Real Space Renormalization Group Calculations Research Articles

A real-space renormalization-group calculation for the quantum gauge theory on a square lattice

We revisit Fradkin and Raby’s real-space renormalization-group method to study the quantum gauge theory defined on links forming a two-dimensional square lattice. Following an old suggestion of theirs, a systematic perturbation expansion developed by Hirsch and Mazenko is used to improve the algorithm to second order in an intercell coupling, thereby incorporating the effects of discarded higher energy states. A careful derivation of gauge-invariant effective operators is presented in the Hamiltonian formalism. Renormalization group equations are analyzed near the nontrivial fixed point, reaffirming old work by Hirsch on the dual transverse field Ising model. In addition to recovering Hirsch’s previous findings, critical exponents for the scaling of the spatial correlation length and energy gap in the electric free (deconfined) phase are compared. Unfortunately, their agreement is poor. The leading singular behavior of the ground state energy density is examined near the critical point: we compute both a critical exponent and estimate a critical amplitude ratio.

MonteCarlo Renormalization Group for Classical Lattice Models with Quenched Disorder.

We extend to quenched-disordered systems the variational scheme for real-space renormalization group calculations that we recently introduced for homogeneous spin Hamiltonians. When disorder is present our approach gives access to the flow of the renormalized Hamiltonian distribution, from which one can compute the critical exponents if the correlations of the renormalized couplings retain finite range. Key to the variational approach is the bias potential found by minimizing a convex functional in statistical mechanics. This potential reduces dramatically the MonteCarlo relaxation time in large disordered systems. We demonstrate the method with applications to the two-dimensional dilute Ising model, the random transverse field quantum Ising chain, and the random field Ising in two- and three-dimensional lattices.

Wilson-Like Real-Space Renormalization Group and Low-Energy Effective Spectrum of the XXZ Chain in the Critical Regime

We present a novel real-space renormalization group(RG) for the one-dimensional XXZ model in the critical regime, reconsidering the role of the cut-off parameter in Wilson's RG for the Kondo impurity problem. We then demonstrate the RG calculation for the XXZ chain with the free boundary. Comparing the hierarchical structure of the obtained low-energy spectrum with the Bethe ansatz result, we find that the proper scaling dimension is reproduced as a fixed point of the RG transformation.

Dynamical real-space renormalization group calculations with a highly connected clustering scheme on disordered networks

We have defined a type of clustering scheme preserving the connectivity of the nodes in a network, ignored by the conventional Migdal-Kadanoff bond moving process. In high dimensions, our clustering scheme performs better for correlation length and dynamical critical exponents than the conventional Migdal-Kadanoff bond moving scheme. In two and three dimensions we find the dynamical critical exponents for the kinetic Ising model to be z=2.13 and z=2.09 , respectively, at the pure Ising fixed point. These values are in very good agreement with recent Monte Carlo results. We investigate the phase diagram and the critical behavior of randomly bond diluted lattices in d=2 and 3 in the light of this transformation. We also provide exact correlation exponent and dynamical critical exponent values on hierarchical lattices with power-law and Poissonian degree distributions.

Dynamical properties of three component Fibonacci quasicrystal

We present a real space renormalization group (RSRG) method to study the lattice dynamics of a three component Fibonacci (3CF) quasicrystal. Phonon dispersion relations corresponding to different models of this lattice are obtained. Some features of the phonon dispersion curves are compared with experiments on real quasicrystal. It is observed that the positions of the strongest Bragg peaks calculated analytically are in perfect agreement with our RSRG calculations.

Tricritical behavior in deterministic aperiodic ising systems

We use a mixed-spin model, with aperiodic ferromagnetic exchange interactions and crystalline fields, to investigate the effects of deterministic geometric fluctuations on first-order transitions and tricritical phenomena. The interactions and the crystal-field parameters are distributed according to some two-letter substitution rules. From a Migdal-Kadanoff real-space renormalization-group calculation, which turns out to be exact on a suitable hierarchical lattice, we show that the effects of aperiodicity are qualitatively similar for tricritical and simple critical behavior. In particular, the fixed point associated with tricritical behavior becomes fully unstable beyond a certain threshold dimension (which depends on the aperiodicity), and is replaced by a two-cycle that controls a weakened and temperature-depressed tricritical singularity.

Low temperature static behavior of the two-dimensional quantum easy-axis Heisenberg model

We use the self-consistent harmonic approximation (SCHA) to study static properties of the two-dimensional quantum Heisenberg model with easy-axis anisotropy. We calculate the critical temperature as a function of the spin value, and compare with classical results. Specifically, we compare how the ratio of critical temperatures varies as a function of the spin S in the quantum and classical cases, for afixed anisotropy parameter. We see that, for values of spin near 5/2, the classical result approximates to the quantum results and the classical calculation is justified. We have also studied the behavior of the magnetization for very small anisotropies. We have shown that our magnetization curves do not present a plateau in the limit of very small anisotropies, as predicted by the real-space renormalization-group calculations.

AN EXACT FINITE FIELD RENORMALIZATION GROUP CALCULATION ON A TWO-DIMENSIONAL FRACTAL LATTICE

A two-dimensional fractal lattice, herein referred to as the tetrahedral gasket, is used as a model for an exact two-dimensional real-space renormalization group calculation. It is shown that this Ising spin-system has exact solutions which includes a nontrivial phase transition even in the presence of a finite field. The calculation also introduces a new analysis tool, the free energy surface plot, which gives further insight into the phase diagram of the spin-system. The discussion includes comments concerning the apparent preference of the system to maintain some finite entropy, even in the presence of an extremely large spin–spin coupling.

Surface Phase Transition in Anisotropic Heisenberg Models

The behavior of an anisotropic Heisenberg model on a semi-infinite cubic lattice is studied. We employ the Green's function formalism and the random-phase approximation in our analysis. We consider different possible configurations for the anisotropy parameters of the surface and bulk. We determine the profile of the magnetization and the renormalized magnon energy spectrum as a function of temperature and of the anisotropy parameters of the surface and bulk. We also find the complete phase diagram for the multicritical points which is in agreement with the real-space renormalization group calculations. The onset of surface ordering takes place when the surface magnons become more energetic than the bulk modes. In the limit where bulk and surface are both described by a pure Heisenberg model, there is no long-range surface magnetic order over the paramagnetic bulk, whatever the values of the exchange couplings are. We relate this result with the Mermin-Wagner theorem. We show that the slope of the surface magnetization is discontinuous at the bulk critical temperature and that the discontinuity increases when the model on the surface varies from Ising to Heisenberg. In the particular case when the exchange couplings become antiferromagnetic, we have found the canted-paramagnetic critical field at the surface which is larger than that of the bulk. At very low temperatures, the critical field at the surface does not follow a T3/2 Bloch law, but instead decreases linearly with temperature.

Evidence for a genuine ferromagnetic to paramagnetic reentrant phase transition in a Potts spin-glass model

uch experimental and theoretical efforts have been devoted in the past twenty years to search for a genuine thermodynamic reentrant phase transition from a ferromagnetic to either a paramagnetic or spin glass phase in disordered ferromagnets. So far, no real system or theoretical model of a short-range spin glass system has been shown to convincingly display such a reentrant transition. We present here results from Migdal-Kadanoff real-space renormalization-group calculations that provide for the first time strong evidence for ferromagnetic to paramagnetic reentrance in Potts spin glasses on hierarchical lattices. Our results imply that there is no fundamental reason ruling out thermodynamic reentrant phase transitions in all non mean-field randomly frustrated systems, and may open the possibility that true reentrance might occur in some yet to be discovered real randomly frustrated materials.

Critical behaviour of the random field Ising model

We present an analytic real-space renormalization group calculation for the random-field Ising model. We apply the Migdal-Kadanoff approximation for the renormalization of a cubic cell in dimensions d, introducing a new field partitioning scheme which allows us to treat the random-field fluctuations in a coherent manner. Our scheme leads naturally to a lower critical dimensionality and allows us to calculate a complete set of three independent exponents in arbitrary dimension. In three dimensions the magnetization exponent and the Schwartz-Soffer inequality is almost satisfied as an equality. We expand analytically in . Further, we show that and the magnitude of the inequality go to zero exponentially with . We calculate the crossover exponent, from pure to the random-field system and find surprisingly good agreement with experimental values. We find that satisfies the Schwartz-Soffer inequality: , the susceptibility exponent of the pure system. We expand in and find that the magnitude of the inequality varies exponentially in . Finally we find that dimensional reduction is satisfied to first order in , with the reduced dimension .

Percolation Transition in the Random Antiferromagnetic Spin-1 Chain

We give a physical description in terms of percolation theory of the phase transition that occurs when the disorder increases in the random antiferromagnetic spin-1 chain between a gapless phase with topological order and a random singlet phase. We study the statistical properties of the percolation clusters by numerical simulations, and we compute exact exponents characterizing the transition by a real-space renormalization group calculation.

The magnetization of adsorbed3He studied by renormalization group techniques

The magnetic and thermodynamic properties of adsorbed3He have drawn considerably attention in the last few years. The experiments have been refined recently using different substrates and applied fields. In the ferromagnetic regime the system has been described by the Heisenberg Hamiltonian and its thermodynamics properties are well described at high temperatures by a Heisenberg series expansion. The very low temperature properties are described by calculations using spin wave theory. We present here real space renormalization group calculations that are valid in both domains and specially forT∼J, and a new approach which takes into account the finite size of the substrate.

The fractal behavior of the shortest path aggregation: simulations and RG calculations

The shortest-path aggregation (SPA) is an irreversible process in which particles are attached to the cluster on the closer perimeter sites to the release site. Randomness is introduced via the choice of sites from which the particles are sequentially released. This model generates random fractals which have dendritic structures. Simulations and kinetic real-space renormalization group calculations on a two-dimensional square lattice are presented. Numerically, we find D f=1.20±0.01 for the fractal dimension of the clusters. The universal behavior of the model against the way of releasing particles is observed. We find a transition from a weak correlation region to a strong correlation region. In particular, we find the fractal dimension D=1.20±0.01 when the released particles are weakly correlated, while a one-dimensional object is obtained when the correlation is strong. Analytically, 2 × 2 and 3 × 3 cell real-space renormalization group calculations yield D f = 1.19 and 1.21, respectively, in good agreement with the simulations. A set of heirarchical dimensions D( q) are also computed for the SPA cluster on the square lattice.

Density matrix formulation for quantum renormalization groups.

A generalization of the numerical renormalization-group procedure used first by Wilson for the Kondo problem is presented. It is shown that this formulation is optimal in a certain sense. As a demonstration of the effectiveness of this approach, results from numerical real-space renormalization-group calculations for Heisenberg chains are presented.

Ising model on a semi-infinite lattice: mean field renormalization group approach

We determine the critical properties of the Ising model on a semi-infinite cubic lattice. We use the mean field renormalization group with clusters of up to 25 spins. We show that the critical surface coupling enhancement at the special transition is monotonic with increasing cluster size and goes towards the recent Monte Carlo calculations. We also show that this critical coupling decreases with the distance which the bulk is taken from the free surface, and we explain the high values found for it by real space renormalization group calculations.

The two-dimensional Heisenberg ferromagnet as an approach to adsorbed 3He magnetism

Recent experiments have measured the magnetisation of one to three 3He atomic layers adsorbed on graphite. The properties of this novel system were interpreted by a two-dimensional ferromagnetic Heisenberg model. The authors perform real-space renormalisation group calculations for the two-dimensional triangular Heisenberg model with a magnetic field. They also make a quantum mechanical generalisation of a method which calculates the magnetisation recursively at any point of the global phase diagram. Their theoretical results for T>J=2.1 mK agree well with the experimental data and the exact high-temperature expansion. For T<J the agreement with the data is only qualitative.

Then-component cubic model and flows: Subgraph break-collapse method

We specialize to then-component cubic model the subgraph break-collapse method which we recently developed for theZ(λ) model. The cubic model has less symmetry than the Potts model, for which the method was originally developed, but nevertheless it is still possible to reduce considerably the computational complexity of the generalZ(λ) model. Our recursive algorithm is similar, forn=2, to the break-collapse method for theZ(4) model proposed by Mariz and co-workers. It allows the exact calculation for the partition function and correlation functions forn-component cubic clusters, withn as a variable, without the need to examine all of the spin configurations. An important application is therefore in real-space renormalization-group calculations.

Some properties of the surface magnetization of bulk magnets.

Rigorous results in the statistical mechanics of the Ising model of a magnet are used to analyze recent experimental data obtained by electron-capture spectroscopy on the surface magnetization of gadolinium as well as to test recent approximate real-space renormalization-group calculations of the surface magnetization of the Ising model. The experimental data are found to be consistent with these exact results, while the approximate renormalization-group calculations are found to be inconsistent.

Nonuniversal critical dynamics on the Fibonacci-chain quasicrystal.

An exact real-space renormalization-group calculation of the dynamic exponent z is given for Ising-Glauber dynamics on the Fibonacci-chain quasicrystal. The method is first illustrated for the simple cases of the uniform and alternating Glauber chains, which reproduce the well-known results z=2 and 1+\ensuremath{\Vert}${J}_{A}$/${J}_{B}$\ensuremath{\Vert} (${J}_{A}$g${J}_{B}$), respectively. For positive exchange interactions the critical Glauber dynamics of the Fibonacci chain is governed by a single, unstable one-cycle (fixed point) of the renormalization-group transformation. However, if one or all the exchange interactions are negative, then the critical dynamics is governed by a three-cycle. In all cases the dynamic exponent z is identical to that obtained for the alternating-bond Glauber chain.