Kollektív viselkedés két dimenzióban = Collective behaviour in two-dimensions

A 2D rendszerek kollektiv viselkedésének formáit és törvényszerűségeit tanulmányoztuk úgy kvantummechanikai, mint klasszikus sokrészecskés esetekben. a) Kvantummechanikai rendszerekre módszereket dolgoztunk ki egzakt alapállapotok levezetésére még nem-integrálható esetekre is. Ezeket 2D-ben alkalmazva fém-szigetelő átmenetet mutattunk ki rendezetlen és kölcsönható rendszerekben, stripe és sakktábla fázisokat vezettünk le illetve normálfázisú nem-Fermi folyadék és szigetelő fázisokat kaptunk. b) Kerámiák esetében nemegyensúlyi körülmények között végzett mérésekkel nőveltük a mérések lokális felbontóképességét és a szupravezető Tc feletti tartományban különböző korrelációs hossz és koherencia élettartammal rendelkező tartományt mutattunk ki. c) Elemeztük granulált anyagok erőláncait, dipolusok aggregációs és kristályosodási folyamatait és mágneses zaj létrejöttét törésben. d) Rendezetlen mágnesek esetében javaslatot tettünk egy optimalizációs algoritmusra. Ezen hiszterézises optimalizáció működőképességét spinüveg modelleken és az utazó ügynök problémáján demonstráltuk. e) Epitaxiális felületnővekedés esetében egy térfogati diffúziót, spinodális dekompoziciót és a felületnövekedést magába foglaló modell segitségével megvizsgáltuk az epitaxiális növekedés során fellépő önszervező spontán kompozició modulációkat. Három különböző növekedési módust kaptunk eredményül, amelyek mindegyikében egydimenziós laterális kompozició modulációk alakulnak ki. | We have studied the possibilities and principles of the 2D collective behavior for many body systems holding both quantum mechanical, or classical properties. a) For the quantum case we elaborated procedures which allow the deduction of exact ground states even in non-integrable cases. These used in 2D led to metal-insulator transition for disordered and interacting systems, stripe and checkerboards, normal phase non-Fermi liquids and insulators. b) For ceramics, by measurements effectuated under non-equilibrium conditions we have increased the local resolution of measurements and we have found above the superconducting Tc several regions described by different correlation lengths and coherence lifetimes. c) We studied the force chains in granular media, aggregation and crystallization in dipolar monolayers and magnetic noise during fracture. d) For the case of random magnet systems we introduced an optimization algorithm. We demonstrated the performances of this hysteretic optimization method on spin glass models and on the traveling salesman problem. e) For the surface epitaxial growth case we constructed a model including bulk diffusion, spinodal decomposition, and surface growth to study the self-organized superlattice formation during the process under consideration. We found three growth regimes in which one dimensional lateral composition modulations occur

Disorder Averaging and Finite Size Scaling

We propose a new picture of the renormalization group (RG) approach in the presence of disorder, which considers the RG trajectories of each random sample (realization) separately instead of the usual renormalization of the averaged free energy. The main consequence of the theory is that the average over randomness has to be taken after finding the critical point of each realization. To demonstrate these concepts, we study the finite-size scaling properties of the two-dimensional random-bond Ising model. We find that most of the previously observed finite-size corrections are due to the sample-to-sample fluctuation of the critical temperature and scaling is more adequate in terms of the new scaling variables.Comment: 4 pages, 6 figures include

Mean Field Theory of the Localization Transition

A mean field theory of the localization transition for bosonic systems is developed. Localization is shown to be sensitive to the distribution of the random site energies. It occurs in the presence of a triangular distribution, but not a uniform one. The inverse participation ratio, the single site Green's function, the superfluid order parameter and the corresponding susceptibility are calculated, and the appropriate exponents determined. All of these quantities indicate the presence of a new phase, which can be identified as the {\it Bose-glass}.Comment: 4 pages, Revtex, 2 figures appende

Direct Mott Insulator-to-Superfluid Transition in the Presence of Disorder

We introduce a new renormalization group theory to examine the quantum phase transitions upon exiting the insulating phase of a disordered, strongly interacting boson system. For weak disorder we find a direct transition from this Mott insulator to the Superfluid phase. In d > 4 a finite region around the particle-hole symmetric point supports this direct transition, whereas for 2=< d <4 perturbative arguments suggest that the direct transition survives only precisely at commensurate filling. For strong disorder the renormalization trajectories pass next to two fixed points, describing a pair of distinct transitions; first from the Mott insulator to the Bose glass, and then from the Bose glass to the Superfluid. The latter fixed point possesses statistical particle-hole symmetry and a dynamical exponent z, equal to the dimension d.Comment: 4 pages, Latex, submitted to Physical Review Letter

Revisiting the Theory of Finite Size Scaling in Disordered Systems: \nu Can Be Less Than 2/d

For phase transitions in disordered systems, an exact theorem provides a bound on the finite size correlation length exponent: \nu_{FS}<= 2/d. It is believed that the true critical exponent \nu of a disorder induced phase transition satisfies the same bound. We argue that in disordered systems the standard averaging introduces a noise, and a corresponding new diverging length scale, characterized by \nu_{FS}=2/d. This length scale, however, is independent of the system's own correlation length \xi. Therefore \nu can be less than 2/d. We illustrate these ideas on two exact examples, with \nu < 2/d. We propose a new method of disorder averaging, which achieves a remarkable noise reduction, and thus is able to capture the true exponents.Comment: 4 pages, Latex, one figure in .eps forma

Self-organized criticality in the hysteresis of the Sherrington - Kirkpatrick model

We study hysteretic phenomena in random ferromagnets. We argue that the angle dependent magnetostatic (dipolar) terms introduce frustration and long range interactions in these systems. This makes it plausible that the Sherrington - Kirkpatrick model may be able to capture some of the relevant physics of these systems. We use scaling arguments, replica calculations and large scale numerical simulations to characterize the hysteresis of the zero temperature SK model. By constructing the distribution functions of the avalanche sizes, magnetization jumps and local fields, we conclude that the system exhibits self-organized criticality everywhere on the hysteresis loop.Comment: 4 pages, 4 eps figure

Random quantum magnets with long-range correlated disorder: Enhancement of critical and Griffiths-McCoy singularities

We study the effect of spatial correlations in the quenched disorder on random quantum magnets at and near a quantum critical point. In the random transverse field Ising systems disorder correlations that decay algebraically with an exponent rho change the universality class of the transition for small enough rho and the off-critical Griffiths-McCoy singularities are enhanced. We present exact results for 1d utilizing a mapping to fractional Brownian motion and generalize the predictions for the critical exponents and the generalized dynamical exponent in the Griffiths phase to d>=2.Comment: 4 pages RevTeX, 1 eps-figure include