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Journal of Physics A Mathematical and General

Crossover in randomly diluted classical two-dimensional Heisenberg magnets

Joseph A. Aronovitz, T. C. Lubensky

Published Jan 21, 1990• 1 citation
Publisher

Abstract

The authors use a truncated replicated nonlinear sigma model to study crossover in the vicinity of the percolation threshold of a randomly diluted classical m-component Heisenberg model with m>2. They find that the order parameter correlation length xi diverges with temperature T for concentration p of occupied bonds greater than the percolation threshold p c as exp(2 pi rho s (p)/T) where rho s (p) approximately mod p-p c mod phi is the spin stiffness and for p=p c as T - phi vp/ where v p is the percolation correlation length exponent. The numerical values of the exponents phi and v p predicted in their truncation scheme, however, differ significantly from their accepted values.

DOI
10.1088/0305-4470/23/2/019

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•Jan 1, 1991•0 citations
The authors use a truncated replicated nonlinear sigma model to study crossover in the vicinity of the percolation threshold of a randomly diluted classical m-component Heisenberg model with m>2. They find that the order parameter correlation length xi diverges with temperature T for concentration p of occupied bonds greater than the percolation threshold p c as exp(2 pi rho s (p)/T) where rho s (p) approximately mod p-p c mod phi is the spin stiffness and for p=p c as T - phi vp/ where v p is the percolation correlation length exponent. The numerical values of the exponents phi and v p predicted in their truncation scheme, however, differ significantly from their accepted values.
Quantum phase transition of site-diluted and bond-diluted Heisenberg antiferromagnets on square lattices is studied. By using the novel continuous-time loop algorithm, we perform quantum Monte Carlo simulations on quite larger lattices at extremely lower temperatures than the previous numerical studies. It is found that the antiferromagnetic long-range order at T = 0 persists so long as a cluster of magnetic sites percolates, that is, the critical concentration is equal to the classical percolation threshold, in both of the site-diluted and bond-diluted cases. Furthermore, we find that some critical exponents, such as the magnetization exponent β, are non-classical and strongly depend on the spin size S. On the other hand, we show that the correlation-length exponent ν is universal and is equal to the classical value (ν=4/3).
Series expansions are used to treat the randomly diluted spin-1/2 Heisenberg antiferromagnet at zero temperature. A series is obtained at zero temperature in powers of the concentration for many correlation functions and for the correlation length from which static, dynamic, and crossover exponents are estimated. The correlation length exponent is found to be 0.77±0.10 in two dimensions. The critical concentration for the appearance of long-range order is indistinguishable from the percolation threshold.
The two-dimensional antiferromagnetic S=1/2S=1/2S=1/2 Heisenberg model with random site dilution is studied using quantum Monte Carlo simulations. Ground-state properties of the largest connected cluster on $L\ifmmode\times\else\texttimes\fi{}L$ lattices, with L up to 64, are calculated at the classical percolation threshold. In addition, clusters with a fixed number Nc{N}_{c}Nc​ of spins on an infinite lattice at the percolation density are studied for Nc{N}_{c}Nc​ up to 1024. The disorder averaged sublattice magnetization per spin extrapolates to the same nonzero infinite-size value for both types of clusters. Hence, the percolating clusters, which are fractal with dimensionality d=91/48,d=91/48,d=91/48, have antiferromagnetic long-range order. This implies that the order-disorder transition driven by site dilution occurs exactly at the percolation threshold and that the exponents are classical. The same conclusion is reached for the bond-diluted system. The full sublattice magnetization versus site dilution curve is obtained in terms of a decomposition into a classical geometrical factor and a factor containing all the effects of quantum fluctuations. The spin stiffness is shown to obey the same scaling as the conductivity of a random resistor network.
The two-dimensional antiferromagnetic S=1/2 Heisenberg model with random bond dilution is studied using quantum Monte Carlo simulation at the percolation threshold (50% of the bonds removed). Finite-size scaling of the staggered structure factor averaged over the largest connected clusters of sites on L*L lattices shows that long-range order exists within the percolating fractal clusters in the thermodynamic limit. This implies that the order-disorder transition driven by bond-dilution occurs exactly at the percolation threshold and that the exponents are classical. This result should apply also to the site-diluted system.
The bond-diluted classical Heisenberg ferromagnet is treated by a position space renormalisation group method valid when the valid behaviour is at low temperatures. The method is outlined and used to obtain exact (one-dimensional) and approximate (two- and three-dimensional) results for the critical behaviour of the correlation length near the percolation limit. The procedure agrees in the pure limit with exact results of Polyakov and Migdal for the two-dimensional case. At general concentrations an exact correspondence with related treatments of random resistor networks is exhibited. This implies that the percolation-to-thermal crossover exponents of the Ising and Heisenberg systems are the same. The generalisations required for a proper comparison with neutron scattering experiments are briefly discussed.
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.