Out-of-equilibrium critical dynamics at surfaces: Cluster dissolution and non-algebraic correlations

We study nonequilibrium dynamical properties at a free surface after the system is quenched from the high-temperature phase into the critical point. We show that if the spatial surface correlations decay sufficiently rapidly the surface magnetization and/or the surface manifold autocorrelations has a qualitatively different universal short time behavior than the same quantities in the bulk. At a free surface cluster dissolution may take place instead of domain growth yielding stationary dynamical correlations that decay in a stretched exponential form. This phenomenon takes place in the three-dimensional Ising model and should be observable in real ferromagnets.Comment: 4 pages, 4 figure

Exact renormalization of the random transverse-field Ising spin chain in the strongly ordered and strongly disordered Griffiths phases

The real-space renormalization group (RG) treatment of random transverse-field Ising spin chains by Fisher ({\it Phys. Rev. B{\bf 51}, 6411 (1995)}) has been extended into the strongly ordered and strongly disordered Griffiths phases and asymptotically exact results are obtained. In the non-critical region the asymmetry of the renormalization of the couplings and the transverse fields is related to a non-linear quantum control parameter, $\Delta$, which is a natural measure of the distance from the quantum critical point. $\Delta$, which is found to stay invariant along the RG trajectories and has been expressed by the initial disorder distributions, stands in the singularity exponents of different physical quantities (magnetization, susceptibility, specific heat, etc), which are exactly calculated. In this way we have observed a weak-universality scenario: the Griffiths-McCoy singularities does not depend on the form of the disorder, provided the non-linear quantum control parameter has the same value. The exact scaling function of the magnetization with a small applied magnetic field is calculated and the critical point magnetization singularity is determined in a simple, direct way.Comment: 11 page

Random antiferromagnetic quantum spin chains: Exact results from scaling of rare regions

We study XY and dimerized XX spin-1/2 chains with random exchange couplings by analytical and numerical methods and scaling considerations. We extend previous investigations to dynamical properties, to surface quantities and operator profiles, and give a detailed analysis of the Griffiths phase. We present a phenomenological scaling theory of average quantities based on the scaling properties of rare regions, in which the distribution of the couplings follows a surviving random walk character. Using this theory we have obtained the complete set of critical decay exponents of the random XY and XX models, both in the volume and at the surface. The scaling results are confronted with numerical calculations based on a mapping to free fermions, which then lead to an exact correspondence with directed walks. The numerically calculated critical operator profiles on large finite systems (L<=512) are found to follow conformal predictions with the decay exponents of the phenomenological scaling theory. Dynamical correlations in the critical state are in average logarithmically slow and their distribution show multi-scaling character. In the Griffiths phase, which is an extended part of the off-critical region average autocorrelations have a power-law form with a non-universal decay exponent, which is analytically calculated. We note on extensions of our work to the random antiferromagnetic XXZ chain and to higher dimensions.Comment: 19 pages RevTeX, eps-figures include

Transverse-field Ising spin chain with inhomogeneous disorder

We consider the critical and off-critical properties at the boundary of the random transverse-field Ising spin chain when the distribution of the couplings and/or transverse fields, at a distance $l$ from the surface, deviates from its uniform bulk value by terms of order $l^{-\kappa}$ with an amplitude $A$. Exact results are obtained using a correspondence between the surface magnetization of the model and the surviving probability of a random walk with time-dependent absorbing boundary conditions. For slow enough decay, $\kappa<1/2$, the inhomogeneity is relevant: Either the surface stays ordered at the bulk critical point or the average surface magnetization displays an essential singularity, depending on the sign of $A$. In the marginal situation, $\kappa=1/2$, the average surface magnetization decays as a power law with a continuously varying, $A$-dependent, critical exponent which is obtained analytically. The behavior of the critical and off-critical autocorrelation functions as well as the scaling form of the probability distributions for the surface magnetization and the first gaps are determined through a phenomenological scaling theory. In the Griffiths phase, the properties of the Griffiths-McCoy singularities are not affected by the inhomogeneity. The various results are checked using numerical methods based on a mapping to free fermions.Comment: 11 pages (Revtex), 11 figure

Disorder Induced Phases in Higher Spin Antiferromagnetic Heisenberg Chains

Extensive DMRG calculations for spin S=1/2 and S=3/2 disordered antiferromagnetic Heisenberg chains show a rather distinct behavior in the two cases. While at sufficiently strong disorder both systems are in a random singlet phase, we show that weak disorder is an irrelevant perturbation for the S=3/2 chain, contrary to what expected from a naive application of the Harris criterion. The observed irrelevance is attributed to the presence of a new correlation length due to enhanced end-to-end correlations. This phenomenon is expected to occur for all half-integer S > 1/2 chains. A possible phase diagram of the chain for generic S is also discussed.Comment: 6 Pages and 6 figures. Final version as publishe

Anisotropic Scaling in Layered Aperiodic Ising Systems

The influence of a layered aperiodic modulation of the couplings on the critical behaviour of the two-dimensional Ising model is studied in the case of marginal perturbations. The aperiodicity is found to induce anisotropic scaling. The anisotropy exponent z, given by the sum of the surface magnetization scaling dimensions, depends continuously on the modulation amplitude. Thus these systems are scale invariant but not conformally invariant at the critical point.Comment: 7 pages, 2 eps-figures, Plain TeX and epsf, minor correction

Griffiths-McCoy singularities in random quantum spin chains: Exact results through renormalization

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.Comment: 4 pages RevTeX, 2 figures include

The McCoy-Wu Model in the Mean-field Approximation

We consider a system with randomly layered ferromagnetic bonds (McCoy-Wu model) and study its critical properties in the frame of mean-field theory. In the low-temperature phase there is an average spontaneous magnetization in the system, which vanishes as a power law at the critical point with the critical exponents $\beta \approx 3.6$ and $\beta_1 \approx 4.1$ in the bulk and at the surface of the system, respectively. The singularity of the specific heat is characterized by an exponent $\alpha \approx -3.1$. The samples reduced critical temperature $t_c=T_c^{av}-T_c$ has a power law distribution $P(t_c) \sim t_c^{\omega}$ and we show that the difference between the values of the critical exponents in the pure and in the random system is just $\omega \approx 3.1$. Above the critical temperature the thermodynamic quantities behave analytically, thus the system does not exhibit Griffiths singularities.Comment: LaTeX file with iop macros, 13 pages, 7 eps figures, to appear in J. Phys.

Logarithmic corrections in the two-dimensional Ising model in a random surface field

In the two-dimensional Ising model weak random surface field is predicted to be a marginally irrelevant perturbation at the critical point. We study this question by extensive Monte Carlo simulations for various strength of disorder. The calculated effective (temperature or size dependent) critical exponents fit with the field-theoretical results and can be interpreted in terms of the predicted logarithmic corrections to the pure system's critical behaviour.Comment: 10 pages, 4 figures, extended version with one new sectio

Crossover between aperiodic and homogeneous semi-infinite critical behaviors in multilayered two-dimensional Ising models

We investigate the surface critical behavior of two-dimensional multilayered aperiodic Ising models in the extreme anisotropic limit. The system under consideration is obtained by piling up two types of layers with respectively $p$ and $q$ spin rows coupled via nearest neighbor interactions $\lambda r$ and $\lambda$, where the succession of layers follows an aperiodic sequence. Far away from the critical regime, the correlation length $\xi_\perp$ is smaller than the first layer width and the system exhibits the usual behavior of an ordinary surface transition. In the other limit, in the neighborhood of the critical point, $\xi_\perp$ diverges and the fluctuations are sensitive to the non-periodic structure of the system so that the critical behavior is governed by a new fixed point. We determine the critical exponent associated to the surface magnetization at the aperiodic critical point and show that the expected crossover between the two regimes is well described by a scaling function. From numerical calculations, the parallel correlation length $\xi_\parallel$ is then found to behave with an anisotropy exponent $z$ which depends on the aperiodic modulation and the layer widths.Comment: LaTeX file, 9 pages, 8 eps figures, to appear in Phys. Rev.