Adatom diffusion on a square lattice. Theoretical and numerical studies

A three-dimensional simple cubic lattice gas is investigated by using the real-space renormalization group (RSRG) approach with blocks of different size and symmetries. We have calculated absorption isotherms and density dependencies of the pair correlation function, isothermal susceptibility, chemical and jump diffusion coefficients for different temperatures. All these quantities have also been obtained by Monte Carlo (MC) simulations. We have also obtained the density dependencies for the tracer diffusion coefficient by MC simulations. Despite the fact that both methods constitute very different approaches, the correspondence of numerical data and theoretical results is rather good. We conclude that the RSRG method can be applied to the three-dimensional lattice gas system discussed here to characterize its thermodynamic and kinetic properties.

The influence of adatom diffusion on coverage and emission current fluctuations

The integer quantum Hall (QH) transition is described well in terms of a delocalization-localization transition of the electron wavefunction. In contrast to a usual metalinsulator transition (MIT), the QH transition is characterized by a single extended state located exactly at the center = 0 of each Landau band. When approaching = 0, the localization length ξ of the electron wavefunction diverges according to a power law −ν , where defines the distance to the MIT for a suitable control parameter, e.g., the electron energy. On the theoretical side, the value of ν has been extracted from various numerical simulations, e.g., ν = 2.5 ± 0.5, 2.4 ± 0.2, 2.35± 0.03, and 2.39± 0.01. In experiments ν ≈ 2.3 has been obtained, e.g., from the frequency or the sample size dependence of the critical behavior of the resistance in the transition region at strong magnetic field. We study the critical properties of the integer QH transition by employing the real-space renormalization-group (RG) approach to the Chalker-Coddington (CC) network model. We calculate the critical distribution Pc(G) of the conductance and the critical exponent ν of the QH transition for two different RG units. This allows to demonstrate that the quality of the results crucially depends on the choice of the RG unit. The CC model describes a single QH transition using a chiral network consisting of electron trajectories along equipotential lines (links) and saddle points (SP’s) of the potential (nodes). Each SP acts as a scatterer and relates the wavefunction amplitudes in two incoming and two outgoing channels. It can be characterized by a 2×2 S matrix, which depends only on the transmission and reflection coefficients ti and ri. The links correspond to random phases Φj and reflect the randomness of the potential disorder in a sample. We consider two previously studied, different RG units on a regular 2D square lattice as shown in Fig. 1. One is constructed from 4 SP’s, the other consists of 5 SP’s. The RG unit should be chosen in a way such that the essential properties of the network are taken into account. In the course of our RG approach an RG unit is then mapped onto a new single super-SP using the analytical dependence t′ = f ({ti, ri}, {Φj}) (1)

U(1) lattice gauge theory with $\\theta$-term is investigated by real space\nrenormalization group approach. Flows of renormalized coupling constants are\nanalyzed. For each $\\theta$, renormalization flows converge to a single\ntrajectory irrespective of bare coupling constants of real action. For $\\theta\n\\not= \\pi$ the system is in the confinement phase controlled by an infrared\nfixed point. We found a phase transition at $\\theta = \\pi$. Imaginary part of\nthe action given by $\\theta$-term stays fixed under renormalization group\ntransformations leading to deconfinement.\n

Z(6) model: Criticality and break-collapse method

The Maier-Saupe-Zwanzig model for the nematic phase transitions in liquid crystals is investigated in a diamond hierarchical lattice. The model takes into account a parameter to describe the biaxiality of the microscopic units. Also, a suitably chosen external field is added to the Hamiltonian to allow the determination of critical parameters associated with the nematic phase transitions. Using the transfer-matrix technique, the free energy and its derivatives are obtained in terms of recursion relations between successive generations of the hierarchical lattice. In addition, a real-space renormalization-group approach is developed to obtain the critical parameters of the same model system. Results of both methods are in excellent agreement. There are indications of two continuous phase transitions. One of them corresponds to a uniaxial-isotropic transition, in the class of universality of the three-state Potts model on the diamond hierarchical lattice. The transition between the biaxial and the uniaxial phases is in the universality class of the Ising model on the same lattice.

The authors study an extended version of the discrete N-vector (or cubic) ferromagnetic model within a real space renormalisation group approach which preserves the two-spin correlation function. The N-evolution (for real values of N) of the Wheatstone-bridge hierarchical lattice phase diagram, which presents paramagnetic intermediate (nematic-like) and ferromagnetic phases, as well as of the thermal ( nu ) and crossover ( phi ) critical exponents, is presented. The self-avoiding walk problem is recovered in the N to O limit, and the so-called 'corner rule' is re-obtained in a larger context. The Ising, N- and 2N-state Potts ferromagnets are recovered as particular cases. An interchange of stability occurs at N=N* approximately=6.9 in such a way that the 2N-state Potts special point (where all three existing phases join) is multicritical if N(N*, but only critical if N)N* (consistently phi (N*)=O). For the cubic model, nu (N) presents a maximum at N=Nmax approximately=1.5. The results are exact, for all N, for the Wheatstone-bridge hierarchical lattice, and approximate, for N<or=2, for the square lattice. Last but not least, they discuss the connection between the present approach and the phenomenological renormalisation group.

The low energy behaviour of the two-dimensional antiferromagnetic Heisenberg model is studied in the sector with total spins S = 0,1,2 by means of a renormalization group procedure, which generates a recursion formula for the interaction matrix ΔS (n+1) of 4 neighbouring “n clusters” of size 2n × 2n, n = 1,2,3,... from the corresponding quantities ΔS (n). Conservation of total spin S is implemented explicitly and plays an important role. It is shown, how the ground state energies ES (n+1), S = 0,1,2 approach each other for increasing n, i.e. system size. The most relevant couplings in the interaction matrices are generated by the transitions 〈S’,m’;n+1|Sq *|S,m;n+1〉 between the ground states |S,m;n+1〉 (m = -S,...,S) on an (n+1)-cluster of size 2n+1 × 2n+1, mediated by the staggered spin operator Sq *.

Random spin-1 quantum chains

A renormalization group study of the Anderson Hamiltonian for arbitrary band-filling

Renormalization group approach as an approximate solution of diophantine system of equations: Application to the Ising model

Quasi-one-dimensional spin-Peierls and spin-ladder systems are characterized by a gap in the spin-excitation spectrum, which can be modeled at low energies by that of Dirac fermions with a mass. In the presence of disorder these systems can still be described by a Dirac fermion model, but with a random mass. Some peculiar properties, like the Dyson singularity in the density of states, are well known and attributed to creation of low-energy states due to the disorder. We take one step further and study single-particle correlations by means of Berezinskii's diagram technique. We find that, at low energy $\epsilon$, the single-particle Green function decays in real space like $G(x,\epsilon) \propto (1/x)^{3/2}$. It follows that at these energies the correlations in the disordered system are strong -- even stronger than in the pure system without the gap. Additionally, we study the effects of boundaries on the local density of states. We find that the latter is logarithmically (in the energy) enhanced close to the boundary. This enhancement decays into the bulk as $1/\sqrt{x}$ and the density of states saturates to its bulk value on the scale $L_\epsilon \propto \ln^2 (1/\epsilon)$. This scale is different from the Thouless localization length $\lambda_\epsilon\propto\ln (1/\epsilon)$. We also discuss some implications of these results for the spin systems and their relation to the investigations based on real-space renormalization group approach.

The contractor renormalization group formalism (CORE) is a real-space renormalization group method which is the Hamiltonian analogue of the Wilson exact renormalization group equations. In an earlier paper [Phys. Rev. D 61, 034505 (2000)] I showed that the CORE method could be used to map a theory of free quarks and quarks interacting with gluons into a generalized frustrated Heisenberg antiferromagnet (HAF) and proposed using CORE methods to study these theories. Since generalizations of HAF's exhibit all sorts of subtle behavior which, from a continuum point of view, are related to topological properties of the theory, it is important to know that CORE can be used to extract this physics. In this paper I show that despite the folklore which asserts that all real-space renormalization group schemes are necessarily inaccurate, simple CORE computations can give highly accurate results even if one only keeps a small number of states per block and a few terms in the cluster expansion. In addition I argue that even very simple CORE computations give a much better qualitative understanding of the physics than naive renormalization group methods. In particular I show that the simplest CORE computation yields a first-principles understanding of how the famous Haldane conjecture works for the case of the $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}1/2$ and $\mathrm{s}\mathrm{p}\mathrm{i}\mathrm{n}\ensuremath{-}1$ HAF.

We study the Blume-Emery-Griffiths model in a random crystal field in two and three dimensions, through a real-space renormalization-group approach and a mean-field approximation, respectively. According to the two-dimensional renormalization-group calculation, non-symmetry-breaking first-order phase transitions are eliminated and symmetry-breaking discontinuous transitions are replaced by continuous ones, when disorder is introduced. On the other hand, the mean-field calculation predicts that first-order transitions are not eliminated by disorder, although some changes are introduced in the phase diagrams. We make some comments on the consequences of a degeneracy parameter, which may be relevant in martensitic transitions.

A real-space dynamic renormalization-group scheme is used to evaluate static and dynamic correlation functions for a kinetic Ising model on a two-dimensional square lattice. The critical exponents obtained from the correlation functions calculated using this method satisfy the proper static and dynamic scaling relations and are in excellent agreement with known values.