Ising model on self-avoiding walk chains

The phase transitions of nearest-neighbour interacting Ising models on self-avoiding walk (SAW) chains on square and triangular lattices have been studied using Monte Carlo technique. To estimate the transition temperature (T _c) bounds, the average number of nearest-neighbours (Z _eff) of spins on SAWs have been determined using the computer simulation results, and the percolation thresholds (p _c) for site dilution on SAWs have been determined using Monte Carlo methods. We find, for SAWs on square and triangular lattices respectively,Z _eff=2.330 and 3.005 (which compare very well with our previous theoretically estimated values) andp _c=0.022±0.003 and 0.045±0.005. When put in Bethe-Peierls approximations, the above values ofZ _eff givekT _c/J<1.06 and 1.65 for Ising models on SAWs on square and triangular lattices respectively, while, using the semi-empirical relation connecting the Ising modelT _c's andp _c's for the same lattices, we findkT _c/J0.57 and 0.78 for the respective models. Using the computer simulation results for the shortest connecting path lengths in SAWs on both kinds of lattices, and integrating the spin correlations on them, we find the susceptibility exponent =1.024±0.007, for the model on SAWs on two dimensional lattices.     

Asymptotic behaviour of self-avoiding walks in continuous space

The asymptotic behaviour of self-avoiding walks that do not lie on a lattice has been investigated by a Monte-Carlo procedure. Its dependence on the fine structure of the walk is discussed.     

The effective coordination number of self-avoiding two-dimensional random walks

It is demonstrated that the effective coordination number of self-avoiding random walks in a plane is given by exp(−π²/24) = 0.66283.     

Thermodynamics of a dense self-avoiding walk with contact interactions

A lattice model is used to study the properties of an infinite self-avoiding linear polymer chain that occupies a fraction, 01, of sites on ad-dimensional hypercubic lattice. The model introduces an (attractive or repulsive) interaction energy between nonbonded monomers that are nearest neighbors on the lattice. The lattice cluster theory enables us to derive a double series expansion in and d^–1 for the chain free energy per segment while retaining the full dependence. Thermodynamic quantities, such as the entropy, energy, and mean number of contacts per segment, are evaluated, and their dependences on, , andd are discussed. The results are in good accordance with known limiting cases.     

On the critical properties of the Edwards and the self-avoiding walk model of polymer chains

We study the two- and three-dimensional, superrenormalizable Edwards model and the self-avoiding walk model of polymers. Using a Schwinger-Dyson equation and upper and lower bounds on correlations in terms of “skeleton diagrams” [6] we establish the existence of a non-trivial continuum limit in the two- and three-dimensional, superrenormalizable Edwards model. We also prove that perturbation theory is asymptotic for the continuum correlations of these models.A fairly detailed analysis of the approach to the critical point in the self-avoiding walk model is presented. In particular, we show that η<1. In dimension d?4, we discuss rigorous consequences of the conjecture that η is non-negative: among other implications, we derive that the continuum limit is trivial and that γ=1, in d?5 dimensions, and that corrections to mean-field scaling laws are at most logarithmic in four dimensions.     

Effective interactions are effective interactions

In recent years it has been realized that effective interactions—those valid over a limited range of energy-momentum—are often preferable even when more complete theories are available. We shall here present examples of such methods over a range of physical applications—from classical and quantum mechanics, to quantum gravity.     

Effective effective interactions

The use of effective field theory techniques allows the basic physics to be extracted from otherwise complex systems. We cite examples from quantum mechanics, condensed-matter physics, QCD, and quantum gravity.Received: 30 September 2002, Published online: 22 October 2003PACS: 04.60.-m Quantum gravity - 12.20.-m Quantum electrodynamics     

Real space renormalization group with effective interactions: applications to 2-D spin lattices

The Blochs theory of effective Hamiltonians has been used to improve the Real Space Renormalization Group approach. The effective interactions between elementary blocks of a periodic lattice can be extracted from the knowledge of the spectrum of the dimers or trimers of blocks. The potentialities of the method are illustrated on a series of quasi 1-D and 2-D problems. The spin gap of two-leg ladders is calculated and an estimate of the impact of ferromagnetic couplings between two-leg ladders on the gap is presented. The method satisfactorily identifies the phase transitions in the 1/5-depleted square lattice as well as in the spin-frustrated Shastry-Sutherland lattice. The J ₂/J ₁ checkerboard lattice is studied and a location of the phase transition between the Néel phase and the dimer phase is proposed.Received: 11 June 2004, Published online: 30 September 2004PACS: 71.10.-W Theories and models of many-electron systems - 71.15.Nc Total energy and cohesive energy calculations - 75.10.-b General theory and models of magnetic ordering     

An equation of motion method for deriving microscopic effective interactions in a correlated model space

Starting from a diagrammatic analysis of the equation of motion method, we derive an effective interaction theory for a correlated model space where the basis vectors correspond physically to the addition of valence particles and/or holes to the true ground state of the core nucleus. The resulting effective interaction $\text{V}$ is valence linked and connected, energy independent, and contains folded diagrams. In addition, it gives directly model eigenvectors with amplitudes that correspond to spectroscopic factors. With terms having the same number of folds grouped together, the general structure of $\text{V}$ is very simple. This is very useful in the application of the present theory to actual microscopic nuclear structure calculations. The treatment of core projection insertions is discussed in some detail. A proof of the cancellation of the disconnected diagrams is given. When folded diagrams are summed up using a partial summation method, the present effective interaction theory is shown to be consistent with the usual Green function theory for many-body problems.     

Particles inside electrolytes with ion-specific interactions,their effective charge distributions,and effective interactions

In this work, we explore the statistical physics of colloidal particles that interact with electrolytes via ion-specific interactions. Firstly we study particles interacting weakly with electrolyte using linear response theory. We find that the mean potential around a particle is linearly determined by the effective charge distribution of the particle, which depends both on the bare charge distribution and on ion-specific interactions. We also discuss the effective interaction between two such particles and show that, in the far field regime, it is bilinear in the effective charge distributions of two particles. We subsequently generalize the above results to the more complicated case where particles interact strongly with the electrolyte.Our results indicate that in order to understand the statistical physics of non-dilute electrolytes, both ion-specific interactions and ionic correlations have to be addressed in a single unified and consistent framework.     

Free energy of multiple overlapping chains

How accurate is pair additivity in describing interactions between soft polymer-based nanoparticles? Using numerical simulations we compute the free energy cost required to overlap multiple chains in the same region of space, and provide a quantitative measure of the effectiveness of pair additivity as a function of chain number and length. Our data suggest that pair additivity can indeed become quite inadequate as the chain density in the overlapping region increases. We also show that even a scaling theory based on polymer confinement can only partially account for the complexity of the problem. In fact, we unveil and characterize an isotropic to star-polymer crossover taking place for large number of chains, and propose a revised scaling theory that better captures the physics of the problem.     

Nuclear effective interactions and spin excitations

In view of the one-boson-exchange model for the nucleon-nucleon interaction and the Hartree-Fock (HF) interaction, we formulate the effective interactions for particle-hole states in terms of the exchange of the fields which are confined in the nucleus. This theory, as an extension to the nuclear field theory (NFT), takes into account the propagation of the fields which is neglected in NFT. The effective interactions thus obtained reproduce the energies of a sequence of electric giant resonances and the core polarizabilities associated with the resonances. It is found that the coupling constants of the σ- and ω-fields are suppressed for the particle-hole interaction by 60% with respect to the HF interaction. As for the effective interactions involving nucleon spins, we consider the fields coupled to nucleon spins. The effective interactions obtained, essentially different from those in NFT, have a tensor component. We analyse the energies and cross sections for excitation of stretched spin particle-hole states which are the most sensitive to the tensor force. The effective interaction responsible for the stretched spin states is shown to be consistent with that for the magnetic resonances observed in the (p, n) reactions.     

Quasiconfigurations and the theory of effective interactions

Perturbation theory is reformulated. Schrödinger's equation is recast as a non linear integral equation which yields by Neumann expansion a linked cluster series for the degenerate, quasi degenerate or non degenerate problem. An effective interaction theory emerges that can be formulated in a biorthogonal basis leading to a non Hermitian secular problem. Hermiticity can be recovered in a clear and rigorous way. As the mathematical form of the theory is dictated by the request of physical clarity the latter is obtained naturally. When written in diagrammatic many body language, the integral equation produces a set of linked coupled equations for the degenerate case. The classic summations (Brueckner, Bethe-Faddeev and RPA) emerge naturally. Possible extensions of nuclear matter theory are suggested.     

Padé approximants and realistic effective interactions

The convergence of the J = 0, T = 1 effective interaction for sd-shell nuclei is student in terms of separate summations of infinite subseries. It is suggested that first the subset of non-folded diagrams be separately summed to all orders and that the remaining, folded diagrams be grouped according to the number of folds and expressed in terms of the former sum of non-folded diagrams and its energy derivatives. The sum of non-folded diagrams is extrapolated from the lowest three orders using the Padé approximation, and encouraging results are obtained.     

Two-body and three-body effective interactions in nuclei

Finite-range two-body and zero-range three-body effective forces for use in spin unsaturated systems are determined so as to reproduce the total binding energies, rms radii, and single-particle energies of ¹⁶O, ⁴⁰Ca, ⁴⁸Ca and ⁹⁰Zr; the saturation of nuclear matter; and experimental two-body matrix elements extracted by Schiffer and True. In addition to the Skyrme three-body force which acts only in spatially even states, a spatially odd force is introduced to obtain sufficient generality. The Landau parameters and the effective mass specified by this force are also discussed.     

Spin chains with nearest and next nearest neighbour interactions

The 1/n-expansion is applied to a classical one dimensionaln-vector model in an external field with nearest and next nearest neighbour exchange interactions of both signs. Various effects caused by helical spin structures are discussed for the Heisenberg magnet (n=3).—For the spherical model with multi-neighbour interactions Lifshitz points of higher order are found.