Position-space Renormalisation Group Research Articles

Topological crystalline phases in a disordered inversion-symmetric chain

When translational symmetry is broken by bulk disorder, the topological nature of states in topological crystalline systems may change depending on the type of disorder that is applied. In this work, we characterize the phases of a one-dimensional (1D) chain with inversion and chiral symmetries, where every disorder configuration is inversion-symmetric. By using a basis-independent formulation for the inversion topological invariant, chiral winding number, and bulk polarization, we are able to construct phase diagrams for these quantities when disorder is present. We show that unlike the chiral winding number and bulk polarization, the inversion topological invariant can fluctuate when the bulk spectral gap closes at strong disorder. Using the position-space renormalization group, we are able to compare how the inversion topological invariant, chiral winding number and bulk polarization behave at low energies in the strong disorder limit. We show that with inversion symmetry-preserving disorder, the value of the inversion topological invariant is determined by the inversion eigenvalues of the states at the inversion centers, while quantities such as the chiral winding number and the bulk polarization still have contributions from every state throughout the chain. We also show that it is possible to alter the value of the inversion topological invariant in a clean system by occupying additional states at the inversion centers while keeping the bulk polarization fixed. We discuss the implications of our results for topological crystalline phases in higher-dimensional electronic systems, and in ultra-cold atomic systems.

Renormalization Group Circuits for Weakly Interacting Continuum Field Theories

AbstractWe develop techniques to systematically construct local unitaries which map scale‐invariant, product state wavefunctionals to the ground states of weakly interacting, continuum quantum field theories. More broadly, we devise a “quantum circuit perturbation theory” to construct local unitaries which map between any pair of wavefunctionals which are each Gaussian with arbitrary perturbative corrections. Further, we generalize cMERA to interacting continuum field theories, which requires reworking the existing formalism which is tailored to non‐interacting examples. Our methods enable the systematic perturbative calculation of cMERA circuits for weakly interacting theories, and as a demonstration we compute the 1‐loop cMERA circuit for scalar ϕ4 theory and analyze its properties. In this case, we show that Wilsonian renormalization of the spatial momentum modes is equivalent to a local position space cMERA circuit. This example provides new insights into the connection between position space and momentum space renormalization group methods in quantum field theory. The form of cMERA circuits derived from perturbation theory suggests useful ansatzes for numerical variational calculations.

Random Crystal Field in a Mixed Spin S = 1/2 and S = 3/2 Ising Model by Renormalization Group Theory

In this work, we study the Ising model with mixed spins S = 1/2 and S = 3/2 on the hypercubic lattice and the random crystal field at the sublattice with S = 3/2 described by a two peaks law. To achieve this, we use an approximation of position space renormalization group (PSRG) namely Migdal-Kadanoff in which we use both the free energy derivative and the flow in the parameter space of the Hamiltonian. For all values of the random probability, the critical behavior is determined via the critical exponents at the second-order fixed points while at low temperatures; the discontinuities of the of the free energy derivative provide the positions of the first-order transitions. The introduction of a minimal amount of disorder causes a change in the phase diagram showing the relevance of disorder for d = 2 and d = 3. The second-order transition remains always with the same critical exponents as those of the pure model, and a new first-order transition appears at very low temperature. Also, a comparison with other similar works is given.

The spin-2 Blume-Capel model by position space renormalization group

Using the renormalization group approximation and specifically the Migdal-Kadanoff technique, we study the spin-2 Blume-Capel model on d-dimensional hypercubic lattice. The phase diagram at the ground state of this model is determined. For T≠0K, we give also the phase diagram in the (temperature, anisotropy) plane, with first and second-order phase transitions and tricritical points. The associated fixed points are drawn up in a table, and by linearizing the transformation at the vicinity of these points, we determine the critical exponents for d=2 and d=3. Finally, this work is completed by a discussion and comparison with other approximation.

The two- and three-dimensional spin-3/2 random Blume–Capel model by the position space renormalization group

We use the Migdal–Kadanoff renormalization group technique to study the spin-3/2 Blume–Capel model under a random crystal field, in the two- and three-dimensional cases. Studying the fixed points and the phase diagrams established, we find interesting results allowing us to understand the critical behavior of the system. In the two-dimensional case, the randomness, even in small amounts, removes completely the first order transition between the two ferromagnetic phases present, replacing it by a smooth continuation. Only the second order phase transitions occur. In the three-dimensional case, the first order phase transition disappears only at a certain threshold of randomness. Below this threshold, we observe the presence of an end-point where finishes the first order transition line inside the ferromagnetic phases. This end-point reaches T=0K at a critical value of probability, beyond which only the second order transitions occur.

Non-equilibrium phase transitions in the two-temperature Ising model with Kawasaki dynamics

Phase transitions of the two-finite temperature Ising model on a square lattice are investigated by using a position space renormalization group (PSRG) transformation. Different finite temperatures, T x and T y , and also different time-scale constants, α x and α y for spin exchanges in the x and y directions define the dynamics of the non-equilibrium system. The critical surface of the system is determined by RG flows as a function of these exchange parameters. The Onsager critical point (when the two temperatures are equal) and the critical temperature for the limit when the other temperature is infinite, previously studied by the Monte Carlo method, are obtained. In addition, two steady-state fixed points which correspond to the non-equilibrium phase transition are presented. These fixed points yield the different universality class properties of the non-equilibrium phase transitions.

Position space renormalization group study of the spin-1 random semi-infinite Blume–Capel model

We study the spin-1 Blume–Capel model under a random crystal field in the tridimensional semi-infinite case. This has been done by using the real-space renormalization group approximation and specifically the Migdal–Kadanoff technique. Interesting results are obtained, which tell us that the randomness destroys the first order phase transitions and only those of the second order occur. We give the list of nine fixed points and their topology describing the surface critical behavior. Five new types of phase diagram are found with a rich variety of phase transitions, in accordance with the values of the bulk and surface probabilities and the ratios linking bulk and surface interactions.

The tensor renormalization group for pure and disordered two-dimensional lattice systems

The tensor renormalization group (TRG) is a powerful new approach for coarse-graining classical two-dimensional (2D) lattice Hamiltonians. It uses the intuitive framework of traditional position space renormalization group methods–analyzing flows in the space of Hamiltonian parameters–but can be systematically improved to yield thermodynamic properties at much higher precision. We present initial results demonstrating that the TRG can be generalized to quenched random systems, applying it to obtain the phase diagram of a bond-diluted triangular lattice Ising ferromagnet. This opens a variety of potential future applications, most prominently spin glasses.

High-precision thermodynamic and critical properties from tensor renormalization-group flows

The recently developed tensor renormalization-group (TRG) method provides a highly precise technique for deriving thermodynamic and critical properties of lattice Hamiltonians. The TRG is a local coarse-graining transformation, with the elements of the tensor at each lattice site playing the part of the interactions that undergo the renormalization-group flows. These tensor flows are directly related to the phase diagram structure of the infinite system, with each phase flowing to a distinct surface of fixed points. Fixed-point analysis and summation along the flows give the critical exponents, as well as thermodynamic functions along the entire temperature range. Thus, for the ferromagnetic triangular lattice Ising model, the free energy is calculated to better than 10(-5) along the entire temperature range. Unlike previous position-space renormalization-group methods, the truncation (of the tensor index range D) in this general method converges under straightforward and systematic improvements. Our best results are easily obtained with D=24, corresponding to 4624-dimensional renormalization-group flows.

A non-equilibrium Monte Carlo renormalization-group approach based upon the microscopic master equation applied to the three-state driven lattice gas

We present a general position-space renormalization-group approach for systems in steady states far from equilibrium and illustrate its application to the three-state driven lattice gas. The method is based upon the possibility of a closed form representation of the parameters controlling transition rates of the system in terms of the steady state probability distribution of small clusters, arising from the application of the master equations to small clusters. This probability distribution on various length scales is obtained through a Monte Carlo algorithm on small lattices, which then yields a mapping between parameters on different length scales. The renormalization-group flows indicate the phase diagram, analogous to equilibrium treatments. For the three-state driven lattice gas, we have implemented this procedure and compared the resulting phase diagrams with those obtained directly from simulations. Results in general show the expected topology with one exception. For high densities, an unexpected additional fixed point emerges, which can be understood qualitatively by comparing it with the fixed point of the fully asymmetric exclusion process.

Position-space renormalization-group approach for driven diffusive systems applied to the asymmetric exclusion model.

This paper introduces a position-space renormalization-group approach for nonequilibrium systems and applies the method to a driven stochastic one-dimensional gas with open boundaries. The dynamics are characterized by three parameters: the probability alpha that a particle will flow into the chain to the leftmost site, the probability beta that a particle will flow out from the rightmost site, and the probability p that a particle will jump to the right if the site to the right is empty. The renormalization-group procedure is conducted within the space of these transition probabilities, which are relevant to the system's dynamics. The method yields a critical point at alpha(c)=beta(c)=1/2, in agreement with the exact values, and the critical exponent nu=2.71, as compared with the exact value nu=2.00.

Elastic properties of Sierpinski-like carpets: finite-element-based simulation.

The elastic properties of two-dimensional continuous composites of fractal structures are studied with the set of Sierpinski-like carpets filled by voids or rigid inclusions. The effective elastic moduli of these carpets are calculated numerically using the finite-element and position-space renormalization group techniques. The fixed-point problem is analyzed by flow diagrams in the plane of the current Poisson ratios and coefficients of anisotropy of the composites. It is found that in the general case the effective elastic moduli asymptotically approach a power-law behavior. Moreover, the common exponent characterizes the scaling behavior of each component of the elastic modulus tensor of a definite carpet. The values of the scaling exponents and positions of the fixed points are shown to be independent of the elastic properties of the host and depend significantly on the fractal dimension of the composite.

Elastic properties of a disordered continuum of regular fractal structure: Self-consistent approach and finite-element method simulations

Macromolecular structures, as well as aggregation of filler in polymer-based composites, often may be described properly as fractals. Scaling behavior of the elastic moduli of a modeled fractal, the Sierpinski carpet, was the subject of this study. Sheng and Tao [1] and Patlazhan [2] found that, in the case of voids in on elastic host, axial and shear moduli exhibit distinct scaling dependencies on the size of the system. Nevertheless, it is widely accepted that moduli of random isotropic fractals (percolation clusters) scale with the same exponents. Explanation of the discrepancy is one of the main targets of the paper. The self-consistent approach and position space renormalization group technique (PSRG) have been applied for this goal. The mapping, corresponding to PSRG, was constructed numerically using the finite-element method (FEM) in the cases of voids and rigid inclusions. The self-consistent approach gives scaling behavior with exponents of values of about 0.11, independent of the modulus and type of inclusion, at developed stages of the fractal. It has been shown that mappings of PSRG on the plane, for two ratios of three independent moduli, have stable fixed points. This means that different elastic moduli exhibit scaling behavior with the same exponents (0.29 for voids and 0.17 for rigid squares) for developed fractal structure. The discrepancy in the exponent values obtained in the previous simulations is caused by the analysis of the initial stages of the structure. We believe that analogous results are valid for the wide class of self-similar fractals, and the dimension is the main parameter that governs the exponents and fixed point values.

Novel position-space renormalization group for bond directed percolation in two dimensions

A new position-space renormalization group approach is investigated for bond directed percolation in two dimensions. The threshold value for the bond occupation probabilities is found to be p c =0.6443. Correlation length exponents on time (parallel) and space (transverse) directions are found to be ν ∥=1.719 and ν ⊥=1.076, respectively, which are in very good agreement with the best-known series expansion results.

Double transitions in the fully frustrated XY model

The fully frustrated XY model is studied via the position-space renormalization group approach. The model is mapped into two coupled XY models, for which the scaling equations are derived. By integrating directly the scaling equations, we observe that there exists a narrow temperature range in which both the vortex and coupling charge fugacities grow large, suggesting double transitions in the system. While the transition at lower temperature is identified to be of the Kosterlitz-Thouless type, the higher-temperature one appears not to be of the Ising universality class.

CRITICAL DYNAMICS OF THE XY-MODEL ON THE ONE-DIMENSIONAL SUPERLATTICE BY POSITION SPACE RENORMALIZATION GROUP

The critical dynamics of the isotropic XY-model on the one-dimensional superlattice is considered in the framework of the position space renormalization group theory. The decimation transformation is introduced by considering the equations of motion of the operators associated to the excitations of the system, and it corresponds to an extension of the procedure introduced by Stinchcombe and dos Santos (J. Phys. A18, L597 (1985)) for the homogeneous lattice. The dispersion relation is obtained exactly and the static and dynamic scaling forms are explicitly determined. The dynamic critical exponent is also obtained and it is shown that it is identical to the one of the XY-model on the homogeneous chain.

On position-space renormalization group approach to percolation

In a position-space renormalization group (PSRG) approach to percolation one calculates the probabilityR(p,b) that a finite lattice of linear sizeb percolates, wherep is the occupation probability of a site or bond. A sequence of percolation thresholdsp c (b) is then estimated fromR(p c ,b)=p c (b) and extrapolated to the limitb→∞ to obtainp c =p c (∞). Recently, it was shown that for a certain spanning rule and boundary condition,R(p c ,∞)=R c is universal, and sincep c is not universal, the validity of PSRG approaches was questioned. We suggest that the equationR(p c ,b)=α, where α isany number in (0,1), provides a sequence ofp c (b)'s thatalways converges top c asb→∞. Thus, there is anenvelope from any point inside of which one can converge top c . However, the convergence is optimal if α=R c . By calculating the fractal dimension of the sample-spanning cluster atp c , we show that the same is true aboutany critical exponent of percolation that is calculated by a PSRG method. Thus PSRG methods are still a useful tool for investigating percolation properties of disordered systems.

Regularity properties and pathologies of position-space renormalization-group transformations: Scope and limitations of Gibbsian theory

We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Regarding regularity, we show that the RG map, defined on a suitable space of interactions (= formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension $d \ge 3$, these pathologies occur in a full neighborhood $\{ \beta > \beta_0 ,\, |h| < \epsilon(\beta) \}$ of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension $d \ge 2$, the pathologies occur at low temperatures for arbitrary magnetic-field strength. Pathologies may also occur in the critical region for Ising models in dimension $d \ge 4$. We discuss in detail the distinction between Gibbsian and non-Gibbsian measures, and give a rather complete catalogue of the known examples. Finally, we discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems.

Influence of morphology on the optical properties of nanocermets and granular inhomogeneous media

The macroscopic dielectric function of 2-dimensional (2D) and 3-dimensional (3D) inhomogeneous media depends on the morphology of the sample. Usual effective medium theories (namely the Maxwell Garnett theory and the mean field theories of Bruggeman, Ping Sheng) do not take into account the actual morphology of the clusters. We proposed earlier a position space renormalization group approach, based on the Kadanoff blocks method, to calculate the optical properties of cermets and granular materials. The aim of this work is to show the influence of the morphology of these media on their optical properties by applying this model to various samples that only differ in their morphological parameters (same materials, same concentration and same thickness).

Nonuniversality and analytical continuation in moments of directed polymers on hierarchical lattices

We prove the moments of the directed polymer partition function GZ, using an exact position space renormalization group scheme on a hierarchical lattice. After sufficient iteration the characteristic functionf(n)=ln〈GZn〉 of the probability ℘(Z) converges to a stable limitf*(n). For smalln the limiting behavior is independent of the initial distribution, while for largen,f*(n) is completely determined by it and is thus nonuniversal. There is a smooth crossover between the two regimes for small effective dimensions, and the nonlinear behavior of the small moments can be used to extract information on the universal scaling properties of the distribution. For large effective dimensions there is a sharp transition between the two regimes, and analytical continuation from integer moments ton→0 is not possible. Replica arguments can account for most features of the observed results.