Screening of charged singularities of random fields

Many types of point singularity have a topological index, or 'charge', associated with them. For example the phase of a complex field depending on two variables can either increase or decrease on making a clockwise circuit around a simple zero, enabling the zeros to be assigned charges of plus or minus one. In random fields we can define a correlation function for the charge-weighted density of singularities. In many types of random fields, this correlation function satisfies an identity which shows that the singularities 'screen' each other perfectly: a positive singularity is surrounded by an excess of concentration of negatives which exactly cancel its charge, and vice-versa. This paper gives a simple and widely applicable derivation of this result. A counterexample where screening is incomplete is also exhibited.Comment: 12 pages, no figures. Minor revision of manuscript submitted to J. Phys. A, August 200

An alternative field theory for the Kosterlitz-Thouless transition

We extend a Gaussian model for the internal electrical potential of a two-dimensional Coulomb gas by a non-Gaussian measure term, which singles out the physically relevant configurations of the potential. The resulting Hamiltonian, expressed as a functional of the internal potential, has a surprising large-scale limit: The additional term simply counts the number of maxima and minima of the potential. The model allows for a transparent derivation of the divergence of the correlation length upon lowering the temperature down to the Kosterlitz-Thouless transition point.Comment: final version, extended discussion, appendix added, 8 pages, no figure, uses IOP documentclass iopar

Microscopic mechanism for the 1/8 magnetization plateau in SrCu_2(BO_3)_2

The frustrated quantum magnet SrCu_2(BO_3)_2 shows a remarkably rich phase diagram in an external magnetic field including a sequence of magnetization plateaux. The by far experimentally most studied and most prominent magnetization plateau is the 1/8 plateau. Theoretically, one expects that this material is well described by the Shastry-Sutherland model. But recent microscopic calculations indicate that the 1/8 plateau is energetically not favored. Here we report on a very simple microscopic mechanism which naturally leads to a 1/8 plateau for realistic values of the magnetic exchange constants. We show that the 1/8 plateau with a diamond unit cell benefits most compared to other plateau structures from quantum fluctuations which to a large part are induced by Dzyaloshinskii-Moriya interactions. Physically, such couplings result in kinetic terms in an effective hardcore boson description leading to a renormalization of the energy of the different plateaux structures which we treat in this work on the mean-field level. The stability of the resulting plateaux are discussed. Furthermore, our results indicate a series of stripe structures above 1/8 and a stable magnetization plateau at 1/6. Most qualitative aspects of our microscopic theory agree well with a recently formulated phenomenological theory for the experimental data of SrCu_2(BO_3)_2. Interestingly, our calculations point to a rather large ratio of the magnetic couplings in the Shastry-Sutherland model such that non-perturbative effects become essential for the understanding of the frustrated quantum magnet SrCu_2(BO_3)_2.Comment: 24 pages, 24 figure

Distributions of absolute central moments for random walk surfaces

We study periodic Brownian paths, wrapped around the surface of a cylinder. One characteristic of such a path is its width square, $w^2$, defined as its variance. Though the average of $w^2$ over all possible paths is well known, its full distribution function was investigated only recently. Generalising $w^2$ to $w^{(N)}$, defined as the $N$-th power of the {\it magnitude} of the deviations of the path from its mean, we show that the distribution functions of these also scale and obtain the asymptotic behaviour for both large and small $w^{(N)}$

Magnetization of SrCu2(BO3)2 in ultrahigh magnetic fields up to 118 T

The magnetization process of the orthogonal-dimer antiferromagnet SrCu2(BO3)2 is investigated in high magnetic fields of up to 118 T. A 1/2 plateau is clearly observed in the field range 84 to 108 T in addition to 1/8, 1/4 and 1/3 plateaux at lower fields. Using a combination of state-of-the-art numerical simulations, the main features of the high-field magnetization, a 1/2 plateau of width 24 T, a 1/3 plateau of width 34 T, and no 2/5 plateau, are shown to agree quantitatively with the Shastry-Sutherland model if the ratio of inter- to intra-dimer exchange interactions J'/J=0.63. It is further predicted that the intermediate phase between the 1/3 and 1/2 plateau is not uniform but consists of a 1/3 supersolid followed by a 2/5 supersolid and possibly a domain-wall phase, with a reentrance into the 1/3 supersolid above the 1/2 plateau.Comment: 5 pages + 10 pages supplemental materia

Critical holes in undercooled wetting layers

The profile of a critical hole in an undercooled wetting layer is determined by the saddle-point equation of a standard interface Hamiltonian supported by convenient boundary conditions. It is shown that this saddle-point equation can be mapped onto an autonomous dynamical system in a three-dimensional phase space. The corresponding flux has a polynomial form and in general displays four fixed points, each with different stability properties. On the basis of this picture we derive the thermodynamic behaviour of critical holes in three different nucleation regimes of the phase diagram.Comment: 18 pages, LaTeX, 6 figures Postscript, submitted to J. Phys.

Random wave functions and percolation

Recently it was conjectured that nodal domains of random wave functions are adequately described by critical percolation theory. In this paper we strengthen this conjecture in two respects. First, we show that, though wave function correlations decay slowly, a careful use of Harris' criterion confirms that these correlations are unessential and nodal domains of random wave functions belong to the same universality class as non critical percolation. Second, we argue that level domains of random wave functions are described by the non-critical percolation model.Comment: 13 page

The Statistics of the Points Where Nodal Lines Intersect a Reference Curve

We study the intersection points of a fixed planar curve $\Gamma$ with the nodal set of a translationally invariant and isotropic Gaussian random field \Psi(\bi{r}) and the zeros of its normal derivative across the curve. The intersection points form a discrete random process which is the object of this study. The field probability distribution function is completely specified by the correlation G(|\bi{r}-\bi{r}'|) = . Given an arbitrary G(|\bi{r}-\bi{r}'|), we compute the two point correlation function of the point process on the line, and derive other statistical measures (repulsion, rigidity) which characterize the short and long range correlations of the intersection points. We use these statistical measures to quantitatively characterize the complex patterns displayed by various kinds of nodal networks. We apply these statistics in particular to nodal patterns of random waves and of eigenfunctions of chaotic billiards. Of special interest is the observation that for monochromatic random waves, the number variance of the intersections with long straight segments grows like $L \ln L$, as opposed to the linear growth predicted by the percolation model, which was successfully used to predict other long range nodal properties of that field.Comment: 33 pages, 13 figures, 1 tabl

Dynamic Scaling of Width Distribution in Edwards--Wilkinson Type Models of Interface Dynamics

Edwards--Wilkinson type models are studied in 1+1 dimensions and the time-dependent distribution, P_L(w^2,t), of the square of the width of an interface, w^2, is calculated for systems of size L. We find that, using a flat interface as an initial condition, P_L(w^2,t) can be calculated exactly and it obeys scaling in the form _\infty P_L(w^2,t) = Phi(w^2 / _\infty, t/L^2) where _\infty is the stationary value of w^2. For more complicated initial states, scaling is observed only in the large- time limit and the scaling function depends on the initial amplitude of the longest wavelength mode. The short-time limit is also interesting since P_L(w^2,t) is found to closely approximate the log-normal distribution. These results are confirmed by Monte Carlo simulations on a `roof-top' model of surface evolution.Comment: 5 pages, latex, 3 ps figures in a separate files, submitted to Phys.Rev.