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Journal of Applied Physics

The dilute quantum Heisenberg antiferromagnet

C. C. Wan, A. Brooks Harris, Joan Adler

Published Apr 15, 1991• 21 citations
Publisher

Abstract

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.

DOI
10.1063/1.348123

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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.
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.
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).
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.
Ground-state magnetic properties of the diluted Heisenberg antiferromagnet on a square lattice are investigated by means of the quantum Monte Carlo method with the continuous-time loop algorithm. It is found that the critical concentration of magnetic sites is independent of the spin size S, and equal to the two-dimensional percolation threshold. However, the existence of quantum fluctuations makes the critical exponents deviate from those of the classical percolation transition. Furthermore, we found that the transition is not universal, i.e., the critical exponents significantly depend on S.
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.
Ground state properties of the randomly diluted two-dimensional Heisenberg antiferromagnet are discussed. Quantum Monte Carlo (stochastic series expansion) data for L×L lattices, with L up to 64, are extrapolated to infinite size for dilution fractions p between 0 (clean system) and p* (the percolation threshold). The sublattice magnetization is calculated using a decomposition into a classical and a quantum mechanical factor, which are evaluated separately. The quantum mechanical factor (the sublattice magnetization of the largest connected cluster of magnetic sites) remains finite at the percolation threshold, implying that the dilution driven order-disorder transition is a classical percolation transition. The spin stiffness is calculated using the winding number fluctuations.