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Journal of Physics A Mathematical and General

Critical dynamics of Heisenberg spins on self-avoiding-walk chains

S. Bhattacharya, Bikas K. Chakrabarti

Published Oct 1, 1984• 3 citations
Publisher

Abstract

The dynamical exponent z for the critical spin-wave dynamics of nearest-neighbour interacting Heisenberg spins on a self-avoiding-walk (SAW) chain is estimated here using a scaling picture and also applying a real-space renormalisation group technique to some quasilinear fractal lattices. The results indicate z=2Dt, where D is the fractal dimensionality of the SAW chain lattice and t is the exponent for the length of the shortest nearest-neighbour connecting path of the SAW.

DOI
10.1088/0305-4470/17/14/007

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The dynamical exponent z for the critical spin-wave dynamics of nearest-neighbour interacting Heisenberg spins on a self-avoiding-walk (SAW) chain is estimated here using a scaling picture and also applying a real-space renormalisation group technique to some quasilinear fractal lattices. The results indicate z=2Dt, where D is the fractal dimensionality of the SAW chain lattice and t is the exponent for the length of the shortest nearest-neighbour connecting path of the SAW.
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.
Recent results concerning non-universality in the dynamical critical behavior of the one-dimensional Ising model, are generalised to several formulations of the Potts model, using arguments based on domain wall motion. By applying real-space renormalisation group methods, the authors also study the dynamical Ising model, with single flips, both for a period distribution of nearest-neighbor exchanges and for the random chain (with nonzero couplings). They show that the critical dynamical exponent is z=1+J M /J m where J M (J m ) is the greatest (smallest) of the exchanges.