, 533 (1998)]. Accurate $(\ensuremath{\sim}{10}^{\ensuremath{-}7})$ independent empirical fits to the respective data are obtained over t ranges spanning 25 orders of magnitude, $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}25}<~t<~5,$ which contain extrapolations to the respective exact t=0t=0t=0 limits. The infinite temperature entropy calculated using our C(t)C(t)C(t) fit function is within 8 parts in 108{10}^{8}108 of the exact value ln2.\mathrm{ln}2.ln2. Quantum Monte Carlo (QMC) simulations and transfer-matrix density-matrix renormalization group (TMRG) calculations of ${\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)$ are presented for 0.002< t< 100.002<~t<~100.002< t< 10 and $0.05<~\ensuremath{\alpha}<~1,$ and an accurate $(2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}4})$ two-dimensional $(\ensuremath{\alpha},t)$ fit to the combined data is obtained for 0.01< t< 100.01<~t<~100.01< t< 10 and $0<~\ensuremath{\alpha}<~1.$ From the low-ttt TMRG data, the spin gap $\ensuremath{\Delta}(\ensuremath{\alpha})$ is extracted for $0.8<~\ensuremath{\alpha}<~0.995$ and compared with previous results, and a fit function is formulated for $0<~\ensuremath{\alpha}<~1$ by combining these data with literature data. We infer from our data that the asymptotic critical regime near the uniform chain limit is only entered for $\ensuremath{\alpha}\ensuremath{\gtrsim}0.99.$ We examine in detail the theoretical predictions of Bulaevskii [Sov. Phys. Solid State 11,11,11, 921 (1969)], for ${\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)$ and compare them with our results. To illustrate the application and utility of our theoretical results, we model our experimental $\ensuremath{\chi}(T)$ and specific heat Cp(T){C}_{\mathrm{p}}(T)Cp​(T) data for NaV2O5{\mathrm{NaV}}_{2}{\mathrm{O}}_{5}NaV2​O5​ single crystals in detail. The $\ensuremath{\chi}(T)$ data above the spin dimerization temperature ${T}_{\mathrm{c}}\ensuremath{\approx}34\mathrm{K}$ are not in quantitative agreement with the prediction for the S=1/2S=1/2S=1/2 uniform Heisenberg chain, but can be explained if there is a moderate ferromagnetic interchain coupling and/or if J changes with T. Fitting the $\ensuremath{\chi}(T)$ data using our ${\ensuremath{\chi}}^{*}(\ensuremath{\alpha},t)$ fit function, we obtain the sample-dependent spin gap and range $\ensuremath{\Delta}{(T=0)/k}_{\mathrm{B}}=103(2)\mathrm{K},$ alternation parameter $\ensuremath{\delta}(0)\ensuremath{\equiv}(1\ensuremath{-}\ensuremath{\alpha})/(1+\ensuremath{\alpha})=0.034(6)$ and average exchange constant J(0)/kB=640(80)K.{J(0)/k}_{\mathrm{B}}=640(80)\mathrm{K}.J(0)/kB​=640(80)K. The $\ensuremath{\delta}(T)$ and $\ensuremath{\Delta}(T)$ are derived from the data. A spin pseudogap with magnitude $\ensuremath{\approx}0.4\ensuremath{\Delta}(0)$ is consistently found just above Tc,{T}_{\mathrm{c}},Tc​, which decreases with increasing temperature. From our Cp(T){C}_{\mathrm{p}}(T)Cp​(T) measurements on two crystals, we infer that the magnetic specific heat at low temperatures $T\ensuremath{\lesssim}15\mathrm{K}$ is too small to be resolved experimentally, and that the spin entropy at Tc{T}_{\mathrm{c}}Tc​ is too small to account for the entropy of the transition. A quantitative analysis indicates that at Tc,{T}_{\mathrm{c}},Tc​, at least 77% of the entropy change due to the transition at Tc{T}_{\mathrm{c}}Tc​ and associated order parameter fluctuations arise from the lattice and/or charge degrees of freedom and less than 23% from the spin degrees of freedom.
The phenomenological expression $\ensuremath{\chi}T∕(N{g}^{2}{\ensuremath{\mu}}_{B}^{2}∕k)={C}_{1n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{1n}∕T)+{C}_{2n}\phantom{\rule{0.2em}{0ex}}\mathrm{exp}(\ensuremath{-}{W}_{2n}∕T)$ describes very accurately the temperature dependence of the magnetic susceptibility computed for antiferromagnetic rings of Heisenberg spins S=1S=1S=1, whose size nnn is even and ranges from 6 to 20. This expression has been obtained through a strategy justified by scaling considerations together with finite size numerical calculations. For nnn large, the coefficients of the expression converge towards C1=0.125{C}_{1}=0.125C1​=0.125, W1=0.451J{W}_{1}=0.451JW1​=0.451J, C2=0.564{C}_{2}=0.564C2​=0.564, W2=1.793J{W}_{2}=1.793JW2​=1.793J (JJJ is the exchange constant), which are appropriate for describing the susceptibility of the spin-1 Haldane chain. The Curie constant, the paramagnetic Curie--Weiss temperature, the correlation length at T=0T=0T=0 and the Haldane gap are found to be closely related to these coefficients. With this expression, a very good description of the magnetic behavior of Y2BaNiO5{\mathrm{Y}}_{2}\mathrm{Ba}\mathrm{Ni}{\mathrm{O}}_{5}Y2​BaNiO5​ and of Ni(C2H8N2)2NO2ClO4\mathrm{Ni}{({\mathrm{C}}_{2}{\mathrm{H}}_{8}{\mathrm{N}}_{2})}_{2}\mathrm{N}{\mathrm{O}}_{2}\mathrm{Cl}{\mathrm{O}}_{4}Ni(C2​H8​N2​)2​NO2​ClO4​ (NENP), the archetype of the Haldane gap systems, is achieved over the whole temperature range.
We use the diagram technique for spin operators to calculate Green's functions and observables of the spin-12\frac{1}{2}21​ quantum Heisenberg antiferromagnet on a square lattice. The first corrections to the self-energy and interaction are taken into account in the chain diagrams. The approximation reproduces main results of Takahashi's modified spin-wave theory [Phys. Rev. B 40, 2494 (1989)] and is applicable in a wider temperature range. The energy per spin calculated in this approximation is in good agreement with the Monte Carlo and small-cluster exact-diagonalization calculations in the range $0<~T\ensuremath{\lesssim}1.2J$ where J is the exchange constant. For the static uniform susceptibility the agreement is good for $T\ensuremath{\lesssim}0.6J$ and becomes somewhat worse for higher temperatures. Nevertheless the approximation is able to reproduce the maximum in the temperature dependence of the susceptibility near T=0.9J.T=0.9J.T=0.9J.