The impact of renormalization group theory on magnetism

The basic issues of renormalization group (RG) theory, i.e. universality, crossover phenomena, relevant interactions etc. are verified experimentally on magnetic materials. Universality is demonstrated on account of the saturation of the magnetic order parameter for T ↦ 0. Universal means that the deviations with respect to saturation at T = 0 can perfectly be described by a power function of absolute temperature with an exponent ε that is independent of spin structure and lattice symmetry. Normally the T^ε function holds up to ~0.85T_c where crossover to the critical power function occurs. Universality for T ↦ 0 cannot be explained on the basis of the material specific magnon dispersions that are due to atomistic symmetry. Instead, continuous dynamic symmetry has to be assumed. The quasi particles of the continuous symmetry can be described by plane waves and have linear dispersion in all solids. This then explains universality. However, those quasi particles cannot be observed using inelastic neutron scattering. The principle of relevance is demonstrated using the competition between crystal field interaction and exchange interaction as an example. If the ratio of crystal field interaction to exchange interaction is below some threshold value the local crystal field is not relevant under the continuous symmetry of the ordered state and the saturation moment of the free ion is observed for T ↦ 0. Crossover phenomena either between different exponents or between discrete changes of the pre-factor of the T^ε function are demonstrated for the spontaneous magnetization and for the heat capacity.     

Surname distribution in population genetics and in statistical physics

Surnames tend to behave like neutral genes, and their distribution has attracted a growing attention from genetists and physicists. We review the century-long history of surname studies and discuss the most recent developments. Isonymy has been regarded as a tool for the measurement of consanguinity of individuals and populations and has been applied to the analysis of migrations. The analogy between patrilineal surname transmission and the propagation of Y chromosomes has been exploited for the genetic characterization of families, communities and control groups. Surname distribution is the result of a stochastic dynamics, which has been studied either as a Yule process or as a branching phenomenon: both approaches predict the asymptotic power-law behavior which has been observed in many empirical researches. Models of neutral evolution based on the theory of disordered systems have suggested the application of field-theoretical techniques, and in particular the Renormalization Group, to describe the dynamics leading to scale-invariant distributions and to compute the related (critical) exponents.     

Wavefunction comparisons for the valence-bond model for conjugated π-networks

Approximate ground-state wavefunctions for valence-bond (or Heisenberg) models are obtained both within Néel-state-based and within Kekulé-state-based resonance-theoretic approaches. Comparisons are made between these and other general approaches, with particular emphasis on organic -network systems. Attention is drawn to the manner in which the quality of the different approximation schemes changes with variations in structural characteristics of the system. It is suggested that resonance-theoretic ideas are most appropriate for (aromatic benzenoid) systems with low coordination number, whereas Néel-state based ideas are most appropriate for (3-dimensional) structures with higher coordination number (and little frustration).Work supported by The Robert A. Welch Foundation of Houston, Texas     

Triviality of φ 4 4 and all that in a hierarchical model approximation

The infrared behavior of lattice _d ⁴ ,d 4, and dipole gases ind 1 is rigorously shown to be Gaussian within the context of a hierarchical approximation. Methods are developed to generalize the result beyond this approximation.On leave of absence from the Department of Mathematical Methods of Physics, University, 00-682 Warsaw, Hoza 74, Poland.     

An exact solution of the renormalization-group equations for the mean field theory of stable and metastable states

An exact solution of the renormalization-group equations corresponding to the mean field theory of stable and metastable states is given which yields the correct free energies for these states. An unusual feature of this solution is that the renormalized Hamiltonian in the two-phase region becomes a multivalued function of the order parameter for all values of the length rescaling parameter beyond a certain critical value. This is closely related to the multivaluedness of the free energy as a function of magnetic field which characterizes the classical theory of metastable and unstable states. As a consequence of this multivaluedness, the trajectory flow in the space of coupling constants exhibits unusual bifurcation. This leads to difficulties in evaluating the metastable and unstable free energies by a trajectory integral of the spin-independent term, which can be resolved by an extension of the standard formalism.This work was supported by NSF grant #550-346-01 (JDG) and a U.S.-Japan Cooperative Science grant (KK and JDG).     

Renormalization group approach for the site-bond percolation in structured stochastic environments

The quenched averaged percolation problem of a lattice with a given structure is analyzed. The structure is described by the static structure factorS(q)q ^–ain the regionq 0. As a result of the renormalization group, it follows that the critical behavior fora < 2 is the same as in the random percolation. In the case ofa=2 second universality class with=0 and=1/2+/8+ ²/32 is predicted.     

Mathematical properties of position-space renormalization-group transformations

Properties of position-space or cell-type renormalization-group transformations from an Ising model object system onto an Ising model image system, of the type introduced by Niemeijer, van Leeuwen, and Kadanoff, are studied in the thermodynamic limit of an infinite lattice. In the case of a KadanofF transformation with finitep, we prove that if the magnetic field in the object system is sufficiently large (i.e., the lattice-gas activity is sufficiently small), the transformation leads to a well-defined set of image interactions with finite norm, in the thermodynamic limit, and these interactions are analytic functions of the object interactions. Under the same conditions the image interactions decay exponentially rapidly with the geometrical size of the clusters with which they are associated if the object interactions are suitably short-ranged. We also present compelling evidence (not, however, a completely rigorous proof) that under other conditions both the finite- and infinite-p (majority rule) transformations exhibit peculiarities, suggesting either that the image interactions are undefined (i.e., the transformation does not possess a thermodynamic limit) or that they fail to be smooth functions of the object interactions. These peculiarities are associated (in terms of their mathematical origin) with phase transitions in the object system governed not by the object interactions themselves, but by a modified set of interactions.Supported in part by NSF Grant No. DMR 76-23071.     

Decimation transformations in lattice systems of continuous spins

Decimation renormalization transformations are investigated for systems of continuous spins. The usual arguments against decimation can be avoided by considering products of decimation and spin scaling transformations. With the simple local types of spin scaling normally used for continuous spins, even these product transformations will have no fixed points for lattice dimension greater than one. A Gaussian fixed point for one-dimensional models with short range (but not only nearest neighbor) interactions is exhibited. A series of scaling transformations of increasing generality is investigated. It is found that a product of a nonlocal spin scaling transformation and a decimation will produce the usual fixed points, but that this type of product transformation is effectively much more a block-type transformation than a pure decimation.     

Fixed points of renormalization group for the hierarchical fermionic model

A fermionic version of Dyson's hierarchical model is defined. An exact renormalization group transformation is given as a rational transformation of two-dimensional parameter space. Two branches of nontrivial fixed points are described, one of which bifurcates from the trivial Gaussian branch. The existence of the thermodynamic limit for these branches of fixed points is investigated.