Monte Carlo Renormalization Group Research Articles

Evidences of the Instability Fixed Points of First-Order Phase Transitions

We study the dynamics of the first-order phase transitions in the two-dimensional Potts model driven by a linearly varying temperature using a finite-time scaling with extended dynamic Monte Carlo renormalization group method. It is found that, for sufficiently large lattice sizes, the flows of apparent exponents of different temperature sweep rates upon renormalization show characteristics that are markedly distinct from those of continuous transitions and are argued to result from the instability fixed points involved and provide a method for estimating the associated instability exponents.

Exotic Gapless Mott Insulators of Bosons on Multileg Ladders

We present evidence for an exotic gapless insulating phase of hard-core bosons on multileg ladders with a density commensurate with the number of legs. In particular, we study in detail a model of bosons moving with direct hopping and frustrating ring exchange on a 3-leg ladder at ν=1/3 filling. For sufficiently large ring exchange, the system is insulating along the ladder but has two gapless modes and power law transverse density correlations at incommensurate wave vectors. We propose a determinantal wave function for this phase and find excellent comparison between variational Monte Carlo and density matrix renormalization group calculations on the model Hamiltonian, thus providing strong evidence for the existence of this exotic phase. Finally, we discuss extensions of our results to other N-leg systems and to N-layer two-dimensional structures.

Conformal or walking? Monte Carlo renormalization group studies ofSU(3)gauge models with fundamental fermions

Strongly coupled gauge systems with many fermions are important in many phenomenological models. I use the 2-lattice matching Monte Carlo renormalization group method to study the fixed point structure and critical indexes of SU(3) gauge models with 8 and 12 flavors of fundamental fermions. With an improved renormalization group block transformation I am able to connect the perturbative and confining regimes of the N_f=8 flavor system, thus verifying its QCD-like nature. With N_f=12 flavors the data favor the existence of an infrared fixed point and conformal phase, though the results are also consistent with very slow walking. I measure the anomalous mass dimension in both systems at several gauge couplings and find that they are barely different from the free field value.

The Wavelet-Analysis at Monte–Carlo Modeling of Phase Transitions in a Ferroelastic Crystal of Lithium Ammonium Sulphate

Using methods of the wavelet—analysis at Monte—Carlo modeling a behavior of thermodynamic parameters of Ising model for the ferroelectric—ferroelastic crystal of lithium ammonium sulphate LiNH4SO4 is studied. The discussed method is similar to Monte—Carlo Renormalization Group (MCRG), but it is more effective, thanks to a presence of algorithm of fast wavelet–transformation (WT). The short review of the wavelet transform and Monte—Carlo modeling by means of MCRG algorithm has been made. Temperature dependences of a specific heat capacity and dielectric permittivity for a set of lattice models with the sizes N × N at N = 64, 128, and 256 were derived as a result of 1, 2, 3-level WT. Critical exponents are found by numerical differentiation of a temperature dependence of thermodynamic parameters near a temperature of phase transition.

Alternative boundary conditions for Monte Carlo simulations based on self-consistent correlations: Application to the two- and three-dimensional Ising models

An alternative to periodic boundary conditions is developed and tested in Monte Carlo simulations of the two- and three-dimensional Ising models. The boundary conditions are based on a mean-field approach that incorporates consistency constraints for the magnetization and correlations between nearest neighbors by means of an effective field and an extra coupling between nearest neighbors at the boundary of the simulation box. During the simulation the self-consistent equations are solved, and statistics are accumulated to obtain thermodynamic averages. In comparison with the standard periodic boundary conditions the method gives a more accurate estimation of nonuniversal magnitudes, such as the transition temperature and the behavior of the magnetization, but it cannot compete with the accuracy of other strategies such as finite-size scaling theory or Monte Carlo renormalization group to obtain critical exponents.

Numerical coarse-graining of fluid field theories

We present a formalism for the systematic numerical coarse-graining of field-theoretic models of fluids that draws upon techniques from both the Monte Carlo renormalization group and particle-based coarse-graining literature. A force-matching technique initially developed for coarse-graining particle-based interaction potentials is adapted to calculate renormalized field-theoretic coupling coefficients in a complex-valued field theory, and a related method is introduced for coarse-graining field-theoretic operators. The viability of this methodology is demonstrated by coarse-graining a field-theoretic model of a Gaussian-core fluid and thereby reducing lattice discretization errors.

Stiffness dependence of critical exponents of semiflexible polymer chains situated on two-dimensional compact fractals

We present an exact and Monte Carlo renormalization group (MCRG) study of semiflexible polymer chains on an infinite family of the plane-filling (PF) fractals. The fractals are compact, that is, their fractal dimension df is equal to 2 for all members of the fractal family enumerated by the odd integer b(3<or=b<infinity). For various values of stiffness parameter s of the chain, on the PF fractals (for 3<or=b<or=9 ), we calculate exactly the critical exponents nu (associated with the mean squared end-to-end distances of polymer chain) and gamma (associated with the total number of different polymer chains). In addition, we calculate nu and gamma through the MCRG approach for b up to 201. Our results show that for each particular b, critical exponents are stiffness dependent functions, in such a way that the stiffer polymer chains (with smaller values of s) display enlarged values of nu, and diminished values of gamma. On the other hand, for any specific s, the critical exponent nu monotonically decreases, whereas the critical exponent gamma monotonically increases, with the scaling parameter b. We reflect on a possible relevance of the criticality of semiflexible polymer chains on the PF family of fractals to the same problem on the regular Euclidean lattices.

Valence Bond and von Neumann Entanglement Entropy in Heisenberg Ladders

We present a direct comparison of the recently proposed valence bond entanglement entropy and the von Neumann entanglement entropy on spin-1/2 Heisenberg systems using quantum Monte Carlo and density-matrix renormalization group simulations. For one-dimensional chains we show that the valence bond entropy can be either less or greater than the von Neumann entropy; hence, it cannot provide a bound on the latter. On ladder geometries, simulations with up to seven legs are sufficient to indicate that the von Neumann entropy in two dimensions obeys an area law, even though the valence bond entanglement entropy has a multiplicative logarithmic correction.

Investigating the critical properties of beyond-QCD theories using Monte Carlo renormalization group matching

Monte Carlo renormalization group (MCRG) methods were designed to study the nonperturbative phase structure and critical behavior of statistical systems and quantum field theories. I adopt the 2-lattice matching method used extensively in the 1980's and show how it can be used to predict the existence of nonperturbative fixed points and their related critical exponents in many flavor SU(3) gauge theories. This work serves to test the method and I study relatively well understood systems: the ${N}_{f}=0$, 4 and 16 flavor models. The pure gauge and ${N}_{f}=4$ systems are confining and chirally broken and the MCRG method can predict their bare step scaling functions. Results for the ${N}_{f}=16$ model indicate the existence of an infrared fixed point with nearly marginal gauge coupling. I present preliminary results for the scaling dimension of the mass at this new fixed point.

Critical dynamics of the two-dimensional random-bond Potts model with nonequilibrium Monte Carlo simulations

We study two-dimensional q -state random-bond Potts models for both q=8 and q=5 with a linearly varying temperature. By applying a successive Monte Carlo renormalization group procedure, both the static and dynamic critical exponents are obtained for randomness amplitudes (the strong to weak coupling ratio) of r_{0}=3 , 10, 15, and 20. The correlation length exponent nu increases with disorder from less than to larger than unity and this variation is justified by the good collapse of the specific heat near the critical region. The specific heat exponent is obtained by the usual hyperscaling relation alpha=2-dnu and thus indicates no possibility of the activated dynamic scaling. Both r_{0} and q have effects on the critical dynamics of the disordered systems, which can be seen from variations of the rate exponent, the hysteresis exponent, and the dynamic critical exponent. Implications of these results are discussed.

Classical and quantum anisotropic Heisenberg antiferromagnets

We study classical and quantum Heisenberg antiferromagnets with exchange anisotropy of XXZ-type and crystal field single-ion terms of quadratic and quartic form in a field. The magnets display a variety of phases, including the spin-flop (or, in the quantum case, spin-liquid) and biconical (corresponding, in the quantum lattice gas description, to supersolid) phases. Applying ground-state considerations, Monte Carlo and density matrix renormalization group methods, the impact of quantum effects and lattice dimension is analysed. Interesting critical and multicritical behaviour may occur at quantum and thermal phase transitions.

Cluster-algorithm renormalization-group study of universal fluctuations in the two-dimensional Ising model

In this paper we propose a method to study critical systems numerically, which combines collective-mode algorithms and renormalization group on the lattice. This method is an improved version of the Monte Carlo renormalization group in the sense that it has all the advantages of cluster algorithms. As an application we considered the 2D Ising model and studied whether scale invariance or universality are possible underlying mechanisms responsible for the approximate "universal fluctuations" close to a so-called bulk temperature T(L) . "Universal fluctuations" were first proposed in the work of Bramwell, Holdsworth, and Pinton [Nature (London) 396, 552 (1998)] and stated that the probability density function of a global quantity for very dissimilar systems, such as a confined turbulent flow and a two-dimensional (2D) magnetic system, properly normalized to the first two moments, becomes similar to the "universal distribution," originally obtained for magnetization in the 2D XY model in the low-temperature region. The results for the critical exponents and the renormalization-group flow of the probability density function are very accurate and show no evidence to support that the approximate common shape of the PDF should be related to both scale invariance or universal behavior.

Critical behavior of the system of two crossing self-avoiding walks on a family of three-dimensional fractal lattices

We study the polymer system consisting of two-polymer chains situated in a fractal container that belongs to the three-dimensional Sierpinski Gasket (3D SG) family of fractals. The two-polymer system is modeled by two interacting self-avoiding walks (SAW) immersed in a good solvent. To conceive the inter-chain interactions we apply the model of two crossing self-avoiding walks (CSAW) in which the chains can cross each other. By applying renormalization group (RG) method, we establish the relevant phase diagrams for b = 2 and b = 3 members of the 3D SG fractal family. Also, at the appropriate transition fixed points we calculate the contact critical exponents φ , associated with the number of contacts between monomers of different chains. For larger b ( 2 ⩽ b ⩽ 30 ) we apply Monte Carlo renormalization group (MCRG) method, and compare the obtained results for φ with phenomenological proposals for the contact critical exponent, as well as with results obtained for other similar models of two-polymer system.

Large magnetoresistance in a manganite spin tunnel junction usingLaMnO3as the insulating barrier

A spin-tunnel-junction based on manganites, with La1 xSrxMnO3 (LSMO) as ferromagnetic metallic electrodes and the undoped parent compound LaMnO3 (LMO) as insulating barrier, is here theoretically discussed using double exchange model Hamiltonians and numerical techniques. For an even number of LMO layers, the ground state is shown to have anti-parallel LSMO magnetic moments. This highly resistive, but fragile, state is easily destabilized by small magnetic fields, which orient the LSMO moments in the direction of the field. The magnetoresistance associated with this transition is very large, according to Monte Carlo and Density Matrix Renormalization Group studies. The influence of temperature, the case of an odd number of LMO layers, and the differences between LMO and SrTiO3 as barriers are also addressed. General trends are discussed. The study of strongly correlated electronic systems (SCES) continues attracting the attention of the Condensed Matter community. These materials present complex phase diagrams that illustrates the competition which exists among phases with very different physical properties, such as d-wave superconductivity, antiferroand ferro-magnetic order, charge- and orbital-order, multiferroic behavior, and several others. Moreover, this complexity and phase competition lead to self-organized nano-scale inhomogeneities which are believed to generate giant responses, as in the famous colossal magnetoresistance (CMR) effect of the Mn-oxides known as manganites. 1

The ferrimagnetic phase of the Blume-Emery-Griffiths model on the cellular automaton

Abstract The Blume-Emery-Griffiths model is simulated using the cooling algorithm which is improved from the Creutz cellular automaton (CCA) under periodic boundary conditions. The simulations are carried out on a simple cubic lattice at K/J = −1.5 in the range of −3.5 &lt; D/J &lt; 0.5, with J and K representing the nearestneighbour bilinear and biquadratic interactions, D being the single-ion anisotropy parameter. The phase diagram characterizing phase transition of the model is obtained. We found different kinds of phase transitions between the ferromagnetic, quadrupolar, staggered quadrupolar and ferrimagnetic phases for K/J = −1.5. In particular, the region of the phase diagram containing a ferrimagnetic phase is explored and compared to those obtained by other methods. The simulations confirm that the ferrimagnetic phase occurs in the narrow interval −3.006 ≤ D/J &lt; −3. This result is in a good agreement with Monte Carlo renormalization group and closer to the cluster variation method result than the mean field approximation result.

STRING TENSION AND REMOVAL OF LATTICE COARSENING EFFECTS IN MONTE CARLO RENORMALIZATION GROUP

We study the computation of the static quark potential under decimations in the Monte Carlo Renormalization Group (MCRG). Employing a multirepresentation plaquette action, we find that fine-tuning the decimation prescription so that the MCRG equilibrium self-consistency condition is satisfied produces dramatic improvement at large distances. In particular, lattice coarsening (change of effective lattice spacing on action-generated lattices after decimation) is nearly eliminated. Failure to correctly tune the decimation, on the other hand, produces large coarsening effects, of order 50% or more, consistent with those seen in previous studies. We also study rotational invariance restoration at short distances, where no particular improvement is seen for this action.

Determination of the dynamic and static critical exponents of the two-dimensional three-state Potts model using linearly varying temperature

We employ a successive Monte Carlo renormalization group procedure in the presence of a linearly varying temperature to study the three-state (q=3) Potts model on square lattices. By matching correlation functions of different lattice sizes at different renormalization levels, a rate exponent r associated with the temperature sweep rate R and the correlation length exponent nu are obtained. The dynamic critical exponent z is then obtained by the scaling law z=r-1/nu derived in our method. The results of z=2.171(62) for q=3 in this work and the previously obtained z=2.15(13) for q=2 seem to support the extension of the "weak universality" hypothesis to dynamic critical behavior. With these calculated exponents, the dynamic scaling forms of both specific heat and order parameter at various R are presented and all the other static critical exponents are determined, which verify our method. Discussions are made on how to improve the accuracy of the estimation of the dynamic critical exponent.

Improving the Improved Action

We investigate the construction of improved actions by the Monte Carlo renormalization group method in the context of SU(2) gauge theory utilizing different decimation procedures and effective actions. We demonstrate that the basic self-consistency requirement for correct application of the Monte Carlo renormalization group, i.e., that the decimated configurations are equilibrium configurations of the adopted form of the effective action, can be achieved only by careful fine-tuning of the choice of decimation prescription and/or action.

Renormalization group therapy

We point out a general problem with the procedures commonly used to obtain improved actions from Monte Carlo Renormalization Group (MCRG) decimated configurations. Straightforward measurement of the couplings from the decimated configurations, by one of the known methods, can result into actions that do not correctly reproduce the physics on the undecimated lattice. This is because the decimated configurations are generally not representative of the equilibrium configurations of the assumed form of the effective action at the measured couplings. Curing this involves fine-tuning of the chosen MCRG decimation procedure, which is also dependent on the form assumed for the effective action. We illustrate this in decimation studies of the $SU(2)$ lattice gauge theory using Swendsen and ``double smeared blocking'' decimation procedures. A single-plaquette improved action involving five group representations and nearly free of this pathology is given.

On the number of contacts of a floating polymer chain cross-linked with a surface adsorbedchain on fractal structures

We study the problem of the interaction of a linear polymer chain, floating in fractal containersthat belong to the three-dimensional Sierpinski gasket (3D SG) family of fractals, with asurface adsorbed linear polymer chain. Each member of the 3D SG fractal family has a fractalimpenetrable 2D adsorbing surface, which appears to be 2D SG fractal. The two-polymersystem is modelled by two mutually crossing self-avoiding walks. By applying the MonteCarlo renormalization group (MCRG) method, we calculate the critical exponents,φ, associated with the number of contacts of the 3D SG floating polymer chain,and the 2D SG adsorbed polymer chain, for a sequence of SG fractals with2 ≤ b ≤ 40. Also, we propose a codimension additivity argument formula forφ, and compare its predictions with our reliable set of MCRG data. We find thatφ decreases monotonicallywith increasing b, that is, with increase of the container fractal dimension. Finally, we discuss the relationsbetween different contact exponents, and analyse their possible behaviour in thefractal-to-Euclidean crossover region b → ∞