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Journal of the Physical Society of Japan

Spin Gap ofS=1/2Heisenberg Ladder: Field Theoretical Study of Magnetic Susceptibility and NMR Relaxation Rate

Jun‐ichiro Kishine, Hidetoshi Fukuyama

Published Jan 15, 1997• 27 citations
Publisher

Abstract

The temperature dependence of the static spin susceptibility, χ, and NMR relaxation rate, T 1 -1 , of S =1/2 Heisenberg two-leg ladder systems have been studied based on the Majorana fermion representation of the S =1/2 Heisenberg spin Hamiltonian introduced by Shelton, Nersesyan and Tsvelik (Phys. Rev. B 53 (1996) 8521). It is shown that in the case of SrCu 2 O 3 , we can fit the experimental data of χ and T 1 -1 in terms of the massive Majorana fermion excitation with mass m =440 K to resolve the apparent discrepancy of the spin gap activation energies between χ and T 1 -1 .

DOI
10.1143/jpsj.66.26

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arXiv:cond-mat/0401477•Journal of the Physical Society of Japan•Mar 15, 2004•9 citations
The temperature dependence of the static spin susceptibility, χ, and NMR relaxation rate, T 1 -1 , of S =1/2 Heisenberg two-leg ladder systems have been studied based on the Majorana fermion representation of the S =1/2 Heisenberg spin Hamiltonian introduced by Shelton, Nersesyan and Tsvelik (Phys. Rev. B 53 (1996) 8521). It is shown that in the case of SrCu 2 O 3 , we can fit the experimental data of χ and T 1 -1 in terms of the massive Majorana fermion excitation with mass m =440 K to resolve the apparent discrepancy of the spin gap activation energies between χ and T 1 -1 .
Magnetic properties of SrCu2{\mathrm{Cu}}_{2}Cu2​O3{\mathrm{O}}_{3}O3​ and Sr2{\mathrm{Sr}}_{2}Sr2​Cu3{\mathrm{Cu}}_{3}Cu3​O5{\mathrm{O}}_{5}O5​, containing two-leg and three-leg S=12S=\frac{1}{2}S=21​ ladders made of antiferromagnetic Cu-O-Cu linear bonds, were investigated. The susceptibility of the two-leg ladder material, SrCu2{\mathrm{Cu}}_{2}Cu2​O3{\mathrm{O}}_{3}O3​, is characteristic of thermal excitation from a nonmagnetic ground state with a spin gap of about 420 K, while the susceptibility of Sr2{\mathrm{Sr}}_{2}Sr2​Cu3{\mathrm{Cu}}_{3}Cu3​O5{\mathrm{O}}_{5}O5​, containing three-leg ladders, reflects a gapless spin excitation spectrum. The temperature dependence of the nuclear-spin lattice relaxation rate, 1T1\frac{1}{{T}_{1}}T1​1​, of 63Cu^{63}\mathrm{Cu}63Cu NMR also indicates the existence of a spin gap only for SrCu2{\mathrm{Cu}}_{2}Cu2​O3{\mathrm{O}}_{3}O3​. The spin gap we observed for SrCu2{\mathrm{Cu}}_{2}Cu2​O3{\mathrm{O}}_{3}O3​ confirms a recent theoretical prediction for the magnetic behavior of spin = \textonehalf{} antiferromagnetically coupled Heisenberg chains.
A comprehensive theoretical and experimental study is presented of the magnetic susceptibility versus temperature χ(T) of spin S = 1/2 two- and three-leg Heisenberg ladders and ladder oxide compounds. Extensive quantum Monte Carlo simulations of χ(T) were carried out for both isolated and coupled two-leg ladders with spatially anisotropic intraladder exchange. Accurate fits to these and related literature QMC data were obtained. We have also calculated the one- and two-magnon dispersion relations and the dynamical spin structure factor for anisotropic isolated 2 x 12 ladders. The exchange constants in the two-leg ladder compound SrCu2O3 are estimated from LDA+U calculations. We report the detailed crystal structure of SrCu2O3 and of the three-leg ladder compound Sr2Cu3O5. New experimental χ(T) data are reported for the two-leg ladder cuprates SrCu2O3 and LaCuO_{2.5}, and for the (nominally) two-leg ladder vanadates CaV2O5 and MgV2O5. The new and literature χ(T) data for these compounds and for Sr2Cu3O5 are modeled using our QMC χ(T) simulation fits, and the exchange coupling constants between the spins-1/2 are thereby estimated for each material. The surpisingly strong spatial anisotropy of the bilinear intraladder exchange constants in the cuprate compounds is discussed together with the results of other experiments sensitive to this anisotropy. Recent theoretical predictions are discussed including those which indicate that a four-spin cyclic exchange interaction within a Cu4 plaquette is important to determining the magnetic properties and which can significantly influence the exchange interactions estimated from χ(T) data assuming the presence of only bilinear exchange.
We study the nuclear spin-lattice relaxation rate 1/T1{1/T}_{1}1/T1​ in the two-leg antiferromagnetic spin-1/2 Heisenberg ladder. More specifically, we consider the contribution to 1/T1{1/T}_{1}1/T1​ from the processes with momentum transfer $(\ensuremath{\pi},\ensuremath{\pi}).$ In the limit of weak coupling between the two chains, this contribution is of activation type with gap $2\ensuremath{\Delta}$ at low temperatures $(\ensuremath{\Delta}$ is the spin gap), but crosses over to a slowly decaying temperature dependence at the crossover temperature $T\ensuremath{\approx}\ensuremath{\Delta}.$ This crossover possibly explains the recent high-temperature NMR results on ladder-containing cuprates by Imai et al.
Extraction of antiferromagnetic (AF) exchange constants J and J\ensuremath{'} in two-leg spin-1/2 nearest-neighbor (NN) Heisenberg ladders from experimental magnetic spin susceptibility \ensuremath{\chi}(T) data is studied, where J\ensuremath{'} is the NN exchange constant in the rungs and J is that in the legs. Two low-T (T\ensuremath{\ll}J/kB{\mathit{k}}_{\mathit{B}}kB​) approximations of Troyer, Tsunetsugu, and W\"urtz [Phys. Rev. B 50, 13 515 (1994)] for \ensuremath{\chi}(T) of the isolated spin-1/2 two-leg ladder with J\ensuremath{'}/J=1 are shown to be accurate to \ensuremath{\sim}10% when extrapolated to T\ensuremath{\sim}J\ensuremath{'}/kB{\mathit{k}}_{\mathit{B}}kB​. The variations in the magnetic excitation dispersion relation parameters [which enter the low-T expression of Troyer et al. for \ensuremath{\chi}(T)] for J\ensuremath{'}/J1 are inferred from previous theoretical results. Next, the values of the maxima in \ensuremath{\chi}(T)\ensuremath{\equiv}${\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}$ from many numerical calculations in the literature for various nonfrustrated spin-1/2 NN Heisenberg antiferromagnets are considered. The values of the product ${\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}$zeff{\mathit{z}}_{\mathrm{eff}}zeff​ are found to depend systematically and almost uniqely on an effective magnetic coordination number zeff{\mathit{z}}_{\mathrm{eff}}zeff​, defined in the text, but are nearly independent of the dimensionality of the spin lattice and of the magnitude of any gap in the magnetic excitation spectrum. This observation allows accurate bounds on the exchange coupling constants in arbitrary quasi-low-dimensional spin-1/2 antiferromagnets to be inferred from experimental ${\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}$ values. Finally, a mean-field-type expression for \ensuremath{\chi}(T) for arbitrary quasi-low-dimensional spin-1/2 Heisenberg antiferromagnets is derived, using the above ${\mathrm{\ensuremath{\chi}}}^{\mathrm{max}}$ phenomenology as input, which allows the influence of intersystem coupling on \ensuremath{\chi}(T) to be evaluated. These results, and the \ensuremath{\chi}(T) calculations of Barnes and Riera [Phys. Rev. B 50, 6817 (1994)] for isolated two-leg spin-1/2 Heisenberg ladders, are used to analyze the experimental \ensuremath{\chi}(T) data of Azuma et al. [Phys. Rev. Lett. 73, 3463 (1994)] for the spin-1/2 two-leg ladder compound SrCu2{\mathrm{SrCu}}_{2}SrCu2​O3{\mathrm{O}}_{3}O3​. The analyses together suggest that J\ensuremath{'}/J\ensuremath{\sim}0.5, contrary to the expectation that J\ensuremath{'}/J\ensuremath{\approxeq}1, and that J is very large (\ensuremath{\sim}2000 K), similar to the value of J in the linear-chain cuprate Sr2{\mathrm{Sr}}_{2}Sr2​CuO3{\mathrm{CuO}}_{3}CuO3​. \textcopyright{} 1996 The American Physical Society.
The critical properties of the S=1/2S=1/2S=1/2 Heisenberg two-leg ladders are investigated in a magnetic field. Combining the exact diagonalization method and the finite-size-scaling analysis based on conformal field theory, we calculate the critical exponents of spin correlation functions numerically. For a strong interchain coupling, magnetization dependence of the critical exponents shows characteristic behavior depending on the sign of the interchain coupling. We also calculate the critical exponents for the S=1/2S=1/2S=1/2 Heisenberg two-leg ladder with a diagonal interaction, which is thought of as a model Hamiltonian of the organic spin ladder compound ${\mathrm{Cu}}_{2}(1,4\ensuremath{-}\mathrm{d}\mathrm{i}\mathrm{a}\mathrm{z}\mathrm{a}\mathrm{c}\mathrm{y}\mathrm{c}\mathrm{l}\mathrm{o}\mathrm{h}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{e}{)}_{2}{\mathrm{Cl}}_{4}.$ Numerical results are compared with experimental results of temperature dependence of the NMR relaxation rate 1/T1.{1/T}_{1}.1/T1​.
In terms of spinless fermions via the Jordan-Wigner transformation along a snake-like path and spin waves modified so as to restore the sublattice symmetry, we investigate static and dynamic properties of two-leg antiferromagnetic Heisenberg ladders. The specific heat is finely reproduced by the spinless fermions, whereas the magnetic susceptibility is well described by the modified spin waves. The nuclear spin-lattice relaxation rate is discussed in detail with particular emphasis on its novel field dependence.