Analytic results for the free energy, magnetic structure factor, and Edwards-Anderson order parameter of a one-dimensional ferromagnetic Ising model in a random magnetic field are obtained. The structure factor consists of both Lorentzian and Lorentzian-squared terms at all temperatures (
T) greater than zero; the Lorentzian-squared terms vanish at
T=0. The calculated correlation length agrees with predictions of the Imry-Ma domain argument.
We present an exact solution of the long-time relaxational behavior of the magnetization in the Ising and
XY chains in a quenched random field. The random field is assumed to be of infinite strength but present at only a fraction of the spin sites. We use Glauber dynamics for the Ising case, and a suitably generalized master equation for the
XY case. In both models we find that for
T=0, as time $t\ensuremath{\rightarrow}\ensuremath{\infty}$ the magnetization decays as exp (${\ensuremath{-}Ct}^{\frac{1}{3}}$), where
C is a constant. At finite temperatures the ultimate asymptotic behavior is purely exponential.
We investigate the temperature dependence of the finite-field magnetization of the S = 1/2 bondalternating XY model in the Random magnetic field along the z-direction having the Lorentzian distribution.The random-averaged free energy can be exactly calculated, which enables us to obtain the thermodynamical quantities.The temperature dependence of the finite-field magnetization shows various behaviors depending on the parameters δ and Γ, where δ and Γ are the bond-alternation parameter and the Lorentzian distribution parameter, respectively.
We have performed Monte Carlo studies of the three-dimensional
XY model with random uniaxial anisotropy, which is a model for randomly pinned spin-density waves. We study $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L$ simple cubic lattices using
L values in the range 16--64, and with random anisotropy strengths of
D/2J=1, 2, 3, 6, and $\ensuremath{\infty}$. There is a well-defined finite-temperature critical point,
Tc, for each of these values of
D/2J. We present results for the angle-averaged magnetic structure factor
S(k) at
Tc for
L=64. We also use finite-size scaling analysis to study scaling functions for the critical behavior of the specific heat, the magnetization, and the longitudinal magnetic susceptibility. Good data collapse of the scaling functions over a wide range of
T is seen for
D/2J=6 and $\ensuremath{\infty}$. For our finite values of
D/2J the scaled magnetization function increases with
L below
Tc and appears to approach an
L-independent limit for large
L. This suggests that the system is ferromagnetic below
Tc.
A real-space renormalization-group treatment of random transverse-field Ising spin chains that was introduced previously is developed and extensively analyzed. It yields results that are asymptotically exact in the critical region near the zero-temperature para-to-ferromagnetic quantum phase transition. In particular, the exact scaling function is obtained for the magnetization as a function of a uniform applied magnetic field and the distance to the critical point, and up to the solution of a linear ordinary differential equation whose solution can be exhaustively analyzed, the scaling function of the average spin-spin correlation function is also obtained. Thus more exact information is obtainable about the critical behavior for this random model than is known for the pure version which is equivalent to the two-dimensional Ising model. The basic reason for this is the extremely broad distribution of energy scales that occurs at low energies near the critical point of the random system. For the random chain the distribution of the magnetization of the first spin in a semi-infinite system is also studied and the results found to agree in the scaling limit with results of McCoy obtained from the exact solution of the closely related McCoy-Wu Ising model; this provides strong justification for the validity of the present approach. The singular properties of the weakly ordered and weakly disordered ``Griffiths' phases'' that occur at zero temperature near the critical point are also studied, as well as the behavior at low but nonzero temperature. Possible extensions of the results and general lessons drawn from them for other random systems are briefly discussed.
The spin-spin correlation function of a one-dimensional random Ising model decays in a manner inversely proportional to the square of the partition function of the part of the system between the spins. At low temperatures and strong coupling there are logarithmic corrections to the scaling power-law dependence of the correlation length on the magnetic field. The correlation of the boundary spins as a function of the system size is evaluated analytically and numerically. At low temperatures and strong coupling this function has a crossover between different behaviors and its Fourier transform oscillates with wave vector.