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Physical review. B, Condensed matter

Relaxation in the neighborhood of a magnetic impurity in thes=1/2 Heisenberg chain at high temperatures

Surajit Sen, Amy Kathleen Bylsma Engebretson, Vanessa L. Gates, Lowell I. McCann

Published Aug 1, 1994• 4 citations
Publisher

Abstract

A Mori-Lee theoretic study of spin relaxation in the neighborhood of a magnetic impurity in the s=1/2 nearest-neighbor (NN) Heisenberg chain at high temperatures is presented. Our calculations suggest that in the limit of zero impurity coupling, the spectral function for the ``surface spin'' is consistent with a set of discrete frequencies given approximately by ${\mathit{w}}_{\mathit{n}}$\ensuremath{\approxeq}[n(${\mathit{w}}_{0}$+1/4)-1/4] against a shallow continuum background, where ${\mathit{w}}_{0}$\ensuremath{\approxeq}3J/4 with 2J being the coupling between NN spins. The frequencies broaden as the impurity coupling is increased to the NN coupling.

DOI
10.1103/physrevb.50.4244

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Physical review. B, Condensed matter•Feb 1, 1996•5 citations
A Mori-Lee theoretic study of spin relaxation in the neighborhood of a magnetic impurity in the s=1/2 nearest-neighbor (NN) Heisenberg chain at high temperatures is presented. Our calculations suggest that in the limit of zero impurity coupling, the spectral function for the ``surface spin'' is consistent with a set of discrete frequencies given approximately by wn{\mathit{w}}_{\mathit{n}}wn​\ensuremath{\approxeq}[n(w0{\mathit{w}}_{0}w0​+1/4)-1/4] against a shallow continuum background, where w0{\mathit{w}}_{0}w0​\ensuremath{\approxeq}3J/4 with 2J being the coupling between NN spins. The frequencies broaden as the impurity coupling is increased to the NN coupling.
By making use of the Lanczos and the density-matrix renormalization-group methods, we study the zero-temperature properties of a spin-1/2 antiferromagnetic Heisenberg chain system with a spin-1/2 magnetic impurity symmetrically coupled to its two nearest-neighbor spins. A comparison is made with the results for a magnetic impurity coupled only to the nearest-neighbor spin in the chain. The binding energy and the spin correlations are calculated. The matrix elements of the impurity spin operators are also calculated between several low-energy states. By carefully analyzing finite-size effects, we show that the dynamical structure factor of the impurity contains a \ensuremath{\delta}-function contribution at zero frequency which does not vanish in the thermodynamic limit for the case of symmetrical coupling. This indicates a partial decoupling of the impurity spin from the spins in the chain, and therefore the impurity susceptibility is predicted to show a Curie behavior. This contradicts the two-channel Kondo behavior suggested from analytic considerations.
We calculate the impurity susceptibility for a spin-1/2 magnetic impurity symmetrically coupled to its two nearest-neighbor spins in an antiferromagnetic Heisenberg spin-1/2 chain. This is done by extending White's density-matrix renormalization group (DMRG) technique to finite temperatures. It is found that the impurity susceptibility satisfies Curie law at low temperatures, and that the Curie constant decreases rapidly with increasing coupling constant between the impurity spin and the spins in the chain. For strong couplings, a logarithmic dependence of the susceptibility on temperature emerges. The Curie law behavior agrees with the DMRG analysis at zero temperature but deviates from the prediction of conformal field theory.
Combining high-temperature expansions with the recursion method and quantum Monte Carlo simulations with the maximum entropy method, we study the dynamics of the spin-1/2 Heisenberg chain at temperatures above and below the coupling J. By comparing the two sets of calculations, their relative strengths are assessed. At high temperatures, we find that there is a low-frequency peak in the momentum-integrated dynamic structure factor, due to diffusive long-wavelength modes. This peak is rapidly suppressed as the temperature is lowered below J. Calculation of the complete dynamic structure factor S(k,\ensuremath{\omega}) shows how the spectral features associated with the two-spinon continuum develop at low temperatures. We extract the nuclear spin-lattice relaxation rate 1/T1{\mathrm{T}}_{1}T1​ from the \ensuremath{\omega}\ensuremath{\rightarrow}0 limit, and compare with recent experimental results for Sr2{\mathrm{Sr}}_{2}Sr2​CuO3{\mathrm{CuO}}_{3}CuO3​ and CuGeO3{\mathrm{CuGeO}}_{3}CuGeO3​. We also discuss the scaling behavior of the dynamic susceptibility, and of the static structure factor S(k) and the static susceptibility \ensuremath{\chi}(k). We confirm the asymptotic low-temperature forms S(\ensuremath{\pi})\ensuremath{\sim}[ln (T)]3/2{]}^{3\mathrm{/}2}]3/2 and \ensuremath{\chi}(\ensuremath{\pi})\ensuremath{\sim}${\mathrm{T}}^{\mathrm{\ensuremath{-}}1}$[ln (T)]1/2{]}^{1\mathrm{/}2}]1/2, expected from previous theoretical studies.
In the framework of the exactly solvable model, using nonperturbative methods, the temperature and magnetic field behavior of the NMR relaxation rate (and low-energy dynamical structure factor) for the antiferromagnetic Heisenberg spin-1/2 chain with magnetic impurities is calculated. For a single magnetic impurity, totally screened at low temperatures by spinons of the chain, the critical exponents, which determine the low-temperature and magnetic field behavior, get renormalized. For the finite concentration of impurities with the random distribution of the impurity-chain coupling the weak disorder in the distribution does not produce essential changes to the dynamical structure factor. Contrary, the strong disorder yields drastically different behavior of the low-energy dynamical characteristics, comparing to the homogeneous chain. Our results are generic, and they qualitatively agree in limiting cases with the results of perturbative calculations for disordered spin chains and with the data of experiments on quasi-one-dimensional spin compounds.
Most of the existing dynamical studies in one dimension on magnetic insulators have considered the simplest spin models with nearest-neighbor interactions. In real systems, however, it is possible that longer range interactions are not entirely negligible. It is expected that the inclusion of next-nearest-neighbor interactions between spins in one-dimensional spin models will introduce a multitude of new frequencies in addition to the ones already present in the dynamics that arises due to nearest-neighbor interactions. We first present an exact solution for the dynamical xx-spin-pair correlations in an Ising chain with both nearest- and next-nearest-neighbor interactions to confirm our expectation. We next show, via an approximate analytical calculation, that the dynamical zz-spin-pair correlations in the next-nearest-neighbor transverse Ising chain when plotted as a function of time is noticeably different with respect to the exactly solvable nearest-neighbor transverse Ising chain at T\ensuremath{\rightarrow}\ensuremath{\infty} when the next-nearest-neighbor interaction is \ensuremath{\gtrsim}1/2 of the magnitude of the nearest-neighbor interaction. The effects could be fairly subtle in the time domain representation and in the spectral function when these additional interactions are weak (i.e., 1/2 of the nearest-neighbor interaction magnitude). The general conclusions reached in this work are expected to be valid for other simple quantum spin models such as the XY and XXZ models in one dimension. \textcopyright{} 1996 The American Physical Society.