Renormalization Group Iterations Research Articles

Energy Correlations of Non-Integrable Ising Models: The Scaling Limit in the Cylinder

We consider a class of non-integrable 2D Ising models whose Hamiltonian, in addition to the standard nearest neighbor couplings, includes additional weak multi-spin interactions which are even under spin flip. We study the model in cylindrical domains of arbitrary aspect ratio and compute the multipoint energy correlations at the critical temperature via a multiscale expansion, uniformly convergent in the domain size and in the lattice spacing. We prove that, in the scaling limit, the multipoint energy correlations converge to the same limiting correlations as those of the nearest neighbor Ising model in a finite cylinder with renormalized horizontal and vertical couplings, up to an overall multiplicative constant independent of the shape of the domain. The proof is based on a representation of the generating function of correlations in terms of a non-Gaussian Grassmann integral, and a constructive Renormalization Group (RG) analysis thereof. A key technical novelty compared with previous works is a systematic analysis of the effect of the boundary corrections to the RG flow, in particular a proof that the scaling dimension of boundary operators is better by one dimension than their bulk counterparts. In addition, a cancellation mechanism based on an approximate image rule for the fermionic Green’s function is of crucial importance for controlling the flow of the (superficially) marginal boundary terms under RG iterations.

Different critical behaviors in perovskites with a structural phase transition from cubic-to-trigonal and cubic-to-tetragonal symmetry

Perovskites like LaAlO3 (or SrTiO3) undergo displacive structural phase transitions from a cubic crystal to a trigonal (or tetragonal) structure. For many years, the critical exponents in both these types of transitions have been fitted to those of the isotropic three-components Heisenberg model. However, field theoretical calculations showed that the isotropic fixed point of the renormalization group is unstable, and renormalization group iterations flow either to a cubic fixed point or to a fluctuation-driven first-order transition. Here we show that these two scenarios correspond to the cubic to trigonal and to the cubic to tetragonal transitions, respectively. In both cases, the critical behavior is described by slowly varying effective critical exponents, which exhibit universal features. For the trigonal case, we predict a crossover of the effective exponents from their Ising values to their cubic values (which are close to the isotropic ones). For the tetragonal case, the effective exponents can have the isotropic values over a wide temperature range, before exhibiting large changes en route to the first-order transition. New renormalization group calculations near the isotropic fixed point in three dimensions are presented and used to estimate the effective exponents, and dedicated experiments to test these predictions are proposed. Similar predictions apply to cubic magnetic and ferroelectric systems.

Maximally random discrete-spin systems with symmetric and asymmetric interactions and maximally degenerate ordering.

Discrete-spin systems with maximally random nearest-neighbor interactions that can be symmetric or asymmetric, ferromagnetic or antiferromagnetic, including off-diagonal disorder, are studied, for the number of states q=3,4 in d dimensions. We use renormalization-group theory that is exact for hierarchical lattices and approximate (Migdal-Kadanoff) for hypercubic lattices. For all d>1 and all noninfinite temperatures, the system eventually renormalizes to a random single state, thus signaling q×q degenerate ordering. Note that this is the maximally degenerate ordering. For high-temperature initial conditions, the system crosses over to this highly degenerate ordering only after spending many renormalization-group iterations near the disordered (infinite-temperature) fixed point. Thus, a temperature range of short-range disorder in the presence of long-range order is identified, as previously seen in underfrustrated Ising spin-glass systems. The entropy is calculated for all temperatures, behaves similarly for ferromagnetic and antiferromagnetic interactions, and shows a derivative maximum at the short-range disordering temperature. With a sharp immediate contrast of infinitesimally higher dimension 1+ε, the system is as expected disordered at all temperatures for d=1.

Critical asymmetry in renormalization group theory for fluids

The renormalization-group (RG) approaches for fluids are employed to investigate critical asymmetry of vapour-liquid equilibrium (VLE) of fluids. Three different approaches based on RG theory for fluids are reviewed and compared. RG approaches are applied to various fluid systems: hard-core square-well fluids of variable ranges, hard-core Yukawa fluids, and square-well dimer fluids and modelling VLE of n-alkane molecules. Phase diagrams of simple model fluids and alkanes described by RG approaches are analyzed to assess the capability of describing the VLE critical asymmetry which is suggested in complete scaling theory. Results of thermodynamic properties obtained by RG theory for fluids agree with the simulation and experimental data. Coexistence diameters, which are smaller than the critical densities, are found in the RG descriptions of critical asymmetries of several fluids. Our calculation and analysis show that the approach coupling local free energy with White's RG iteration which aims to incorporate density fluctuations into free energy is not adequate for VLE critical asymmetry due to the inadequate order parameter and the local free energy functional used in the partition function.

Quantum entanglement and quantum phase transition in theXYmodel with staggered Dzyaloshinskii-Moriya interaction

We study the quantum entanglement and quantum phase transition (QPT) of the anisotropic spin-$1/2$ $XY$ model with staggered Dzyaloshinskii-Moriya (DM) interaction by means of the quantum renormalization group method. The scaling of coupling constants and the critical points of the system are obtained. It is found that when the number of renormalization group iterations tends to infinity, the system exhibit a QPT between the spin-fluid and N\'eel phases which correspond with two saturated values of the concurrence for a given value of the strength of DM interaction. The DM interaction can enhance the entanglement and influence the QPT of the system. To gain further insight, the first derivative of the entanglement exhibit a nonanalytic behavior at the critical point and it directly associates with the divergence of the correlation length. This shows that the correlation length exponent is closely related to the critical exponent, i.e., the scaling behaviors of the system.

Luttinger liquid fixed point for a two-dimensional flat Fermi surface

We consider a system of two-dimensional interacting fermions with a flat Fermi surface. The apparent conflict between Luttinger and non-Luttinger liquid behaviors found through different approximations is resolved by showing the existence of a line of nontrivial fixed points, for the renormalization group (RG) flow, corresponding to Luttinger liquid behavior; the presence of marginally relevant operators can cause flow away from the fixed point. The analysis is nonperturbative and based on the implementation, at each RG iteration, of Ward identities obtained from local phase transformations depending on the Fermi surface side, implying the partial vanishing of the beta function.

Convergence behavior of the density-matrix renormalization group algorithm for optimized orbital orderings

The density-matrix renormalization group algorithm has emerged as a promising new method in ab initio quantum chemistry. However, many problems still need to be solved before this method can be applied routinely. At the start of such a calculation, the orbitals originating from a preceding quantum chemical calculation must be placed in a specific order on a one-dimensional lattice. This ordering affects the convergence of the density-matrix renormalization group iterations significantly. In this paper, we present two approaches to obtain optimized orderings of the orbitals. First, we use a genetic algorithm to optimize the ordering with respect to a low total electronic energy obtained at a predefined stage of the density-matrix renormalization group algorithm with a given number of total states kept. In addition to that, we derive orderings from the one- and two-electron integrals of our test system. This test molecule is the chromium dimer, which is known to possess a complicated electronic structure. For this molecule, we have carried out calculations for the various orbital orderings obtained. The convergence behavior of the density-matrix renormalization group iterations is discussed in detail.

Critical (?4)3,?

The Euclidean (φ4)3,ε model in R 3 corresponds to a perturbation by a φ4 interaction of a Gaussian measure on scalar fields with a covariance depending on a real parameter ε in the range 0≤ε≤1. For ε=1 one recovers the covariance of a massless scalar field in R 3 . For ε=0, φ4 is a marginal interaction. For 0≤ε<1 the covariance continues to be Osterwalder-Schrader and pointwise positive. We consider the infinite volume critical theory with a fixed ultraviolet cutoff at the unit length scale and we prove that for ε>0, sufficiently small, there exists a non-gaussian fixed point (with one unstable direction) of the Renormalization Group iterations. We construct the stable critical manifold near this fixed point and prove that under Renormalization Group iterations the critical theories converge to the fixed point.

Renormalization Group¶and the Melnikov Problem for PDE's

We give a new proof of persistence of quasi-periodic, low dimensional elliptic tori in infinite dimensional systems. The proof is based on a renormalization group iteration that was developed recently in [BGK] to address the standard KAM problem, namely, persistence of invariant tori of maximal dimension in finite dimensional, near integrable systems. Our result covers situations in which the so called normal frequencies are multiple. In particular, it provides a new proof of the existence of small-amplitude, quasi-periodic solutions of nonlinear wave equations with periodic boundary conditions.

Renormalization group flow and fixed point of the lattice topological charge in the 2D O(3)σmodel

We study the renormalization group evolution up to the fixed point of the lattice topological susceptibility in the 2D O(3) nonlinear $\ensuremath{\sigma}$ model. We start with a discretization of the continuum topological charge by a local charge density polynomial in the lattice fields. Among the different choices we propose also a Symanzik-improved lattice topological charge. We check step by step in the renormalization group iteration the progressive dumping of quantum fluctuations, which are responsible for the additive and multiplicative renormalizations of the lattice topological susceptibility with respect to the continuum definition. We find that already after three iterations these renormalizations are negligible and an excellent approximation of the fixed point is achieved. We also check by an explicit calculation that the assumption of slowly varying fields in iterating the renormalization group does not lead to a good approximation of the fixed point charge operator.

DISTRIBUTIONS OF STRUCTURAL PROPERTIES FOR PERCOLATION CLUSTERS

The distributions of structural properties, which span between two terminals on percolation clusters, are studied using the “H-cell” renormalization group (RG). The RG iteration for these distributions is directly related to a simple Galton-Watson branching process, and we therefore apply theorems developed in the mathematical literature to obtain exact expressions for the asymptotic distribution functions. Distributions of the minimal path lengths, average edge-to-edge self-avoiding walk lengths, and of the masses of percolation clusters are found, and we derive exact exponential forms for the asymptotic tail behavior of the scaled probability densities at both small and large arguments. We also find new results for the multifractal distribution of wave functions on these clusters.

Infrared asymptotic freedom of a hierarchical ? 3 6 lattice theory

For the weakly coupled latticeφ 3 6 theory in a hierarchical model approximation a nonperturbative renormalization group analysis in the spirit of Gawedzki and Kupiainen is performed to study the flow of the effective actions. We deduce a domain of attraction to the tricritical (Gaussian) fixed point. The two relevant coupling constants of the problem are controlled by analytic continuation to complex domains, tracing their images under the renormalization group iterations.

First- and second-order transitions in the Potts model near four dimensions

The continuum generalization of the $p$-state Potts model is analyzed in the ordered phase. Renormalization-group iterations in $d=4\ensuremath{-}\ensuremath{\epsilon}$ dimensions are followed by an elimination of the transverse modes and a mapping onto an effective Ising model. This model is then used to show that the transition is first order for $p&gt;{p}_{c}(d)$ and continuous for $p&lt;{p}_{c}(d)$. We find that ${p}_{c}(d)=2$ for $d&gt;4$ and ${p}_{c}(4\ensuremath{-}\ensuremath{\epsilon})=2+\ensuremath{\epsilon}+O({\ensuremath{\epsilon}}^{2})$.