Using the Ferrenberg-Swendsen Monte Carlo method for nonuniversal exponents

We use the method of Monte Carlo evolution in the coupling constant space of Ferrenberg and Swendsen to evaluate the nonuniversal exponent η* associated to a linear defect in a 2d Ising model. © 1989.

Simulação de Monte Carlo: de modelos de spin à teoria de campos na rede

Pós-graduação em Física - IFT

Partially Observed Markov Random Fields Are Variable Neighborhood Random Fields

The present paper has two goals. First to present a natural example of a new class of random fields which are the variable neighborhood random fields. The example we consider is a partially observed nearest neighbor binary Markov random field. The second goal is to establish sufficient conditions ensuring that the variable neighborhoods are almost surely finite. We discuss the relationship between the almost sure finiteness of the interaction neighborhoods and the presence/absence of phase transition of the underlying Markov random field. In the case where the underlying random field has no phase transition we show that the finiteness of neighborhoods depends on a specific relation between the noise level and the minimum values of the one-point specification of the Markov random field. The case in which there is phase transition is addressed in the frame of the ferromagnetic Ising model. We prove that the existence of infinite interaction neighborhoods depends on the phase.

Progress in the mathematical theory of quantum disordered systems

We review recent progress in the mathematical theory of quantum disordered systems: the Anderson transition, including some joint work with Marchetti, the (quantum and classical) Edwards-Anderson (EA) spin-glass model and return to equilibrium for a class of spin-glass models, which includes the EA model initially in a very large transverse magnetic field. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4770066]

Rounding of a first-order quantum phase transition to a strong-coupling critical point

We investigate the effects of quenched disorder on first-order quantum phase transitions on the example of the N-color quantum Ashkin-Teller model. By means of a strong-disorder renormalization group, we demonstrate that quenched disorder rounds the first-order quantum phase transition to a continuous one for both weak and strong coupling between the colors. In the strong-coupling case, we find a distinct type of infinite-randomness critical point characterized by additional internal degrees of freedom. We investigate its critical properties in detail and find stronger thermodynamic singularities than in the random transverse field Ising chain. We also discuss the implications for higher spatial dimensions as well as unusual aspects of our renormalization-group scheme. DOI: 10.1103/PhysRevB.86.214204

Finite-size corrections of the entanglement entropy of critical quantum chains

Using the density matrix renormalization group, we calculated the finite-size corrections of the entanglement alpha-Renyi entropy of a single interval for several critical quantum chains. We considered models with U(1) symmetry such as the spin-1/2 XXZ and spin-1 Fateev-Zamolodchikov models, as well as models with discrete symmetries such as the Ising, the Blume-Capel, and the three-state Potts models. These corrections contain physically relevant information. Their amplitudes, which depend on the value of a, are related to the dimensions of operators in the conformal field theory governing the long-distance correlations of the critical quantum chains. The obtained results together with earlier exact and numerical ones allow us to formulate some general conjectures about the operator responsible for the leading finite-size correction of the alpha-Renyi entropies. We conjecture that the exponent of the leading finite-size correction of the alpha-Renyi entropies is p(alpha) = 2X(epsilon)/alpha for alpha > 1 and p(1) = nu, where X-epsilon denotes the dimensions of the energy operator of the model and nu = 2 for all the models.

Generalized Metropolis dynamics with a generalized master equation: An approach for time-independent and time-dependent Monte Carlo simulations of generalized spin systems

The extension of Boltzmann-Gibbs thermostatistics, proposed by Tsallis, introduces an additional parameter q to the inverse temperature beta. Here, we show that a previously introduced generalized Metropolis dynamics to evolve spin models is not local and does not obey the detailed energy balance. In this dynamics, locality is only retrieved for q = 1, which corresponds to the standard Metropolis algorithm. Nonlocality implies very time-consuming computer calculations, since the energy of the whole system must be reevaluated when a single spin is flipped. To circumvent this costly calculation, we propose a generalized master equation, which gives rise to a local generalized Metropolis dynamics that obeys the detailed energy balance. To compare the different critical values obtained with other generalized dynamics, we perform Monte Carlo simulations in equilibrium for the Ising model. By using short-time nonequilibrium numerical simulations, we also calculate for this model the critical temperature and the static and dynamical critical exponents as functions of q. Even for q not equal 1, we show that suitable time-evolving power laws can be found for each initial condition. Our numerical experiments corroborate the literature results when we use nonlocal dynamics, showing that short-time parameter determination works also in this case. However, the dynamics governed by the new master equation leads to different results for critical temperatures and also the critical exponents affecting universality classes. We further propose a simple algorithm to optimize modeling the time evolution with a power law, considering in a log-log plot two successive refinements.

Exact correlation functions in particle-reaction models with immobile particles

Exact results on particle densities as well as correlators in two models of immobile particles, containing either a single species or else two distinct species, are derived. The models evolve following a descent dynamics through pair annihilation where each particle interacts once at most throughout its entire history. The resulting large number of stationary states leads to a non-vanishing configurational entropy. Our results are established for arbitrary initial conditions and are derived via a generating function method. The single-species model is the dual of the 1D zero-temperature kinetic Ising model with Kimball-Deker-Haake dynamics. In this way, both in finite and semi-infinite chains and also the Bethe lattice can be analysed. The relationship with the random sequential adsorption of dimers and weakly tapped granular materials is discussed.

Spin-glass behaviour on random lattices

The ground-state phase diagram of an Ising spin-glass model on a random graph with an arbitrary fraction w of ferromagnetic interactions is analysed in the presence of an external field. Using the replica method, and performing an analysis of stability of the replica-symmetric solution, it is shown that w = 1/2, corresponding to an unbiased spin glass, is a singular point in the phase diagram, separating a region with a spin-glass phase (w < 1/2) from a region with spin-glass, ferromagnetic, mixed and paramagnetic phases (w > 1/2).

Dynamic symmetry loss of high-frequency hysteresis loops in single-domain particles with uniaxial anisotropy

Understanding how magnetic materials respond to rapidly varying magnetic fields, as in dynamic hysteresis loops, constitutes a complex and physically interesting problem. But in order to accomplish a thorough investigation, one must necessarily consider the effects of thermal fluctuations. Albeit being present in all real systems, these are seldom included in numerical studies. The notable exceptions are the Ising systems, which have been extensively studied in the past, but describe only one of the many mechanisms of magnetization reversal known to occur. In this paper we employ the Stochastic Landau-Lifshitz formalism to study high-frequency hysteresis loops of single-domain particles with uniaxial anisotropy at an arbitrary temperature. We show that in certain conditions the magnetic response may become predominantly out-of-phase and the loops may undergo a dynamic symmetry loss. This is found to be a direct consequence of the competing responses due to the thermal fluctuations and the gyroscopic motion of the magnetization. We have also found the magnetic behavior to be exceedingly sensitive to temperature variations, not only within the superparamagnetic-ferromagnetic transition range usually considered, but specially at even lower temperatures, where the bulk of interesting phenomena is seen to take place. (C) 2011 Elsevier B.V. All rights reserved.

Non-mean-field effects in systems with long-range forces in competition

We investigate the canonical equilibrium of systems with long-range forces in competition. These forces create a modulation in the interaction potential and modulated phases appear at the system scale. The structure of these phases differentiate this system from monotonic potentials, where only the mean-field and disordered phases exist. With increasing temperature, the system switches from one ordered phase to another through a first-order phase transition. Both mean-field and modulated phases may be stable, even at zero temperature, and the long-range nature of the interaction will lead to metastability characterized by extremely long time scales.

Mikrostruktur und Magnetisierungsdynamik feinmodulierter Co/Pt-Heterostrukturen

Gegenstand dieser Arbeit ist die Untersuchung der strukturellen und magnetischen Eigenschaften von (111)-texturierten epitaktischen dünnen Co/Pt-Vielfachschichten und Pt/Co/Pt-Heterostrukturen. Mit Hilfe von Röntgen-Diffraktions-Experimenten wurde der Einfluß der Oberflächenqualität des MgO (111) Substratmaterials auf die Zwischenlagenstruktur und die kristalline Ordnung in den Filmen analysiert. Es konnte nachgewiesen werden, daß die Unordnung an der Co/Pt-Grenzfläche unterhalb einer Längenskala von 6 nm allein durch die Wachstums- und Interdiffusionsprozesse zwischen der Co- und der Pt-Lage bestimmt ist, unabhängig von der Qualität der Substratoberfläche. Demgegenüber zeigte sich, daß durch eine besondere Substratbehandlung eine langreichweitige kristalline Kohärenz der Schichten und eine Unterdrückung der Verzwillingung aus abc- und acb-Wachstumsdomänen des fcc-Platin erzielt werden können. Anhand integraler Messungen des magneto-optischen Kerr-Effektes wurde ein direkter Zusammenhang zwischen der Substrat-induzierten Defektdichte der Filme und der Nukleation magnetischer Domänen während der Ummagnetisierung nachgewiesen. Pt/Co/Pt-Dreifachlagen mit Kobalt-Dicken bis zu 1 nm besitzen eine senkrechte magnetische Anisotropie und zeigen magnetische Domänen mit Größen von bis zu einigen hundert Mikrometern, die mit Hilfe optischer Kerr-Mikroskopie visualisiert wurden. In Pt/Co/Pt-Dreifachschichten mit weniger als drei Monolagen Kobalt, welche auf vicinalen MgO (111)-Substraten aufgebracht wurden, treten während der Ummagnetisierung aufgrund anisotroper Domänenwandbewegung charakteristische dreieckige Domänenformen auf. Es wurde ein mikroskopischer Mechanismus vorgeschlagen, welcher dieses anisotrope Pinning von magnetischen Domänenwänden an mesoskopischen Stufen-Strukturen der Substratoberfläche beschreibt. Zur quantitativen Beschreibung der anisotropen Domänenwandbewegung wurden zweidimensionale numerische Simulationen durchgeführt, basierend auf einem modifizierten Random-Field-Ising-Modell mit einem Ginzburg-Landau-artigen Hamiltonian, in dem der Einfluß der Stufenkanten auf den Ordnungsparamter durch ein neu eingeführtes effektives anisotropes Feld G(r) repräsentiert ist. Unter Annahme einer lateralen Anordnung der Stufenkanten in Form eines Fischgrätenmusters konnten im Rahmen dieses Modells die experimentell beobachteten charakteristischen anisotropen Domänenformen sowie die Skaleneigenschaften der Domänenwände in exzellenter Weise numerisch reproduziert werden.

Quantenkorrekturen an magnetischen Domänenwänden auf Gitterstrukturen

Wir betrachten die eindimensionale Heisenberg-Spinkette aus einem neuen und aktuelleren Blickwinkel. Experimentelle Techniken der Herstellung und selbstverständlich auch experimentelle Meßmethoden erlauben nicht nur die Herstellung von Nanopartikeln und Nanodrähten, sondern gestatten es auch, Domänenwände in diesen Strukturen auszumessen. Die meisten heute verwendeten Theorien und Simulationsmethoden haben ihre Grundlage im mikromagnetischen Kontinuumsmodell, daß schon über Jahrzehnte hinweg erforscht und erprobt ist. Wir stellen uns jedoch die Frage, ob die innere diskrete Struktur der Substrate und die quantenmechanischen Effekte bei der Genauigkeit heutiger Messungen in Betracht gezogen werden müssen. Dazu wählen wir einen anderen Ansatz. Wir werden zunächst den wohlbekannten klassischen Fall erweitern, indem wir die diskrete Struktur der Materie in unseren Berechnungen berücksichtigen. Man findet in diesem Formalismus einen strukturellen Phasenübergang zwischen einer Ising-artigen und einer ausgedehnten Wand. Das führt zu bestimmten Korrekturen im Vergleich zum Kontinuumsfall. Der Hauptteil dieser Arbeit wird sich dann mit dem quantenmechanischen Fall beschäftigen. Wir rotieren das System zunächst mit einer Reihe lokaler Transformationen derart, daß alle Spins in die z-Richtung ausgerichtet sind. Im Rahmen einer 1/S-Entwicklung läßt sich der erhaltene neue Hamilton-Operator diagonalisieren. Setzt man hier die klassische Lösung ein, so erhält man Anregungsmoden in diesem Grenzfall. Unsere Resultate erweitern und bestätigen frühere Berechnungen. Mit Hilfe der Numerik wird schließlich der Erwartungswert der Energie minimiert und somit die Form der Domänenwand im quantenmechanischen Fall berechnet. Hieraus ergeben sich auch bestimmte Korrekturen zum kritischen Verhalten des Systems. Diese Ergebnisse sind vollkommen neu.

Statistical Mechanics of Monomer-Dimer Models on Complete and Erdös-Rényi Graphs

Nella tesi sono trattate due famiglie di modelli meccanico statistici su vari grafi: i modelli di spin ferromagnetici (o di Ising) e i modelli di monomero-dimero. Il primo capitolo è dedicato principalmente allo studio del lavoro di Dembo e Montanari, in cui viene risolto il modello di Ising su grafi aleatori. Nel secondo capitolo vengono studiati i modelli di monomero-dimero, a partire dal lavoro di Heilemann e Lieb,con l'intento di dare contributi nuovi alla teoria. I principali temi trattati sono disuguaglianze di correlazione, soluzioni esatte su alcuni grafi ad albero e sul grafo completo, la concentrazione dell'energia libera intorno al proprio valor medio sul grafo aleatorio diluito di Erdös-Rényi.

Computer simulations of two-dimensional colloidal crystals in confinement

Monte Carlo simulations are used to study the effect of confinement on a crystal of point particles interacting with an inverse power law potential in d=2 dimensions. This system can describe colloidal particles at the air-water interface, a model system for experimental study of two-dimensional melting. It is shown that the state of the system (a strip of width D) depends very sensitively on the precise boundary conditions at the two ``walls'' providing the confinement. If one uses a corrugated boundary commensurate with the order of the bulk triangular crystalline structure, both orientational order and positional order is enhanced, and such surface-induced order persists near the boundaries also at temperatures where the system in the bulk is in its fluid state. However, using smooth repulsive boundaries as walls providing the confinement, only the orientational order is enhanced, but positional (quasi-) long range order is destroyed: The mean-square displacement of two particles n lattice parameters apart in the y-direction along the walls then crosses over from the logarithmic increase (characteristic for $d=2$) to a linear increase (characteristic for d=1). The strip then exhibits a vanishing shear modulus. These results are interpreted in terms of a phenomenological harmonic theory. Also the effect of incommensurability of the strip width D with the triangular lattice structure is discussed, and a comparison with surface effects on phase transitions in simple Ising- and XY-models is made