Small Ising lattices with both ferromagnetic (F) and antiferromagnetic (AF) exchange interactions (or bonds) and increasing numbers of spins are studied by means of two independent methods: computational solutions to the Hamiltonian problem and topological counting of frustration paths. Equal magnitudes and concentrations are assumed for both types of bonds. Two different geometries are considered: square lattices (SL's) with coordination number 4 and triangular lattices (TL's) with coordination number 6. Two-dimensional samples with a total number of spins N between 4 and 64 are considered for SL's, while N is varied between 4 and 44 for TL's. They are distributed in two-dimensional arrays where periodic boundary conditions are imposed. After an array is selected, bond distributions (samples) are independently and randomly generated in fixed positions. The physical parameters are then calculated exactly for each sample. The emphasis here is on the ground-state properties and their dependence with size and shape for the two kinds of lattices. All magnitudes correspond to a basic statistics over a large number of samples for each array. The following magnitudes are reported: ground-state energy per bond, frustration segment, abundance of first excited states, remnant entropy, low-temperature specific heat, and site order parameters q, p, and h. Parameters p and h are introduced here, showing advantages over other similar magnitudes. The results are in good correspondence with analytic studies for the thermodynamic limit. This means that the spin site correlation (p) tends to vanish as N grows. However, we have found that the shape dependence modulates the behavior of these systems toward the thermodynamic limit. There is no tendency to vanish for the bond correlation parameter (h). For both kinds of lattices h might be a constant independent of size and shape.
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.
We study ground-state properties of the two-dimensional random-bond Ising model with couplings having a concentration
p∊[0,1] of antiferromagnetic and $(1\ensuremath{-}p)$ of ferromagnetic bonds. We apply an exact matching algorithm which enables us the study of systems with linear dimension
L up to
700. We study the behavior of the domain-wall energies and of the magnetization. We find that the paramagnet-ferromagnet transition occurs at ${p}_{c}\ensuremath{\sim}0.103$ compared to the concentration ${p}_{n}\ensuremath{\sim}0.109$ at the Nishimori point, which means that the phase diagram of the model exhibits a re-entrance. Furthermore, we find no indications for an (intermediate) spin-glass ordering at finite temperature.
At low temperatures, the classical two-dimensional random bond Ising model undergoes a frustration-driven ferromagnet-to-paramagnet transition controlled by a zero-temperature fixed point separating ferromagnet and spin glass phases. We show that this critical point can be understood through a renormalization group transformation that constructs the ground state of the Ising model through a sequence of Hamiltonians that, starting with an unfrustrated model, iteratively adds in frustration until the target Hamiltonian is reached. Via a mapping of the thermodynamics of the 2d Ising model to the spectral properties of a related Hermitian matrix -- the Hamiltonian of a noninteracting quantum problem -- this RG procedure corresponds to an iterative diagonalization of the quantum Hamiltonian. The flow toward zero temperature in the Ising picture manifests as a flow toward infinite randomness in the spectrum of the quantum Hamiltonian, with the log gap of the Hamiltonian scaling as a power of the system size:
logεmin−1∼Lψ. The tunneling exponent
ψ is equal to the spin stiffness exponent
θc characterizing the zero-temperature fixed point.