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Journal of Physics A Mathematical and General

Critical dynamic response of the dilute antiferromagnetic chain

Anthony C. Maggs, Robin B. Stinchcombe

Published May 11, 1984• 13 citations
Publisher

Abstract

The transverse dynamic response of the dilute classical Heisenberg antiferromagnetic chain is calculated near the percolation threshold as k to 0 and k to pi . In the critical limit the universal dynamic structure function, which is calculated in closed form, exhibits dynamic scaling with exponents z=1, eta T =1. This function displays the crossover from spin wave (k- pi ) xi p >>1 to hydrodynamic (k- pi ) xi p <<1 response. The relevance of the results to the experimental situation is discussed.

DOI
10.1088/0305-4470/17/7/023

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•Jan 1, 1991•0 citations
The transverse dynamic response of the dilute classical Heisenberg antiferromagnetic chain is calculated near the percolation threshold as k to 0 and k to pi . In the critical limit the universal dynamic structure function, which is calculated in closed form, exhibits dynamic scaling with exponents z=1, eta T =1. This function displays the crossover from spin wave (k- pi ) xi p >>1 to hydrodynamic (k- pi ) xi p <<1 response. The relevance of the results to the experimental situation is discussed.
Critical spin wave dynamics in the dilute Heisenberg chain near the percolation threshold is treated by two complementary approaches. The first exploits the renormalisation group transformation of parameters under a length scaling achieved by decimation on the equations of motion of the random system, the second obtains from the contributions from chain segments of all possible sizes the average dynamic response using a continuum approach valid for small wavevector k and long percolation correlation length xi p. Both approaches yield identical dynamic scaling forms for the dynamic response, with dynamic exponent z=2, and details of the crossover of characteristic frequency between hydrodynamic and critical form as k xi p varies from 0 to infinity . A detailed expression for the scaling function for the dynamic response is also obtained.
The damping of spin waves in bond-diluted Heisenberg antiferromagnets in d dimensions is considered using a method developed for the ferromagnetic case by the authors. They treat the hydrodynamic limit q xi identical to (k- pi ) xi <<1 and derive an expression for the damping Gamma c (q, xi ) varies as (p-p c ) - mu q d+1 , where mu = nu d- 1 / 2 (2t- beta -(d-2) nu ) which is consistent with the dynamic scaling principle. They also obtain a closed-form expression for the response function and consider finite-cluster response. The critical dynamic properties of the corresponding site-diluted system are then examined. Finally, they contrast d=1 and d>1 and then generalise the results to the ferrimagnet.
The critical spin-wave dynamics of dilute ferromagnetic and antiferromagnetic Heisenberg systems near the percolation threshold is studied on a two-dimensional regular (fractal) model for bond percolation, which captures the relevant geometric features of the cluster structure. The dynamic critical exponent and spectral dimension for both the ferromagnetic and antiferromagnetic systems are calculated. The treatment of the dynamics in these systems requires the use of generalised scaling techniques for sublattice systems, which involve an extension of the parameter space.
The decay of zero temperature spin wave excitations near the percolation threshold in bond-diluted d-dimensional (d>1) Heisenberg ferromagnets is investigated using a Green function equation-of-motion method. The diagrammatic perturbation theory is convergent in the hydrodynamic limit (k xi <<1) and the following expression is obtained there for the damping, or decay rate: Gamma c (k, xi ) varies as (p-p c ) -vd+(t- beta ) k d+2 ,(k xi <<1). This form satisfies the requirements of the dynamic scaling principle, by which Gamma c (k, xi )=k z g(k xi ) for general k xi , and z=2+(t- beta )/ nu . An explicit expression, applicable in the hydrodynamic regime and consistent with dynamic scaling, is also obtained for the infinite cluster transverse dynamic structure function. The approach is extended to incorporate the ensemble of finite clusters and to treat the case of non-zero temperature and also the experimentally more relevant situation of site dilution.
We use linear spin wave theory and series expansion method to study the diluted quantum antiferromagnetic Heisenberg model at zero temperature near the p = 1 and p = psbQp\\sb{Q}psbQ respectively, where p is the concentration of the spins or bonds, and psbQp\\sb{Q}psbQ is the transition point from order to disorder of the quantum spin system. For the first case, our complete calculation to the first order in x equiv\\equivequiv 1 −-− p gives a dispersion relation for long-wavelength spin waves in the form omega(vecq)\\omega(\\vec q)omega(vecq) = c(x)q+igamma(x)qsptauc(x)q + i\\gamma(x)q\\sp\\tauc(x)q+igamma(x)qsptau. Our results for c(x)c(x)c(x) agrees with previous work, where c(x)c(x)c(x) = c(0)lbrack1−alphax+O(xsp2)rbrackc(0)\\lbrack 1 - \\alpha x + O(x\\sp2)\\rbrackc(0)lbrack1−alphax+O(xsp2)rbrack. In particular the coefficient alpha\\alphaalpha, which can be related to the helicity modulus and the uniform perpendicular susceptibility, diverges in the limit dto2d\\to 2dto2. One major new results is that tau\\tautau = d−1d - 1d−1, which means that d = 2 is the lower critical dimension at which infrared divergences affect the dynamical response due to vacancies. For the second case, the critical behavior of the quantum system near psbQp\\sb{Q}psbQ is found to be different from that of the percolation and pure systems. Thus there exists a new fixed point in the phase diagram, which specifies a new universality class. The correlation length exponents nu\\nunu (which comes from the relation xisim(psbQ−p)spnu\\xi\\sim(p\\sb{Q} - p)\\sp\\nuxisim(psbQ−p)spnu, where xi\\xixi is the correlation length) are estimated, which are 0.65 pm\\pmpm 0.2 and 0.55 pm\\pmpm 0.2 in 2 and 3 dimensions respectively. The exponents of crossover with temperature and time are determined. We also find evidence that psbQ>psbcp\\sb{Q} > p\\sb{c}psbQ>psbc.