Linear perturbation renormalization group method for Ising-like spin systems

The linear perturbation group transformation (LPRG) is used to study the thermodynamics of the axial next-nearest-neighbor Ising model with four spin interactions (extended ANNNI) in a field. The LPRG for weakly interacting Ising chains is presented. The method is used to study finite field para-ferrimagnetic phase transitions observed in layered uranium compounds, UAs_{1-x}Se_x, UPd_2Si_2 or UNi_2Si_2. The above-mentioned systems are made of ferromagnetic layers and the spins from the nearest-neighbor and next-nearest-neighbor layers are coupled by the antiferromagnetic interactions J_1<0 and J_2<0, respectively. Each of these systems exhibits a triple point in which two ordered phases (ferrimagnetic and incommensurate) meet the paramagnetic one, and all undergo the high field phase transition from para- to ferrimagnetic (++-) phase. However, if in UAs_{1-x}Se_x the para-ferri phase transition is of the first order as expected from the symmetry reason, in UT_2Si_2 (T=Pd,Ni) this transition seems to be a continuous one, at least in the vicinity of the multicritical point. Within the MFA, the critical character of the finite field para-ferrimagnetic transition at least at one isolated point can be described by the ANNNI model supplemented by an additional, e.g., four-spin interaction. However, in LPRG approximation for the ratio \kappa=J_2/J_1 around 0.5 there is a critical value of the field for which an isolated critical point also exists in the original ANNNI model. The positive four-spin interaction shifts the critical point towards higher fields and changes the shape of the specific heat curve. In the latter case for the fields small enough, the specific heat exhibits two-peak structure in the paramagnetic phase.Comment: 8 pages, 7 figure

Langevin processes, agent models and socio-economic systems

We review some approaches to the understanding of fluctuations in some models used to describe socio and economic systems. Our approach builds on the development of a simple Langevin equation that characterises stochastic processes. This provides a unifying approach that allows first a straightforward description of the early approaches of Bachelier. We generalise the approach to stochastic equations that model interacting agents. Using a simple change of variable, we show that the peer pressure model of Marsilli and the wealth dynamics model of Solomon are closely related. The methods are further shown to be consistent with a global free energy functional that invokes an entropy term based on the Boltzmann formula. A more recent approach by Michael and Johnson maximised a Tsallis entropy function subject to simple constraints. We show how this approach can be developed from an agent model where the simple Langevin process is now conditioned by local rather than global noise. The approach yields a BBGKY type hierarchy of equations for the system correlation functions. Of especial interest is that the results can be obtained from a new free energy functional similar to that mentioned above except that a Tsallis like entropy term replaces the Boltzmann entropy term. A mean field approximation yields the results of Michael and Johnson. We show how personal income data for Brazil, the US, Germany and the UK, analysed recently by Borgas can be qualitatively understood by this approach.Comment: 1 figur

Thermodynamics, geometrical frustration and quantum fluctuations in coupled spin chains

The linear-perturbation real space renormalization transformation (LPRG) is presented and applied to the study of quantum spin chains coupled by interchain interaction (k₁) weaker than intrachain one (k). The method is examined in two exact solvable cases: Ising chains on the square and triangular lattices and quantum XY chain. For the Ising model, in the second order in the cumulant epansion, the deviation of the critical temperature from the exact value is less than 1% for 0.5k > k₁ > 0.15k, but even in the case of the standard Ising model (k₁ = k) we found the value of Tc which differs by 2% from the exact one. For the quantum XY chain the deviation of the free energy value found by using LPRG from the exact Katsura result is less than 1% for T/J > 1, and for rather low temperature T/J = 0.08 is about 6%. The LPRG is used to study the effects of interchain frustration on the phase transition in 2D Heisenberg spin chains with easy axis along the z direction. It is shown that contrary to the pure Ising model in systems with in-plane interactions (XY), the interchain frustration does not destroy the finite-temperature transition. However, such a frustration changes the character of the phase transition from Ising-like to, probably, Kosterlitz-Thouless-like. We have also applied the LPRG method to the calculation of the isothermal magnetocaloric coef cient (Mт) for several spin models in disordered phases. Is is demonstrated that in the presence of antiferromagnetic fluctuations, Mт changes sign at some value of the magnetic field. Generally, Mт is negative if magnetic field competes with a short-range order, and consequently it can be an indicator of the change in the short-range correlation.Сформульовано лiнiйно-пертурбативне ренормалiзацiйне перетворення у дiйсному просторi (ЛПРП), що використовується для вивчення квантових спiнових ланцюжкiв, зв’язаних мiжланцюжковою взаємодiєю (k₁), яка є слабшою за взаємодiю (k) мiж спiнами у ланцюжку. Метод протестовано для двох точно розв’язуваних моделей: Iзингiвських ланцюжкiв на квадратнiй та трикутнiй ґратках i квантових XY ланцюжкiв. Для моделi Iзинга у другому порядку кумулянтного розкладу показано, що вiдхилення критичної температури вiд точного значення є меншим 1% для 0.5k > k₁ > 0.15k; однак навiть у випадку стандартної моделi Iзинга (k₁ = k) отримано значення Tc, що вiдрiзняється вiд точного на 2%. Для квантового XY ланцюжка вiдхилення вiльної енергiї, що знайдена методом ЛПРП, вiд точного результату Кацура не перевищує 1% для T/J > 1, а для доволi низької температури T/J = 0.08 складає бiля 6%. Метод ЛПРП використовується для вивчення впливу мiжланцюжкової фрустрацiї на фазовий перехiд у двовимiрних Гайзенбергiвських спiнових ланцюжках з вiссю легкого намагнiчення вздовж напрямку z. Показано, що на вiдмiну вiд чисто Iзингiвської моделi, у системах iз планарною XY взаємодiєю мiжланцюжкова фрустрацiя не порушує фазовий перехiд при скiнчених температурах. Однак, така фрустрацiя змiнює характер фазових переходiв вiд iзингiвського типу до, iмовiрно, переходiв типу Костерлiца-Таулса. Ми використали також метод ЛПРП для розрахунку iзотермiчного магнетокалоричного коефiцiєнта (Mт) кiлькох спiнових моделей у невпорядкованих фазах. Показано, що за наявностi антиферомагнiтних флуктуацiй,Mт змiнює свiй знак при певному значеннi магнiтного поля. Загалом Mт – вiд’ємна величина, якщо магнiтне поле конкурує з близьким порядком, i тому це може служити iндикатором змiни короткосяжних кореляцiй

Magnetic and quadrupolar order in a one-dimensional ferromagnet with cubic crystal-field anisotropy

The zero temperature phase diagram of a one-dimensional S=2 Heisenberg ferromagnet with single-ion cubic anisotropy is studied numerically using the density-matrix renormalization group method. Evidence is found that although the model does not involve quadrupolar couplings, there is a purely quadrupolar phase for large values of the anisotropy. The phase transition between the magnetic and quadrupolar phases is continuous and it seems to be characterized by Ising critical exponents.Comment: 11 pages, 7 figures, REVTeX, accepted in Phys. Rev. B (scheduled on June 99

Outflow Dynamics in Modeling Oligopoly Markets: The Case of the Mobile Telecommunications Market in Poland

In this paper we introduce two models of opinion dynamics in oligopoly markets and apply them to a situation, where a new entrant challenges two incumbents of the same size. The models differ in the way the two forces influencing consumer choice -- (local) social interactions and (global) advertising -- interact. We study the general behavior of the models using the Mean Field Approach and Monte Carlo simulations and calibrate the models to data from the Polish telecommunications market. For one of the models criticality is observed -- below a certain critical level of advertising the market approaches a lock-in situation, where one market leader dominates the market and all other brands disappear. Interestingly, for both models the best fits to real data are obtained for conformity level $p \in (0.3,0.4)$. This agrees very well with the conformity level found by Solomon Asch in his famous social experiment

Initiating a Mexican wave: An instantaneous collective decision with both short and long range interactions

An interesting example for collective decision making is the so-called Mexican wave during which the spectators in a stadium leap to their feet with their arms up and then sit down again following those to their left (right) with a small delay. Here we use a simple, but realistic model to explain how the combination of the local and global interactions of the spectators produces a breaking of the symmetry resulting in the replacement of the symmetric solution -- containing two propagating waves -- by a single wave moving in one of the two possible directions. Our model is based on and compared to the extensive observations of volunteers filling out the related questionnaire we have posted on the Internet. We find that, as a function of the parameter controlling the strength of the global interactions, the transition to the single wave solution has features reminiscent of discontinuous transitions. After the spontaneous symmetry breaking the two directions of propagation are still statistically equivalent. We investigate also how this remaining symmetry is broken in real stadia by a small asymmetrical term in the perception of spectators.Comment: Main text: 12 pages, 3 figures. Appendices: 18 pages (incl. answers from online survey on Mexican waves). Supplementary website: http://angel.elte.hu/localgloba

Ferromagnetic fluid as a model of social impact

The paper proposes a new model of spin dynamics which can be treated as a model of sociological coupling between individuals. Our approach takes into account two different human features: gregariousness and individuality. We will show how they affect a psychological distance between individuals and how the distance changes the opinion formation in a social group. Apart from its sociological aplications the model displays the variety of other interesting phenomena like self-organizing ferromagnetic state or a second order phase transition and can be studied from different points of view, e.g. as a model of ferromagnetic fluid, complex evolving network or multiplicative random process.Comment: 8 pages, 5 figure

Chaotic, staggered and polarized dynamics in opinion forming: the contrarian effect

We revisit the no tie breaking 2-state Galam contrarian model of opinion dynamics for update groups of size 3. While the initial model assumes a constant density of contrarians a for both opinions, it now depends for each opinion on its global support. Proportionate contrarians are thus found to indeed preserve the former case main results. However, restricting the contrarian behavior to only the current collective majority, makes the dynamics more complex with novel features. For a density a<a_c=1/9 of one-sided contrarians, a chaotic basin is found in the fifty-fifty region separated from two majority-minority point attractors, one on each side. For 1/9<a< 0.301 only the chaotic basin survives. In the range a>0.301 the chaotic basin disappears and the majority starts to alternate between the two opinions with a staggered flow towards two point attractors. We then study the effect of both, decoupling the local update time sequence from the contrarian behavior activation, and a smoothing of the majority rule. A status quo driven bias for contrarian activation is also considered. Introduction of unsettled agents driven in the debate on a contrarian basis is shown to only shrink the chaotic basin. The model may shed light to recent apparent contradictory elections with on the one hand very tied results like in US in 2000 and in Germany in 2002 and 2005, and on the other hand, a huge majority like in France in 2002.Comment: 17 pages, 10 figure

Fermiology via the electron momentum distribution

Investigations of the Fermi surface via the electron momentum distribution reconstructed from either angular correlation of annihilation radiation (or Compton scattering) experimental spectra are presented. The basis of these experiments and mathematical methods applied in reconstructing three-dimensional densities from line (or plane) projections measured in these experiments are described. The review of papers where such techniques have been applied to study the Fermi surface of metallic materials with showing their main results is also done.Comment: 22 pages, 9 Figures, 4 Table