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arXiv:cond-mat/9907117 · Physical Review Letters

Nuclear Spin Relaxation Rates in Two-Leg Spin Ladders

Félix Naef, Xiaoqun Wang

Published Feb 7, 2000• 18 citations
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Abstract

Using the transfer-matrix density-matrix renormalization group method, we study the nuclear spin relaxation rate ${1/T}_{1}$ in the two-leg $s\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}$ ladder as a function of the interchain $({J}_{\ensuremath{\perp}})$ and intrachain $({J}_{\ensuremath{\parallel}})$ couplings. In particular, we separate the ${q}_{y}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ and $\ensuremath{\pi}$ contributions and show that the latter contribute significantly to the copper relaxation rate ${}^{63}({1/T}_{1})$ in the experimentally relevant coupling and temperature range. We compare our results to both theoretical predictions and experimental measures on ladder materials.

DOI
10.1103/physrevlett.84.1320

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Using the transfer-matrix density-matrix renormalization group method, we study the nuclear spin relaxation rate 1/T1{1/T}_{1}1/T1​ in the two-leg s=12s\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\frac{1}{2}s=21​ ladder as a function of the interchain (J\ensuremath⊥)({J}_{\ensuremath{\perp}})(J\ensuremath⊥​) and intrachain (J\ensuremath∥)({J}_{\ensuremath{\parallel}})(J\ensuremath∥​) couplings. In particular, we separate the qy=0{q}_{y}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0qy​=0 and \ensuremathπ\ensuremath{\pi}\ensuremathπ contributions and show that the latter contribute significantly to the copper relaxation rate 63(1/T1){}^{63}({1/T}_{1})63(1/T1​) in the experimentally relevant coupling and temperature range. We compare our results to both theoretical predictions and experimental measures on ladder materials.
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.
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​.
We study the ground-state spin correlations in the gapless incommensurate regime of a S=1/2S=1/2S=1/2 XXZ\mathrm{XXZ}XXZ chain and a two-leg antiferromagnetic ladder under a magnetic field, in which the gapless excitations form a Tomonaga-Luttinger (TL) liquid. We calculate numerically the two-spin correlation functions and the local magnetization in the two models using the density-matrix renormalization-group method. By fitting the numerical results for an open XXZ\mathrm{XXZ}XXZ chain of 100 spins to correlation functions of a Gaussian model, we determine the TL-liquid parameter K and the amplitudes of the correlation functions. The value of K estimated from the fits is in excellent agreement with the exact value obtained from the Bethe ansatz. We apply the same method to the open ladder consisting of 200 spins and determine the dependence of K on the magnetization M. The $K\ensuremath{-}M$ relation changes drastically depending on the ratio of the coupling constants in the leg and rung directions. We also discuss implications of these results to experiments on the nuclear spin relaxation rate 1/T1{1/T}_{1}1/T1​ and dynamical spin structure factors.
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.
The effect of antiferromagnetic interchain coupling in alternating spin-(1,1/2) chains is studied by means of spin-wave theory and density-matrix renormalization group (DMRG). In particular, two limiting cases are investigated, the two-leg ladder and its two-dimensional (2D) generalization. Results of the ground-state properties like energy, spin gap, magnetizations, and correlation functions are reported for the whole range of the interchain coupling ${J}_{\ensuremath{\perp}}.$ For the 2D case the spin-wave results predict a smooth dimensional crossover from 1D to 2D keeping the ground state always ordered. For the ladder system, the DMRG results show that any ${J}_{\ensuremath{\perp}}>0$ drives the system to a gapped ground state. Furthermore the behavior of the correlation functions closely resemble the uniform spin-1/2 ladder. For ${J}_{\ensuremath{\perp}}$ lower than 0.3, however, the gap behaves quadratically as $\ensuremath{\Delta}\ensuremath{\sim}{0.6J}_{\ensuremath{\perp}}^{2}.$ Finally, it is argued that the behavior of the spin gap for an arbitrary number of mixed coupled spin chains is analogous to that of the uniform spin-1/2 chains.
Recent developments in density matrix renormalization group/matrix product state techniques allow today for a quantitative computation of dynamical correlations at finite temperature in low-dimensional quantum spin systems. By combining numerical results and field theoretical asymptotic methods, e.g., bosonization, it is now possible to give a full description of the dynamical quantity one is interested in for the range of temperatures experimentally accessible. This enhances of course the possibility of one-to-one comparisons with experiments without having to resort to uncontrolled approximations. Here, the authors focus on the temperature dependence of the NMR spin-lattice relaxation rate 1/T11/{T}_{1}1/T1​ in spin-12\frac{1}{2}21​ chain systems. This quantity is directly related to local, dynamical spin-spin correlation functions. The results agree in the low-temperature regime with bosonization predictions, and show interesting deviations from this behavior at intermediate temperatures.