Attilio Stella

Differential renormalization of van der Waals spin models

An exact renormalization group equation in differential form is derived for spin systems with general many-body interactions in the van der Waals limit. This equation is solved and the free energy is determined as an integral along the renormalization trajectory in the interaction space. It is shown that the transformation can always be modified in such a way that an undetermined integration constant for the free energy vanishes exactly, also below the critical temperature. We also demonstrate how the invariance of the free energy under a parameterdependent equivalence transformation can provide information about the critical behavior of the system. In this alternative approach “dangerous” irrelevant variables play an essential role.

Random field singularities in renormalization group transformations on Van der Waals-Ising models

Griffiths-Pearce “peculiarities” of renormalization transformations acting on nonrandom mean field Ising models are studied exactly and shown to occur in connection with critical singularities in the quenched free energies of systems with random magnetic fields. The result applies to the transformations of the full hamiltonian, considered as a function of spin configurations, and not only to the recurrence relations for its interaction constants.

Cell-to-cell scaling approach to the dielectric breakdown model

A new ansatz is proposed to implement finite size scaling analysis of the dielectric breakdown model. Calculations on very small cells already allow to obtain good qualitative determinations of the fractal dimensions d in a wide range of dimensionalities and model parameters. Exact enumerations or high accuracy Monte Carlo calculations on larger cells (SI2 x 12) in d =2 show a remarkable degree of convergence towards values known from large scale simulations for d, and, to a lesser extent, for the multifractal dimension D(q) with qa-I. General difficulties inherent to Monte Carlo determinations of D(q) for negative q are also pointed out. For calculating the electrostatic potential inside these cells we use a new exact and fast algorithm, explained in detail in the appendix

Thermodynamics of the Hubbard chain: temperature extension of ground-state renormalisation for fermions

A recently proposed ground-state renormalisation-group approach to the Hubbard model is generalised to finite temperatures. This new method provides a global and consistent description of the thermodynamics of the model at all temperatures. Application to the Hubbard chain with a half-filled band gives values for various thermodynamic functions which compare remarkably well with existing exact or numerical data, over all temperature ranges. The difficulties posed by Fermi-Dirac statistics in the set-up of the transformation are discussed to some extent and it is shown that these do not prevent one, at least in principle, from extending the approach to higher dimensions.

Potential-moving, Migdal's recursion formula, differential renormalization and duality

Migdals original recursion formula is rederived as a low-temperature approximation by an isotropic type of potential-moving. For self-dual spin or gauge systems this transformation is shown to be differentiably conjugate to another one, which is obtained as a high-temperature approximation. The conjugation relation is established through the duality mapping.

Dynamical Renormalization of Kinetic van der Waals Spin Models

An exact dynamical renormalization approach in differential form is proposed for kinetic van der Waals spin systems with general many-body interactions. The problem of restoring covariance in the evolution equation after renormalization of the model is solved by introducing a suitable renormalized time parameter, which depends also on the magnetization of the spin configuration. The study of the behavior of this renormalized time near criticality leads to a scaling relation for the linear relaxation time. This relation can be shown to imply the exact results for the dynamical critical behavior of the system.

Differential real space renormalization: The linear Ising chain

The differential real space renormalization method, recently introduced by Hillhorst et al., is applied to the linear Ising chain. It is shown that chains with spatially homogeneous as well as inhomogeneous or quenched random interactions can be treated. For the first two cases the free energy is computed by renormalization. The discussion includes also the case with a magnetic field, higher order interactions and the behavior of correlation functions under renormalization.

A correlation function renormalization group analysis of the ferromagnetic transition in theZ n spin model

SummaryIn theZn spin model on a finite square lattice of only eight sites, correlation functions are calculated and used for implementing a renormalization transformation for 2≤n≤10. Estimates for critical points and thermal and magnetic exponents are found in fair agreement with exact or Monte Carlo results. Forn≥5 the method describes rather accurately the ferromagnetic critical point at the bottom of the massless phase.RiassuntoPer il modello di spinZn, con 2≤n≤10, in un reticolo di soli 8 siti, alcune funzioni di correlazione sono calcolate ed usate per, costruire una transformazione di rinormalizzazione. Sono così ottenute stime delle temperature critiche e degli esponenti termici e magnetici in buon accordo con risultati esatti o Monte Carlo. Pern≥5 il metodo descrive in modo piuttosto accurato il punto critico al limite inferiore della fase con correlazioni a portata infinita.РезюмеВ рамкахZn спиновой модели на конечной квадратной решетке, для случая восьми узлов, вычисляются корреляционные функции, которые используютсся для перенормированных преобразований для 2≤n≤10. Получено, что оценки для критических точек и для термических и магнитных показателей довольно хорошо согласуются с точными результатами и вычислениями Монте Карло. Дляn≥5 предложенный метод описывает довольно точно ферромагнитную критическую точку на дне безмассовой фазы.