We investigate the ground state properties of the ± J Ising model with asymmetric probability weight in two dimensions. We propose a numerical transfer matrix method which enables us to calculate directly the ground-state properties such as correlation functions, defect energy and entropy at T =0. We find that the spin glass correlation function decays in a power law like
g~(r)∼r−η~ with
η~≃0.21 in the range
0.5≤p≲0.85, suggesting the weak universality of spin glass-the concentration of + J bond is denoted by p . At the ferromagnetic phase boundary locating around p =0.89, the ferromagnetic correlation function decays as
g(r)∼r−η~ with η≃0.240±0.016, which suggests the weak universality of ferromagnetic critical exponent from the non-random critical point ( p =1, T =2.269···) to the ground sate counterpart. The behavior of correlation functions, and also that of defect energies indicate the existence of an extra-phase, called the random antiphase state, in
0.85≲p≲0.89.
We introduce a one dimensional Ising model with two competing interactions: nearest neighbor random couplings +/-J with equal probability and a positive infinite range coupling Lambda. At low temperature T the model exhibits a first order phase transition between a ferromagnetic state (with magnetization m(1) = 1 at T = 0) and a << ferrimagnetic >> state (with m(2) = 2/3 at T = 0), when the disorder strength J/Lambda is increased. For 5/12 < J/Lambda < 1, a whole spectrum of ferrimagnetic ground states with magnetization m(n) = 2/(n + 1) (n = 2, ..., infinity) is present while for J/lambda > 1 the ground state is given by a trivial one dimensional spin glass with m = 0. The main qualitative features of the model can be described by a simplified annealed model where the random couplings can arrange themselves to minimize free energy with the constraint that the number of positive couplings is fixed by the law of large numbers in the thermodynamic limit. This model is exactly solved at all temperatures and the diagram of phase is calculated.
We consider the XY model with ferromagnetic (FM) and antinematic (AN) nearest-neighbor interactions on a square lattice for a varying interaction strength ratio. Besides the expected FM and AN quasi-long-range order (QLRO) phases we identify at low temperatures another peculiar canted ferromagnetic (CFM) QLRO phase, resulting from the competition between the collinear FM and noncollinear AN ordering tendencies. In the CFM phase neighboring spins that belong to different sublattices are canted by a nonuniversal (dependent on the interaction strength ratio) angle and the ordering is characterized by a fast-decaying power-law intrasublattice correlation function. Compared to the FM phase, in the CFM phase correlations are significantly diminished by the presence of zero-energy domain walls due to the inherent degeneracy caused by the AN interactions. We present the phase diagram as a function of the interaction strength ratio and discuss the character of the respective phases as well as the transitions between them.
The spin-spin correlation function of the two-dimensional (2D)
XY model decays as a power law at all temperatures below the Berezinskii-Kosterlitz-Thouless transition point with a temperature-dependent exponent $\ensuremath{\eta}=\ensuremath{\eta}(T/J)$ (
J is the ferromagnetic coupling strength). It is known from computer experiments that in the 2D
XY model with site or bond dilution this exponent depends on the concentration
p of removed sites or bonds as well. Knowing the slope $\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p$ at point
p=0, one can predict the value of the exponent for small dilution concentrations: ${\ensuremath{\eta}(p)\ensuremath{\simeq}\ensuremath{\eta}(0)+p(\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p)|}_{p=0}$. As shown in this paper, the spin-wave Hamiltonian allows one to obtain exact results for this slope: ${(\ensuremath{\partial}\ensuremath{\eta}/\ensuremath{\partial}p)|}_{p=0}=T/(2J)+O({(T/J)}^{2})$ and $T/(\ensuremath{\pi}J)+O({(T/J)}^{2})$ for site and for bond dilution, respectively.
By a recursive method numerically exact free energies are calculated for square $L\ifmmode\times\else\texttimes\fi{}L$ Ising lattices, with
6< L< 18, for several kinds of frozen-in bond disorder: (i) bonds $\ifmmode\pm\else\textpm\fi{}J$ with various concentrations of negative bonds; (ii) bonds distributed according to a Gaussian distribution. Ground states of these systems are identified, the response to "ordering fields" is studied, and the correlation function
〈S0SR〉T2 is calculated as a function of temperature for various distances
R in the lattice. This correlation is found to decay strongly (presumably exponentially) with increasing
R even at temperatures distinctly below the apparent freezing temperature
Tf of previous Monte Carlo simulations; this "freezing transition" is hence unambiguously identified as a nonequilibrium effect. However, the correlation length is found to become long ranged at low temperatures, and it is suggested that a phase transition still occurs at
T=0; while in the Gaussian model the spin-glass order parameter
q(T=0)=1, it is found that $q\ensuremath{\equiv}0$ in the $\ifmmode\pm\else\textpm\fi{}J$ model where rather a power-law decay of correlations
〈S0SR〉T=02 occurs. Performing Monte Carlo simulations for precisely the same systems, the cooling times necessary to reach the true ground states of the system are identified, as well as the simulation times necessary to reach thermal equilibrium for the correlation functions. These times are found to increase so strongly with
L that for systems of macroscopic size the correct thermal equilibrium is probably irrelevant for experimental purposes. Rather a statistical mechanics based on the many long-lived metastable states would be required.
In a Hartree-Fock picture, itinerant ferromagnetism results from a competition between kinetic and exchange energy, with the magnetized state being favored at large interaction strength. In a recent paper [1] we showed that in contrast to this average effect, fluctuations of off-diagonal interaction matrix elements systematically reduce the ground-state magnetization. When the interaction dominates, the occurence of a non-zero ground-state magnetization depends on the ratio λ between average exchange energy and the fluctuation amplitude of the off-diagonal matrix elements, and a nonzero critical value λc is necessary to magnetize the ground state. We extend these results by numerical studies of λ for standard tight-binding models which indicate a regime of intermediate disorder where off-diagonal fluctuations should play an important role for ground-state magnetization. We also emphasize the presence of strong correlations between minimal eigenvalues of different magnetization blocks which further reduce the probability of a nonzero ground-state spin.