Marginal Perturbations Research Articles

Extended surface disorder in the quantum Ising chain

We consider random extended surface perturbations in the transverse field Ising model decaying as a power of the distance from the surface towards a pure bulk system. The decay may be linked either to the evolution of the couplings or to their probabilities. Using scaling arguments, we develop a relevance-irrelevance criterion for such perturbations. We study the probability distribution of the surface magnetization, its average and typical critical behaviour for marginal and relevant perturbations. According to analytical results, the surface magnetization follows a log-normal distribution and both the average and typical critical behaviours are characterized by power-law singularities with continuously varying exponents in the marginal case and essential singularities in the relevant case. For enhanced average local couplings, the transition becomes first order with a nonvanishing critical surface magnetization. This occurs above a positive threshold value of the perturbation amplitude in the marginal case.

Extended massless phase and the Haldane phase in a spin-1 isotropic antiferromagnetic chain

We study the phase transition of isotropic spin-1 models in the vicinity of the Uimin-Lai-Sutherland model by using the SU(3)_1 WZW model with certain marginal perturbations. The unstable RG trajectory by a marginally relevant perturbation generates a mass gap for the Haldane phase, and thus the universality class of the transition from the massless phase to the Haldane phase at ULS point becomes the BKT type. Our results support recent numerical studies by F\'ath and S\'olyom. In the massless phase, we calculate logarithmic finite-size corrections of the energy for the SU(\nu)-symmetric and asymmetric models.

Is a truly marginal perturbation of the Gk × Gk WZNW model at k = −2 cV( G) an exception to the rule?

It is shown that there exists a truly marginal deformation of the direct sum of two G k WZNW models at k = −2 c V ( G) (where c V ( G) is the eigenvalue of the quadratic Casimir operator in the adjoint representation of the group G) which does not seem to fit the Chaudhuri-Schwartz criterion for truly marginal perturbations. In addition, a continuous family of lagrangians describing the WZNW model is constructed.

Anisotropic scaling in layered aperiodic Ising systems

The influence of a layered aperiodic modulation of the couplings on the critical behaviour of the two-dimensional Ising model is studied in the case of marginal perturbations. The aperiodicity is found to induce anisotropic scaling. The anisotropy exponent z, given by the sum of the surface magnetization scaling dimensions, depends continuously on the modulation amplitude. Thus these systems are scale invariant but not conformally invariant at the critical point.

A NEW SET OF EXACT FORM FACTORS

We present form factors for a wide range of integrable models which include marginal perturbations of the SU(2) WZNZ model for arbitrary central charge and the principal chiral field model. The interesting structure of these form factors is discussed.

On the complete solution of a perturbed topological conformal field theory at c=3

The problem of determination of the flat coordinates for a model of a topological conformal field theory corresponding to the 'twisted' version of an N=2 superconformal Landau-Ginzburg field theory with central charge c=3 is analysed. The model is characterized by a Landau-Ginzburg superpotential of the form: W= 1/4 x4+ 1/4 y4. All possible relevant and marginal perturbations with their corresponding couplings (expressed as functions of the dimensionless flat-coordinate) are added to the above superpotential to give the perturbed topological field theory model. It is seen that the couplings can be completely determined (and hence also the dependence of the perturbations on the flat coordinate) by imposing the conditions of flatness on the space of couplings of the perturbed theory.

Duality, marginal perturbations, and gauging.

We study duality transformations for two-dimensional $\ensuremath{\sigma}$ models with Abelian chiral isometries and prove that generic such transformations are equivalent to integrated marginal perturbations by bilinears in the chiral currents, thus confirming a recent conjecture by Hassan and Sen formulated in the context of Wess-Zumino-Witten models. Specific duality transformations instead give rise to coset models plus free bosons.

Marginal extended perturbations in two dimensions and gap-exponent relations

The most general form of a marginal extended perturbation in a two-dimensional system is deduced from scaling considerations. It includes as particular cases extended perturbations decaying either from a surface, a line or a point for which exact results have been previously obtained. The first-order corrections to the local exponents, which are functions of the amplitude of the defect, are deduced from a perturbation expansion of the two-point correlation functions. Assuming covariance under conformal transformation, the perturbed system is mapped onto a cylinder. Working in the Hamiltonian limit, the first-order corrections to the lowest gaps are calculated for the Ising model. The results confirm the validity of the gap-exponent relations for the perturbed system.

Higher twisted sector couplings of [formula omitted] orbifolds

We derive the basic correlation functions of twist fields coming from arbitrary twisted sectors in symmetric Z N orbifold conformal field theories, keeping all the admissible marginal perturbations, in particular those corresponding to the antisymmetric-tensor background field. This allows a thorough investigation of modular symmetries in this type of string compactification. Such a study is explicitly carried out for the group generated by duality transformations. Thus, apart from being of phenomenological use, our couplings are also interesting from the mathematical point of view as they represent automorphic functions for a large class of discrete groups.

CORRELATION FUNCTIONS FOR TOPOLOGICAL LANDAU-GINZBURG MODELS WITH c≤3

We discuss c≤3 topological Landau-Ginzburg models. In particular we give the potential for the three exceptional models E6,7,8 in the constant metric coordinates of coupling constant space and derive the generating function F for correlation functions. For the c=3 torus cases with one marginal deformation and relevant perturbations, we derive and solve the differential equation resulting from flatness of coupling constant space. We perform the transformation to constant metric coordinates and calculate the generating function F. Comparing the three-point correlation functions with those of orbifold superconformal field theory, we find agreement. We finally demonstrate that the differential equations derived from flatness of coupling constant space are the same as the ones satisfied by the periods of the tori.

Multiple crossover phenomena and scale hopping in two dimensions

We study the renormalization group for nearly marginal perturbations of a minimal conformal field theory M_p with p >> 1. To leading order in perturbation theory, we find a unique one-parameter family of ``hopping trajectories'' that is characterized by a staircase-like renormalization group flow of the C-function and the anomalous dimensions and that is related to a recently solved factorizable scattering theory. We argue that this system is described by interactions of the form t phi_{(1,3)} - t' \phi_{(3,1)} . As a function of the relevant parameter t, it undergoes a phase transition with new critical exponents simultaneously governed by all fixed points M_p, M_{p-1}, ..., M_3. Integrable lattice models represent different phases of the same integrable system that are distinguished by the sign of the irrelevant parameter t'.

Differential equations for periods and flat coordinates in two-dimensional topological matter theories

We consider two-dimensional topological Landau-Ginzburg models. In order to obtain the free energy of these models, and to determine the Kähler potential for the marginal perturbations, one needs to determine flat or “special” coordinates that can be used to parametrize the perturbations of the superpotentials. This paper describes the relationship between the natural Landau-Ginzburg parametrization and these flat coordinates. In particular we show how one can explicitly obtain the differential equations that relate the two. We discuss the problem for both Calabi-Yau manifolds and for general topological matter models (with arbitrary central charges) with relevant and marginal perturbations. We also give a number of examples.

Reminimization of Energy Integral and Transition Point of Mode for Relaxed States of Nonideal MHD Plasma

The sufficient condition for relaxed minimum-energy states is derived from the energy principle for a nonideal MHD plasma using two different processes of formulation. The Euler equation for the reminimization of the energy integral is derived and shown to yield the marginal stable perturbations for the sufficient condition. To determine the transition point of the relaxed mode, an extended stability limit for the β=0 relaxed states is derived from the sufficient condition, using the eigenvalue analysis for the Euler equation. The extended stability limit is solved in the 1st-order approximation to explain the deviation of the experimental stability limit from the idealized stability limit obtained by Taylor [J. B. Taylor, Rev. Mod. Phys. 58 (1986) 741]. The stability limit for the finite β relaxed states is also discussed.

Till beneath ice stream B: 4. A coupled ice‐till flow model

A nonsteady model of an ice stream flowing on deforming till shows that the system responds rapidly and in a stable manner to reasonable marginal perturbations. For the model we require one‐dimensional flow and continuity of ice and till, and we assume linear viscous till rheology and balance between the driving stress for ice flow and the resistive stress at the bed. This allows us to write coupled equations for the time rate of change of the ice thickness and the till thickness in terms of these thicknesses and the till viscosity. An analytic, steady state solution for ice stream B, West Antarctica, shows that till viscosity decreases slowly downstream, probably in response to decreasing effective pressure downstream. Nonsteady numerical experiments with fixed ice thicknesses at the ends show that a marginal perturbation causes a wave of adjustment to travel the length of the ice stream in about 50 years, with a new steady state in about 200 years. Changes in till thickness tend to moderate perturbations in the ice and thus stabilize the system.