The Random-bond Potts model in the large-q limit

We study the critical behavior of the q-state Potts model with random ferromagnetic couplings. Working with the cluster representation the partition sum of the model in the large-q limit is dominated by a single graph, the fractal properties of which are related to the critical singularities of the random Potts model. The optimization problem of finding the dominant graph, is studied on the square lattice by simulated annealing and by a combinatorial algorithm. Critical exponents of the magnetization and the correlation length are estimated and conformal predictions are compared with numerical results.Comment: 7 pages, 6 figure

Resonant tunneling and the multichannel Kondo problem: the quantum Brownian motion description

We study mesoscopic resonant tunneling as well as multichannel Kondo problems by mapping them to a first-quantized quantum mechanical model of a particle moving in a multi-dimensional periodic potential with Ohmic dissipation. From a renormalization group analysis, we obtain phase diagrams of the quantum Brownian motion model with various lattice symmetries. For a symmorphic lattice, there are two phases at T=0: a localized phase in which the particle is trapped in a potential minimum, and a free phase in which the particle is unaffected by the periodic potential. For a non-symmorphic lattice, however, there may be an additional intermediate phase in which the particle is neither localized nor completely free. The fixed point governing the intermediate phase is shown to be identical to the well-known multichannel Kondo fixed point in the Toulouse limit as well as the resonance fixed point of a quantum dot model and a double-barrier Luttinger liquid model. The mapping allows us to compute the fixed-poing mobility $\mu^*$ of the quantum Brownian motion model exactly, using known conformal-field-theory results of the Kondo problem. From the mobility, we find that the peak value of the conductance resonance of a spin-1/2 quantum dot problem is given by $e^2/2h$. The scaling form of the resonance line shape is predicted

Evidence for softening of first-order transition in 3D by quenched disorder

We study by extensive Monte Carlo simulations the effect of random bond dilution on the phase transition of the three-dimensional 4-state Potts model which is known to exhibit a strong first-order transition in the pure case. The phase diagram in the dilution-temperature plane is determined from the peaks of the susceptibility for sufficiently large system sizes. In the strongly disordered regime, numerical evidence for softening to a second-order transition induced by randomness is given. Here a large-scale finite-size scaling analysis, made difficult due to strong crossover effects presumably caused by the percolation fixed point, is performed.Comment: LaTeX file with Revtex, 4 pages, 4 eps figure

gl(N|N) Super-Current Algebras for Disordered Dirac Fermions in Two Dimensions

We consider the non-hermitian 2D Dirac Hamiltonian with (A): real random mass, imaginary scalar potential and imaginary gauge field potentials, and (B) arbitrary complex random potentials of all three kinds. In both cases this Hamiltonian gives rise to a delocalization transition at zero energy with particle-hole symmetry in every realization of disorder. Case (A) is in addition time-reversal invariant, and can also be interpreted as the random-field XY Statistical Mechanics model in two dimensions. The supersymmetric approach to disorder averaging results in current-current perturbations of $gl(N|N)$ super-current algebras. Special properties of the $gl(N|N)$ algebra allow the exact computation of the beta-functions, and of the correlation functions of all currents. One of them is the Edwards-Anderson order parameter. The theory is `nearly conformal' and possesses a scale-invariant subsector which is not a current algebra. For N=1, in addition, we obtain an exact solution of all correlation functions. We also study the delocalization transition of case (B), with broken time reversal symmetry, in the Gade-Wegner (Random-Flux) universality class, using a GL(N|N;C)/U(N|N) sigma model, as well as its PSL(N|N) variant, and a corresponding generalized random XY model. For N=1 the sigma model is shown to be identical to the current-current perturbation. For the delocalization transitions (case (A) and (B)) a density of states, diverging at zero energy, is found.Comment: LaTeX, 40 page

Correlation decay and conformal anomaly in the two-dimensional random-bond Ising ferromagnet

The two-dimensional random-bond Ising model is numerically studied on long strips by transfer-matrix methods. It is shown that the rate of decay of correlations at criticality, as derived from averages of the two largest Lyapunov exponents plus conformal invariance arguments, differs from that obtained through direct evaluation of correlation functions. The latter is found to be, within error bars, the same as in pure systems. Our results confirm field-theoretical predictions. The conformal anomaly $c$ is calculated from the leading finite-width correction to the averaged free energy on strips. Estimates thus obtained are consistent with $c=1/2$, the same as for the pure Ising model.Comment: RevTeX 3, 11 pages +2 figures, uuencoded, IF/UFF preprin

Conformal Field Theory Approach to the 2-Impurity Kondo Problem: Comparison with Numerical Renormalization Group Results

Numerical renormalization group and conformal field theory work indicate that the two impurity Kondo Hamiltonian has a non-Fermi liquid critical point separating the Kondo-screening phase from the inter-impurity singlet phase when particle-hole (P-H) symmetry is maintained. We clarify the circumstances under which this critical point occurs, pointing out that there are two types of P-H symmetry. Only one of them guarantees the occurance of the critical point. Much of the previous numerical work was done on models with the other type of P-H symmetry. We analyse this critical point using the boundary conformal field theory technique. The finite-size spectrum is presented in detail and compared with about 50 energy levels obtained using the numerical renormalization group. Various Green's functions, general renormalization group behaviour, and a hidden $SO(7)$ are analysed.Comment: 38 pages, RevTex. 2 new sections clarify the circumstances under which a model will exhibit the non-trivial critical point (hence potentially resolving disagreements with other Authors) and explain the hidden SO(7) symmetry of the model, relating it to an alternative approach of Sire et al. and Ga

Explicit Renormalization Group for D=2 random bond Ising model with long-range correlated disorder

We investigate the explicit renormalization group for fermionic field theoretic representation of two-dimensional random bond Ising model with long-range correlated disorder. We show that a new fixed point appears by introducing a long-range correlated disorder. Such as the one has been observed in previous works for the bosonic ($\phi^4$) description. We have calculated the correlation length exponent and the anomalous scaling dimension of fermionic fields at this fixed point. Our results are in agreement with the extended Harris criterion derived by Weinrib and Halperin.Comment: 5 page

Critical behavior of weakly-disordered anisotropic systems in two dimensions

The critical behavior of two-dimensional (2D) anisotropic systems with weak quenched disorder described by the so-called generalized Ashkin-Teller model (GATM) is studied. In the critical region this model is shown to be described by a multifermion field theory similar to the Gross-Neveu model with a few independent quartic coupling constants. Renormalization group calculations are used to obtain the temperature dependence near the critical point of some thermodynamic quantities and the large distance behavior of the two-spin correlation function. The equation of state at criticality is also obtained in this framework. We find that random models described by the GATM belong to the same universality class as that of the two-dimensional Ising model. The critical exponent $\nu$ of the correlation length for the 3- and 4-state random-bond Potts models is also calculated in a 3-loop approximation. We show that this exponent is given by an apparently convergent series in $\epsilon=c-\frac{1}{2}$ (with $c$ the central charge of the Potts model) and that the numerical values of $\nu$ are very close to that of the 2D Ising model. This work therefore supports the conjecture (valid only approximately for the 3- and 4-state Potts models) of a superuniversality for the 2D disordered models with discrete symmetries.Comment: REVTeX, 24 pages, to appear in Phys.Rev.

Numerical Replica Limit for the Density Correlation of the Random Dirac Fermion

The zero mode wave function of a massless Dirac fermion in the presence of a random gauge field is studied. The density correlation function is calculated numerically and found to exhibit power law in the weak randomness with the disorder dependent exponent. It deviates from the power law and the disorder dependence becomes frozen in the strong randomness. A classical statistical system is employed through the replica trick to interpret the results and the direct evaluation of the replica limit is demonstrated numerically. The analytic expression of the correlation function and the free energy are also discussed with the replica symmetry breaking and the Liouville field theory.Comment: 5 pages, 4 figures, REVTe