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 =
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.
We compute all dynamical spin–spin correlation functions for the spin-1/2 XXZ anisotropic Heisenberg model in the gapless antiferromagnetic regime at zero temperature, using numerical sums of exact determinant representations for form factors of spin operators on the lattice. Contributions from intermediate states containing many particles and string (bound) states are included. We present modified determinant representations for the form factors valid in the general case with string solutions to the Bethe equations. Our results are such that the available sum rules are saturated to high precision. We Fourier transform our results back to real space, allowing us in particular to make a comparison with known exact formulae for equal-time correlation functions for small separations in zero field, and with predictions for the zero-field asymptotics from conformal field theory.
Rigorous results are presented for the dynamics of the nearest-neighbour Heisenberg chain, for fixed but arbitrary wavevector and frequency, to lowest non-trivial order in the temperature. The authors obtain the two-spin correlation function, in the antiferromagnet for all wavevectors, and in the ferromagnet for wavelengths shorter than a coherence length. The energy correlation function at T=0 is obtained exactly, and is identical in the two systems. They find a violation of dynamical scaling in the form of the spectral function in the antiferromagnet for a range of wavevectors. An extension of the perturbative results for the ferromagnet to treat wavelengths longer than a coherence length shows that the diffusion coefficient vanishes logarithmically, in contrast to the scaling prediction that it is constant. An approximate value for the coefficient of the logarithmic term is obtained which agrees well with the results of simulations.
Magnetic and superconducting pairing correlation functions in a general class of Hubbard models, the t - J model and a single-band Hubbard model with additional bond - charge interaction are investigated. Some rigorous upper bounds of the corresponding correlation functions are obtained. It is found that the decay of the spin - spin correlation function with temperature in the general Hubbard models cannot be slower than the inverse square law at low temperatures and the inverse law at high temperatures, while the on-site pairing correlation function cannot be slower than the inverse law. An upper bound for the average energy of the t - J model is found. The upper bounds for the spin - spin and the electron pairing correlation functions in the t - J model as well as in the Hubbard model with bond - charge interaction are also obtained. These bounds are expected to provide certain standards for approximate methods. In some special cases, these bounds rule out the possibility of corresponding magnetic and pairing long-range order.
We study the low-temperature properties of a spin-1 Heisenberg antiferromagnetic chain in an external magnetic field by an extended self-consistent mean-field approach developed earlier based on Anderson's spin-wave theory. At fixed magnetic field, the excitation energies increase, while the correlation length decreases with the increasing temperatures. The magnetic field leads to the Zeeman splitting of the excitations with the upper branch going up and the lower branch going down with the increasing fields. At zero temperature the correlation length is a constant and the gap closes at a critical magnetic field, while at finite temperature the correlation length decreases and a small gap seems to survive at the critical magnetic field. These results are consistent with those of other theoretical studies and agree with the experiments quite well. Also, we show that the system shares very similar properties with the valence-bond-solid state even at finite temperatures. The magnetization is studied and the coexistence of the hidden long-range order and the field-induced ferromagnetic order is discussed.