The Ma-Dasgupta-Hu renormalization group (RG) scheme is used to study singular quantities in the Griffiths phase of random quantum spin chains. For the random transverse-field Ising spin chain we have extended Fisher's analytical solution to the off-critical region and calculated the dynamical exponent exactly. Concerning other random chains we argue by scaling considerations that the RG method generally becomes asymptotically exact for large times, both at the critical point and in the whole Griffiths phase. This statement is checked via numerical calculations on the random Heisenberg and quantum Potts models by the density matrix renormalization group method.In a random quantum system at zero temperature several physical quantities are singular not only at the critical point, but in a whole region, as well, which extends on both sides of the transition point [1]. In this Griffiths phase [2] a random quantum system is non-critical in the space direction (spatial correlations decay exponentially), whereas it is critical in the time direction and the corresponding behavior due to Griffiths-McCoy singularities [2,3] is controlled by a line of semi-critical fixed points characterized by the dynamical exponent z(δ), which depends on the value of the quantum control-parameter, δ. For example average autocorrelations decay in (imaginary) time, τ , as ∼ τ −z(δ) ; for a small magnetic field, H → 0, the magnetization behaves as ∼ H 1/z(δ) ; the low temperature susceptibility and specific heat are singular as ∼ T 1/z(δ)−1 and ∼ T 1/z(δ) , respectively. Let us mention that Griffiths-McCoy singularities are relevant in experiments on quantum spin glasses [4] and provides a theoretical explanation about the non-Fermi liquid behavior in U and Ce intermetallics [5].Among the theoretical methods developed to study random quantum systems the renormalization group (RG) scheme introduced by Ma, Dasgupta and Hu [6] plays a special rôle. For a class of systems, the critical behavior of those is controlled by an infiniterandomness fixed point [7] (IRFP), the RG method becomes asymptotically exact during iteration. For some one-dimensional models, random transverse-field Ising model [8] (RTIM) and the random XXZ-model [9], Fisher has obtained analytical solution of the RG equations and in this way many new exact results and new physical insight about the critical behavior of these models have been gained. Subsequent analytical [10] and numerical [11,10] investigations of the models are in agreement with Fisher's results. The RG-scheme has been numerically implemented in higher dimensions [7,12], as well, to study the critical behavior of the RTIM and reasonable agreement with the results of quantum MonteCarlo simulations [13] has been found. Considering the Griffiths-phase of random quantum spin chains here the RG-scheme has been rarely used [12], mainly due to the general belief that the method looses its asymptotically exact properties by leaving the vicinity of the scale invariant critical point.Our aim in the present Letter is to ...

The effect of quenched disorder on the low-energy properties of various antiferromagnetic spin ladder models is studied by a numerical strong disorder renormalization group method and by density matrix renormalization. For strong enough disorder the originally gapped phases with finite topological or dimer order become gapless. In these quantum Griffiths phases the scaling of the energy, as well as the singularities in the dynamical quantities are characterized by a finite dynamical exponent, z, which varies with the strength of disorder. At the phase boundaries, separating topologically distinct Griffiths phases the singular behavior of the disordered ladders is generally controlled by an infinite randomness fixed point.

Critical properties of quantum spin chains with varying degrees of disorder are studied at zero temperature by analytical and extensive density matrix renormalization methods. Generally the phase diagram is found to contain three phases. The weak disorder regime, where the critical behavior is controlled by the fixed points of the pure system, and the strong disorder regime, which is attracted by an infinite randomness fixed point, are separated by an intermediate disorder regime, where dynamical scaling is anisotropic and the static and dynamical exponents are disorder dependent.

We use extensive density matrix renormalization group (DMRG) calculations to explore the phase diagram of the random S = 1 antiferromagnetic Heisenberg chain with a power-law distribution of the exchange couplings. We use open chains and monitor the lowest gaps, the end-to-end correlation function and the string order parameter. For this distribution, at weak disorder the system is in the gapless Haldane phase with a disorder dependent dynamical exponent, z, and z = 1 signals the border between the nonsingular and singular regions of the local susceptibility. For strong enough disorder, which approximately corresponds to a uniform distribution, a transition into the random singlet phase is detected, at which the string order parameter as well as the average end-toend correlation function are vanishing and at the same time the dynamical exponent is divergent. Singularities of physical quantities are found to be somewhat different in the random singlet phase and in the critical point.

Ising models with nearest-neighbor ferromagnetic random couplings on a square lattice with a (1,1) surface are studied, using Monte Carlo techniques and a star-triangle transformation method. In particular, the critical exponent of the surface magnetization is found to be close to that of the perfect model, β s =1/2. The crossover from surface to bulk critical properties is discussed.KEY WORDS: random Ising model; surface magnetization; Monte Carlo simulations Model and methodsThe bulk critical behavior of the two-dimensional dilute Ising model has been studied extensively in recent years. [1][2][3][4] According to renormalization group calculations, the randomness leads, at least in the limit of weak dilution, to logarithmic modifications of the asymptotic power-laws for various quantities in the perfect model, in agreement with results of Monte Carlo simulations (however, also conflicting interpretations of numerical results have been suggested and discussed 3,5 ). In particular, the bulk magnetization, m b , is expected to vanish aswhere t is the reduced temperature, t = (T c − T )/T c . 1In this Communication we shall present findings on surface critical properties of nearestneighbor random spin-1/2 Ising models on a square lattice with a surface. Randomness is introduced by allowing the nearest-neighbor ferromagnetic couplings to take two values, J 1 and J 2 , where J 1 is greater or equal to J 2 . If both couplings occur with the same probability, then the model is self-dual. 6 The self-dual point is located atdetermining the critical temperature, if the model undergoes one phase transition. Indeed, results of simulations 7 strongly support that assumption.Most of our findings are based on extensive Monte Carlo (MC) simulations, using singlespin and cluster-flip algorithms. To facilitate comparison of the simulational data with those of our numerical evaluation of the star-triangle transformation (ST) method 8,9 , we study the Ising model with a surface in the diagonal or (1,1) direction. In that case, the coordination number of the surface spins is two. (Indeed, we believe the critical properties at this ordinary surface transition to be the same for the (1,1) and the (1,0) direction, as it is known to be the case in the perfect model). In the simulations, we consider lattices consisting of K columns and L rows, where the first and last columns are surface lines; the first and last rows are connected by periodic boundary conditions. Usually, we set L = K/2, with K ranging from 40 to 1280. The ST method, which was originally developed for layered lattices 8 , is generalized here to treat general inhomogeneous systems. In these calculations, K is proportional to the number of iterations and goes to infinity, while L, the number of surface sites, remains finite. In both methods, MC and ST, one has to average over an ensemble of bond configurations. Typically, the number of realizations ranged, in the simulations, from about 20 to several hundreds, taking more configurations for smaller system sizes. ...