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By a recursive method numerically exact free energies are calculated for square L && L Ising lattices, with 6~L~18, for several kinds of frozen-in bond disorder: (i) bonds +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 (SaStt ) r 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; awhile in the Gaussian model the spin-glass order parameter q ( T =0) =1, it is found that q = 0 in the +J model where rather a power-law decay of correlations (SoSR) r a 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 rnetastable states would be required.

We implement a Quantum Monte Carlo calculation for a repulsive Hubbard In the present QMC study of a generalized HM (which includes next-nearest neighbor hopping interactions as well as the usual nearest-neighbor ones), we present evidence for a superconducting tendency in the The SP feature may be incorporated into the Hubbard model within the metallic regime by introducing a next-nearest neighbor interaction. This allows the SP to lie at the Fermi level at a doping of, say 15-25%, while the insulating point, at which the antiferromagnetic instability occurs, lies at 0% doping. These features are characteristic of real cuprate materials [13], [14]. It is in the former situation (15-25% doping) that the model is found to support superconductivity.The model is specified as followsIn (1), U is the repulsive on-site Coulomb interaction, t is the nearest-neighbor hopping integral ( ij denotes nearest neighbor interactions), t ′ is the next-nearest neighbor hopping integral ( ij denotes next-nearest neighbor interactions); t and t ′ are defined to be positive. 2The noninteracting band structure of the tt ′ -Hubbard model (1) has saddle points at energy −4t ′ , and at k = (0, π) and (π, 0). If we take the hole doping x to be of order x ≈ t ′ , then the saddle points in the noninteracting band structure lie near E F (see Fig. 1). In the absence of t ′ , the required doping would be zero, making the sample insulating. The electronic effective mass below the SP's is heavier than the mass above, the ratio being (t + 2t where Θ is a projection parameter and |Ψ 0 a single determinant taken as the ground state of the noninteracting band structure of 1.The exponent in (2) demonstrated [7] in models such as the attractive Hubbard model and simplified models with electronphonon coupling, where a pairing tendency is anticipated to occur. While studying finite systems one has always to extrapolate to the thermodynamic limit. Due to the unsystematic finite level structure of the finite Hubbard model this has turned out to be very difficult. We use two different approaches: a) we are able to analyse the plateau in terms of the effective pairing interaction J, and hence deduce a value for T c in the infinite-sample limit. b) we find analogous finite size scaling behaviour for the superconducting correlations for the repulsive and the attractive Hubbard model.The condition as to whether the finite cluster of size L is 'superconducting' (correlation length ξ > L), or whether superconducitivity is suppressed by finite size effects (ξ < L), does not enter into these considerations and is not relevant for this paper. to derive the single-band U formally from a multi-band model [19] give a value of order 6eV, i.e. 6t, in the case t = 1 eV. We have observed clear evidence for the superconducting tendency for U in the range 0.5t < U < 3t; for larger U-values unnacceptable error bars are obtained.The superconducting correlation function χ(R j ) for d x 2 −y 2 symmetry is defined aswhereHere, p implies a sum over nearest neighbors...

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