The 3-d Random Field Ising Model at zero temperature

We study numerically the zero temperature Random Field Ising Model on cubic lattices of various linear sizes $L$ in three dimensions. For each random field configuration we vary the ferromagnetic coupling strength $J$. We find that in the infinite volume limit the magnetization is discontinuous in $J$. The energy and its first $J$ derivative are continuous. The approch to the thermodynamic limit is slow, behaving like $L^{-p}$ with $p \sim .8$ for the gaussian distribution of the random field. We also study the bimodal distribution $h_{i} = \pm h$, and we find similar results for the magnetization but with a different value of the exponent $p \sim .6$. This raises the question of the validity of universality for the random field problem.Comment: 8 pages, 3 PostScript Figure

The Spectrum of the two dimensional Hubbard model at low filling

Using group theoretical and numerical methods we have calculated the exact energy spectrum of the two-dimensional Hubbard model on square lattices with four electrons for a wide range of the interaction strength. All known symmetries, i.e.\ the full space group symmetry, the SU(2) spin symmetry, and, in case of a bipartite lattice, the SU(2) pseudospin symmetry, have been taken explicitly into account. But, quite remarkably, a large amount of residual degeneracies remains giving strong evidence for the existence of a yet unknown symmetry. The level spacing distribution and the spectral rigidity are found to be in close to but not exact agreement with random matrix theory. In contrast, the level velocity correlation function presents an unexpected exponential decay qualitatively different from random matrix behavior.Comment: 4 pages, latex (revtex), 3 uuencoded postscript figure

Critical and tricritical singularities of the three-dimensional random-bond Potts model for large qq

We study the effect of varying strength, $\delta$, of bond randomness on the phase transition of the three-dimensional Potts model for large $q$. The cooperative behavior of the system is determined by large correlated domains in which the spins points into the same direction. These domains have a finite extent in the disordered phase. In the ordered phase there is a percolating cluster of correlated spins. For a sufficiently large disorder $\delta>\delta_t$ this percolating cluster coexists with a percolating cluster of non-correlated spins. Such a co-existence is only possible in more than two dimensions. We argue and check numerically that $\delta_t$ is the tricritical disorder, which separates the first- and second-order transition regimes. The tricritical exponents are estimated as $\beta_t/\nu_t=0.10(2)$ and $\nu_t=0.67(4)$. We claim these exponents are $q$ independent, for sufficiently large $q$. In the second-order transition regime the critical exponents $\beta_t/\nu_t=0.60(2)$ and $\nu_t=0.73(1)$ are independent of the strength of disorder.Comment: 12 pages, 11 figure

Excess entropy and central charge of the two-dimensional random-bond Potts model in the large-Q limit

We consider the random-bond Potts model in the large-$Q$ limit and calculate the excess entropy, $S_{\Gamma}$, of a contour, $\Gamma$, which is given by the mean number of Fortuin-Kasteleyn clusters which are crossed by $\Gamma$. In two dimensions $S_{\Gamma}$ is proportional to the length of $\Gamma$, to which - at the critical point - there are universal logarithmic corrections due to corners. These are calculated by applying techniques of conformal field theory and compared with the results of large scale numerical calculations. The central charge of the model is obtained from the corner contributions to the excess entropy and independently from the finite-size correction of the free-energy as: $\lim_{Q \to \infty}c(Q)/\ln Q =0.74(2)$, close to previous estimates calculated at finite values of $Q$.Comment: 6 pages, 7 figure