Renormalization-group Recursion Relations Research Articles

Puzzle of bicriticality in the XXZ antiferromagnet

Renormalization-group theory predicts that the XXZ antiferromagnet in a magnetic field along the easy Z-axis has asymptotically either a tetracritical phase-diagram or a triple point in the field-temperature plane. Neither experiments nor Monte Carlo simulations procure such phase diagrams. Instead, they find a bicritical phase-diagram. Here this discrepancy is resolved: after generalizing a ubiquitous condition identifying the tetracritical point, we employ new renormalization-group recursion relations near the isotropic fixed point, exploiting group-theoretical considerations and using accurate exponents at three dimensions. These show that the experiments and simulations results can only be understood if their trajectories flow towards the fluctuation-driven first order transition (and the associated triple point), but reach this limit only for prohibitively large system sizes or correlation lengths. In the crossover region one expects a bicritical phase diagram, as indeed is observed. A similar scenario may explain puzzling discrepancies between simulations and renormalization-group predictions for a variety of other phase diagrams with competing order parameters.

Magnetic phase transition induced by an electric field in coupled spin chains with magnetoelectric interaction

Abstract A magnetization reversal and a magnetic phase transition induced by an electric field are studied in the coupled XY spin chains model with the next-nearest-neighbor, Dzyaloshinskii-Moriya ( S n x S n + 1 y - S n y S n + 1 x ) , and magnetoelectric ( S n x S n + 1 y - S n y S n + 1 x ) S n + 2 z interactions. Using the linear perturbation renormalization group recursion relations the phase diagram in the plane (temperature, electric field) is found. The phase transitions to two different low temperature phases and a reentrance behavior caused by the electric field are observed. At finite temperature for some range of the interchain interaction values in addition to changing the magnetization sign with reversal of the electric field, there is a possibility to the magnetization sign switch in finite electric field values with no applied magnetic field.

Remarks on disorder and aperiodicity in a model for interacting polymers

We present a comparative study of the effects of random and aperiodically distributed interactions on the critical behavior of a model for two interacting polymers on a diamond hierarchical lattice. The problem is formulated in terms of exact renormalization-group (RG) recursion relations. In the disordered case, it is possible to develop a perturbative treatment in order to obtain the fixed points of the moments associated with the random distribution of interactions. Fully uncorrelated disorder may become relevant, driving the system away from a homogeneous fixed point. Layered disorder may lead to a breakdown of the perturbative treatment. In the case of aperiodic interactions, we also show some examples of relevance and irrelevance of geometric fluctuations, and further investigate the models by resorting to an independent transfer-matrix (TM) analysis, which fully corroborates the scaling results.

Critical properties of an aperiodic model for interacting polymers

We investigate the effects of aperiodic interactions on the critical behavior of an interacting two-polymer model on hierarchical lattices (equivalent to the Migadal-Kadanoff approximation for the model on Bravais lattices), via renormalization-group and tranfer-matrix calculations. The exact renormalization-group recursion relations always present a symmetric fixed point, associated with the critical behavior of the underlying uniform model. If the aperiodic interactions, defined by s ubstitution rules, lead to relevant geometric fluctuations, this fixed point becomes fully unstable, giving rise to novel attractors of different nature. We present an explicit example in which this new attractor is a two-cycle, with critical indices different from the uniform model. In case of the four-letter Rudin-Shapiro substitution rule, we find a surprising closed curve whose points are attractors of period two, associated with a marginal operator. Nevertheless, a scaling analysis indicates that this attractor may lead to a new critical universality class. In order to provide an independent confirmation of the scaling results, we turn to a direct thermodynamic calculation of the specific-heat exponent. The thermodynamic free energy is obtained from a transfer matrix formalism, which had been previously introduced for spin systems, and is now extended to the two-polymer model with aperiodic interactions.

FIELD BEHAVIOR OF AN ISING MODEL WITH APERIODIC INTERACTIONS

We derive exact renormalization-group recursion relations for an Ising model, in the presence of external fields, with ferromagnetic nearest-neighbor interactions on Migdal–Kadanoff hierarchical lattices. We consider layered distributions of aperiodic exchange interactions, according to a class of two-letter substitutional sequences. For irrelevant geometric fluctuations, the recursion relations in parameter space display a nontrivial uniform fixed point of hyperbolic character that governs the universal critical behavior. For relevant fluctuations, in agreement with previous work, this fixed point becomes fully unstable, and there appears a two-cycle attractor associated with a new critical universality class.

Influence of long-range interactions on the critical behavior of systems with a negative Fisher exponent

The influence of long-range interactions decaying in d dimensions as ${1/R}^{d+\ensuremath{\sigma}}$ on the critical behavior of systems with negativeFisher's correlation-function exponent for short-range interactions, ${\ensuremath{\eta}}_{\mathrm{SR}}<0,$ is reexamined. Such systems, typically described by ${\ensuremath{\varphi}}^{3}$-field theories, are, e.g., the Potts model in the percolation limit, the Edwards-Anderson spin-glass model, and the Yang-Lee edge singularity. In contrast to preceding studies, it is shown by means of Wilson's momentum-shell renormalization-group recursion relations that the long-range interactions dominate as long as $\ensuremath{\sigma}<2\ensuremath{-}{\ensuremath{\eta}}_{\mathrm{SR}}.$ Exponents change continuously to their short-range values at the boundary of this region.

Critical behavior of the Ising model on a hierarchical lattice with aperiodic interactions

We write the exact renormalization-group recursion relations for nearest-neighbor ferromagnetic Ising models on Migdal–Kadanoff hierarchical lattices with a distribution of aperiodic exchange interactions according to a class of substitutional sequences. For small geometric fluctuations, the critical behavior is unchanged with respect to the uniform case. For large fluctuations, as in the case of the Rudin–Shapiro sequence, the uniform fixed point in the parameter space cannot be reached from any physical initial conditions. We derive a criterion to check the relevance of the geometric fluctuations.

Lévy-flight spreading of epidemic processes leading to percolating clusters

We consider two stochastic processes, the Gribov process and the general epidemic process, that describe the spreading of an infectious disease. In contrast to the usually assumed case of short-range infections that lead, at the critical point, to directed and isotropic percolation respectively, we consider long-range infections with a probability distribution decaying in d dimensions with the distance as 1/R^{d+\sigma}. By means of Wilson's momentum shell renormalization-group recursion relations, the critical exponents characterizing the growing fractal clusters are calculated to first order in an \epsilon-expansion. It is shown that the long-range critical behavior changes continuously to its short-range counterpart for a decay exponent of the infection \sigma =\sigma_c>2.

Equilibrium dynamics of the roughening transition in -dimensional systems

The dynamics of the (1 + 1)-dimensional crystalline surface with long-range interactions is investigated using the renormalization group (RG) technique. The system in question displays a roughening transition which does not occur for systems with short-range interactions. The linear macromobility continuously decreases to zero as temperature decreases to the critical temperature in contrast to the usual two-dimensional roughening transition with a universal jump in the mobility at the transition point. The nonlinear mobility is also derived using the RG recursion relations. Two different RG schemes are employed and their differences are addressed.

Critical Behavior of an Ising Model with Aperiodic Interactions

We write exact renormalization-group recursion relations for a ferromagnetic Ising model on the diamond hierarchical lattice with an aperiodic distribution of exchange interactions according to a class of generalized two-letter Fibonacci sequences. For small geometric fluctuations, the critical behavior is unchanged with respect to the uniform case. For large fluctuations, the uniform fixed point in the parameter space becomes fully unstable. We analyze some limiting cases, and propose a heuristic criterion to check the relevance of the fluctuations.

Application of bond-moving renormalization-group approach to fractal lattices

We investigate the application of the Migdal-Kadanoff bond-moving renormalization group (RG) approach to fractal lattices. We find the following two results: first, for inhomogeneous interaction lattice models, bond moving involving inequivalent bonds is unsuitable because it violates the condition =0 (Δ is the perturbation potential resulting from moving the bonds); second, the condition =0 does not uniquely determine the way to move bonds; different choices of bond moving yield different RG recursion relations and corresponding fixed points, which makes the conclusions concerning the phase transition quite uncertain.

Topological defects on fluctuating surfaces: General properties and the Kosterlitz-Thouless transition.

We investigate the Kosterlitz-Thouless transition for hexatic order on a free fluctuating membrane and derive both a Coulomb gas and a sine-Gordon Hamiltonian to describe it. The Coulomb-gas Hamiltonian includes charge densities arising from disclinations and from Gaussian curvature. There is an interaction coupling the difference between these two densities, whose strength is determined by the hexatic rigidity, and an interaction coupling Gaussian curvature densities arising from the Liouville Hamiltonian resulting from the imposition of a covariant cutoff. In the sine-Gordon Hamiltonian, there is a linear coupling between a scalar field and the Gaussian curvature. We discuss gauge-invariant correlation function for hexatic order and the dielectric constant of the Coulomb gas. We also derive renormalization group recursion relations that predict a transition with decreasing bending rigidity $\kappa$.

Sine-Gordon field theory for the Kosterlitz-Thouless transitions on fluctuating membranes

In the preceding paper, we derived Coulomb-gas and sine-Gordon Hamiltonians to describe the Kosterlitz-Thouless transition on a fluctuating surface. These Hamiltonians contain couplings to Gaussian curvature not found in a rigid flat surface. In this paper, we derive renormalization-group recursion relations for the sine-Gordon model using field-theoretic techniques developed to study flat space problems.

Replica Symmetry Breaking Instability in the 2D XY Model in a Random Field.

We study the 2D vortex-free $\mathrm{XY}$ model in a random field, a model for randomly pinned flux lines in a plane. We construct controlled renormalization group recursion relations which allow for replica symmetry breaking (RSB). The fixed point previously found by Cardy and Ostlund in the glass phase $T<{T}_{c}$ is unstable to RSB. The susceptibility $\ensuremath{\chi}$ associated to infinitesimal RSB perturbation in the high-temperature phase is found to diverge as $\ensuremath{\chi}\ensuremath{\propto}{(T\ensuremath{-}{T}_{c})}^{\ensuremath{-}\ensuremath{\gamma}}$, when $T\ensuremath{\rightarrow}{T}_{c}^{+}$. This provides analytical evidence that RSB occurs in finite dimensional models. The physical consequences for the glass phase are discussed.

Renormalization-group calculations for a mixed-spin Ising model

We use real- and momentum-space renormalization-group techniques to obtain the phase diagram of a mixed-spin Ising model (spin-12 and spin-1) in the presence of a crystal field. A detailed analysis of the fixed points and the flow diagrams of the Migdal-Kadanoff real-space renormalization-group recursion relations indicates the presence of a tricritical point above two dimensions. On the basis of an effective Hamiltonian in terms of continuous spin fields, we perform momentum-space calculations to confirm the existence of this tricritical point in three dimensions. We also present some exact results in one and two dimensions as well as a new mean-field calculation.

Mode-coupling theory and simulation results for the "running-sandpile" model of self-organized criticality.

We examine the model of self-organized criticality investigated by Hwa and Kardar [Phys. Rev. Lett. 62, 1813 (1989)] based on a continuum, nonlinear driven diffusion equation with stochastic noise. We point out some problems in the heuristic arguments used in relating the temporal response functions of the model to the exponents calculated exactly by dynamic-renormalization-group analysis. We propose an alternative, direct calculation of the response function and its power spectrum for the total energy dissipated in the system by a mode-coupling theory, making use of the renormalization-group recursion relations for the renormalized parameters of the theory. We find a significant difference from the previous results. Simulation results of the «running-sandpile» model agree with our mode-coupling-theory predictions

Renormalization group for canted magnets

We use renormalization-group techniques to investigate a model of O(n)/O(n-m) symmetry breaking in 4-\ensuremath{\varepsilon} dimensions. The model describes n-component spin systems in which varying levels of frustration indexed by m\ensuremath{\le}n give rise to canted ground states. The order parameter for these magnets is a reference frame with m orthogonal axes. The model contains the triangular antiferromagnet as a special case. We compute fixed points and critical exponents to scrO(\ensuremath{\varepsilon}) from the renormalization-group recursion relations. We find that a second-order phase transition with O(n)/O(n-m) symmetry breaking occurs for restricted values of m and n. Higher levels of frustration are shown to drive the transition to first order. Finally, we explore a closely related model of U(n)/U(n-m) symmetry breaking, wherein similar results are found.

Percolation thresholds on finitely ramified fractals

Exact renormalisation group recursion relations are used to estimate the effective percolation thresholds for site and bond percolation on finite-generation Sierpinski gaskets, and for bond percolation on branching Koch curves.

Equations of state for the three-state Potts model with symmetry-breaking perturbations

Renormalisation-group recursion relations are used to construct equations of state for the critical region of the three-state Potts model with symmetry-breaking perturbations, to first-order in epsilon =4-d. Explicit scaling forms are obtained for the free energy and the longitudinal susceptibility above and below the critical point, as well as for the order-parameter discontinuity on the first-order phase boundary. Application to the trigonal-to-pseudotetragonal phase transitions in SrTiO3 under stress yields reasonable agreement with the experimental phase boundary.

Low-temperature behavior of the correlation length and the susceptibility of the ferromagnetic quantum Heisenberg chain.

The renormalization-group recursion relations for the spin-S nearest-neighbor quantum Heisenberg ferromagnet recently derived by Chakravarty and the present author are analyzed in one dimension. It is found that the susecptibility \ensuremath{\chi} is given at low temperatures T by T\ensuremath{\chi}=${C}_{\ensuremath{\chi}}^{(0)}$(${\mathrm{JS}}^{2}$/T) [1+${C}_{\ensuremath{\chi}}^{(1)}$\ensuremath{\eta}+O(${\ensuremath{\eta}}^{2}$)], where J is the exchange coupling and the dimensionless expansion parameter \ensuremath{\eta} is given by \ensuremath{\eta}=${\ensuremath{\pi}}^{\mathrm{\ensuremath{-}}1}$(T/${\mathrm{JS}}^{3}$${)}^{1/2}$. For the correlation length \ensuremath{\xi} we find (\ensuremath{\xi}/a)=${C}_{\ensuremath{\xi}}^{(0)}$(${\mathrm{JS}}^{2}$/T) [1+${C}_{\ensuremath{\xi}}^{(1)}$\ensuremath{\eta}+O(${\ensuremath{\eta}}^{2}$)], where a is the lattice spacing. The values of the coefficients ${C}_{\ensuremath{\chi}}^{(j)}$ and ${C}_{\ensuremath{\xi}}^{(j)}$ (j=0,1), have been determined by fitting the above expressions with Monte Carlo data obtained for the S=1/2 Heisenberg chain at temperatures as low as T/J=0.01.