In order to study the influence of quenched disorder on second-order phase transitions, high-temperature series expansions of the susceptibility and the free energy are obtained for the quenched bond-diluted Ising model in
d=3−−5 dimensions. They are analyzed using different extrapolation methods tailored to the expected singularity behaviors. In
d=4 and 5 dimensions we confirm that the critical behavior is governed by the pure fixed point up to dilutions near the geometric bond percolation threshold. The existence and form of logarithmic corrections for the pure Ising model in
d=4 are confirmed, and our results for the critical behavior of the diluted system are in agreement with the type of singularity predicted by renormalization group considerations. In three dimensions we find large crossover effects between the pure Ising, percolation, and random fixed points. We estimate the critical exponent of the susceptibility to be $\ensuremath{\gamma}=1.305(5)$ at the random fixed point.
Generalizing the recent work of Collins, the wave-vector-dependent susceptibility and the static correlation function for a randomly diluted Heisenberg paramagnet on a Bravais lattice with quenched-site, or nearest-neighbor exchange-bond, dilution is studied by high-temperature series-expansion techniques. The first five coefficients for the spin correlation function $S(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})$ and susceptibility $\ensuremath{\chi}(\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}})$ are calculated for arbitrary $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$, the sign of the exchange and the spin magnitude and the magnetic site
c and bond
p concentrations. Numerical results for these functions are presented for various spins and for $\stackrel{\ensuremath{\rightarrow}}{\mathrm{k}}$ along the symmetry directions (1,1,1) and (1,0,0). Because of the generality of the present series, their length is rather limited, and as such they are not suitable for a careful analysis of the critical region. Nonetheless, to get a qualitative feel for the accuracy of the present results near the transition region, we have used Pad\'e approximants for determining the correlation length $\ensuremath{\xi}(T)$ as a function of the reduced temperature $\ensuremath{\epsilon}=\frac{[T\ensuremath{-}{T}_{C(N)}(\mathrm{random})]}{{T}_{C(N)}(\mathrm{random})}$. In this manner, we estimate that the numerical accuracy of our results should be reasonable as long as the system is moderately above the transition temperature, e.g., $\ensuremath{\epsilon}\ensuremath{\gtrsim}0.5$.
New high-density series data for the mean number, percolation probability and 'susceptibility' of finite clusters are presented for bond and site percolation on four standard three-dimensional lattices. A Pade approximant analysis of both high- and low-density series makes particular use of the rather precise unbiased estimates of the percolation threshold, p c , obtained recently by Heermann and Stauffer (1981) for the simple cubic lattice. For this lattice the authors obtain the biased estimate gamma =1.73+or-0.03 for the site problem and a similar estimate but with larger uncertainties for the bond problem. Such a value is significantly larger than earlier series estimates. Assuming gamma to be universal they obtain precise, although biased, estimates of p c for both bond and site percolation on all four lattices. Using the bond estimates of p c they find an overall biased estimate of beta =0.454+or-0.008 for bond percolation on all three-dimensional lattices. (The corresponding site problem requires further study.) Scaling estimates of other critical exponents are alpha =-0.64+or-0.05, delta =4.81+or-0.14, Delta =2.18+or-0.04, nu =0.88+or-0.02 and nu =0.03+or-0.03.
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 =
psbQ respectively, where p is the concentration of the spins or bonds, and
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 1
− p gives a dispersion relation for long-wavelength spin waves in the form
omega(vecq) =
c(x)q+igamma(x)qsptau. Our results for
c(x) agrees with previous work, where
c(x) =
c(0)lbrack1−alphax+O(xsp2)rbrack. In particular the coefficient
alpha, which can be related to the helicity modulus and the uniform perpendicular susceptibility, diverges in the limit
dto2. One major new results is that
tau =
d−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
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 (which comes from the relation
xisim(psbQ−p)spnu, where
xi is the correlation length) are estimated, which are 0.65
pm 0.2 and 0.55
pm 0.2 in 2 and 3 dimensions respectively. The exponents of crossover with temperature and time are determined. We also find evidence that
psbQ>psbc.
The system described by the Hamiltonian shown is investigated on the basis of exact high-temperature series expansions and the Green function technique. Expansions for the zero-field ( B = 0) free energy and susceptibility (valid for arbitrary lattice structure) are developed to fifth order in reciprocal temperature for the S = ½ system with nearest-neighbour forces. The critical behaviour is discussed in two and three dimensions using the random phase approximation and Padé approximants. In particular the variation of the critical temperature with η is accurately determined for common two- and three-dimensional lattices. It is conjectured that the susceptibility index γ remains constant for 0 ⩽ η < 1 but changes discontinuously at η = 1.