Two-dimensional sine-Gordon Model Research Articles

Renormalization of the bilocal sine-Gordon model

The functional renormalization group treatment is presented for the two-dimensional sine-Gordon model including a bilocal term in the potential, which contributes to the flow at the tree level. It is shown that the flow of the bilocal term can substitute the evolution of the wave function renormalization constant, since it can recover the Kosterlitz–Thouless type phase transition. The flows can also reveal the connection between the sine-Gordon and the noninteracting Thirring models at a special value of the wave number parameter.

Bubblelike structures generated by activation of internal shape modes in two-dimensional sine-Gordon line solitons.

Nonlinear waves that collide with localized defects exhibit complex behavior. Apart from reflection, transmission, and annihilation of an incident wave, a local inhomogeneity can activate internal modes of solitons, producing many impressive phenomena. In this work we investigate a two-dimensional sine-Gordon model perturbed by a family of localized forces. We observed the formation of bubblelike and droplike structures due to local internal shape mode instabilities. We describe the formation of such structures on the basis of a one-dimensional theory of activation of internal modes of sine-Gordon solitons. An interpretation of the observed phenomena, in the context of phase transition theory, is given. Implications in Josephson junctions with a current dipole device are discussed.

Asymptotic safety in the sine-Gordon model

In the framework of the functional renormalization group method, it is shown that the phase structure of the two-dimensional sine-Gordon model possesses a nontrivial UV fixed point which makes the model asymptotically safe. The fixed point exhibits strong singularity similar to the scaling found in the vicinity of the infrared fixed point. The singularity signals the upper energy-scale limit to the validity of the model. We argue that the sine-Gordon model with a momentum-dependent wave-function renormalization is in a dual connection with the massive sine-Gordon model.

Numerical solution of two-dimensional sine-Gordon and MBE models using Fourier spectral and high order explicit time stepping methods

In this paper, we consider a Fourier spectral method for spatial variables and three fast and efficient explicit time stepping methods for the numerical solution of two-dimensional (2D) nonlinear partial differential equations (PDEs) including sine-Gordon and molecular beam epitaxy (MBE) models. We discretize the original PDEs with discrete Fourier transform in space and then apply some fourth-order time stepping schemes such as Runge–Kutta integrating factor (IFRK), Runge–Kutta exponential time differencing (ETDRK) and ETDRK method with improved accuracy (ETDRKB), to the resulting systems of ordinary differential equations (ODEs). With regard to stiffness and nonlinearities of the models, in this approach we take advantage of using the fast Fourier transform in combination with explicit fourth-order time stepping methods and expect to get a better accuracy and CPU time in comparison with other reported spatial discretizations and implicit time stepping schemes in literature. Comparing the numerical solutions with analytical solutions and also some reported results in the literature demonstrates the effectiveness of the proposed schemes. We will also show that the schemes of this paper exhibit the conservation of energy for both PDEs.

Quantum censorship in two dimensions

It is pointed out that increasingly attractive interactions, represented by partially concave local potential in the Lagrangian, may lead to the degeneracy of the blocked, renormalized action at the gliding cutoff scale by tree-level renormalization. A quantum counterpart of this mechanism is presented in the two-dimensional sine-Gordon model. The presence of Quantum Censorship is conjectured which makes the loop contributions pile up during the renormalization and thereby realize an approximate semiclassical effect.

Functional Renormalization Group Approach to the Sine-Gordon Model

The renormalization group flow is presented for the two-dimensional sine-Gordon model within the framework of the functional renormalization group method by including the wave-function renormalization constant. The Kosterlitz-Thouless-Berezinski type phase structure is recovered as the interpolating scaling law between two competing IR attractive area of the global renormalization group flow.

Renormalizable parameters of the sine-Gordon model

The well-known phase structure of the two-dimensional sine-Gordon model is reconstructed by means of its renormalization group flow, the study of the sensitivity of the dynamics on microscopic parameters. Such an analysis resolves the apparent contradiction between the phase structure and the triviality of the effective potential in either phases, provides a case where usual classification of operators based on the linearization of the scaling relation around a fixed point is not available and shows that the Maxwell-cut generates an unusually strong universality at long distances. Possible analogies with four-dimensional Yang–Mills theories are mentioned, too.

Renormalization of the periodic scalar field theory by Polchinski's renormalization group method

The renormalization group (RG) flow for the two-dimensional sine-Gordon model is determined by means of Polchinski's RG equation at next-to-leading order in the derivative expansion. In this paper, we have two different goals, (i) to consider the renormalization scheme-dependence of Polchinski's method by matching Polchinski's equation with the Wegner–Houghton equation and with the real space RG equations for the two-dimensional dilute Coulomb-gas, (ii) to go beyond the local potential approximation in the gradient expansion in order to clarify the supposed role of the field-dependent wave-function renormalization. The well-known Coleman fixed point of the sine-Gordon model is recovered after linearization, whereas the flow exhibits strong dependence on the choice of the renormalization scheme when non-linear terms are kept. The RG flow is compared to those obtained in the Wegner–Houghton approach and in the dilute gas approximation for the two-dimensional Coulomb-gas.

Two-loop beta-functions of the sine-Gordon model

We recalculate the two-loop beta functions in the two-dimensional Sine-Gordon model in a two-parameter expansion around the asymptotically free point. Our results agree with those of Amit et al., J. Phys. A13 (1980) 585.

Roughening and super-roughening in the ordered and random two-dimensional sine-gordon models

We present a comparative numerical study of the ordered and the random two-dimensional sine-Gordon models on a lattice. We analytically compute the main features of the expected high-temperature phase of both models, described by the Edwards-Wilkinson equation. We then use those results to locate the transition temperatures of both models in our Langevin dynamics simulations. We show that our results reconcile previous contradictory numerical works concerning the super-roughening transition in the random sine-Gordon model. We also find evidence supporting the existence of two different low-temperature phases for the disordered model. We discuss our results in view of the different analytical predictions available and comment on the nature of these two putative phases.

Observation of whispering gallery resonances in annular Josephson junctions

Experiments indicating the excitation of whispering gallery-type electromagnetic modes by a vortex moving in an annular Josephson junction are reported. At relativistic velocities the Josephson vortex interacts with the modes of the superconducting stripline resonator giving rise to novel resonances on the current–voltage characteristic of the junction. The coupling between the vortex and the linear modes of the junction is explained using a Cherenkov–radiation-like mechanism. We analytically calculate the spectrum of the linear modes taking into account the electromagnetic environment of the junction. The experimental data are in good quantitative agreement with the analysis and numerical calculations based on the two-dimensional sine-Gordon model.

Whispering vortices

Experiments indicating the excitation of whispering-gallery-type electromagnetic modes by a vortex moving in an annular Josephson junction are reported. At relativistic velocities the Josephson vortex interacts with the modes of the superconducting stripline resonator giving rise to novel resonances on the current-voltage characteristic of the junction. The experimental data are in good agreement with analysis and numerical calculations based on the two-dimensional sine-Gordon model.

Symmetry of the BKT transition and twisted boundary condition

The Berezinskii-Kosterlitz-Thouless (BKT) transition, the transition of the two-dimensional sine-Gordon model, plays an important role in low-dimensional physics. We relate the operator content of the BKT transition to that of the SU(2) Wess-Zumino-Witten model, using twisted boundary conditions. With this method, in order k - 1 to determine the BKT critical point, we can use the level crossing of the lower excitations instead of those for the periodic boundary case, thus the convergence to the transition point is highly improved. We verify the efficiency of this method by applying it to the S = 1, 2 spin chains.

Construction of the two-dimensional sine-Gordon model for β<8π

We present a rigorous renormalization group construction of the two-dimensional massless and massive quantum sine-Gordon models in finite volume for the range 0<β<8π. We prove analyticity in the coupling constant ζ, which implies the convergence of perturbation theory. The field correlation functions and their generating functional are analyzed and shown to have the short distance asymptotics of the free field theory. In the massive case the bounds are uniform in volume and we also obtain uniform estimates on the long distance decay of correlations.

Renormalization-group analysis of a class of two-dimensional vortex systems

The statistical-mechanical properties of a class of two-dimensional vortex systems, described by grand-canonical partition functions containing certain Z n- valued adiabatic phases, are investigated. It has been shown previously that such systems describe two-dimensional incompressible superfluids, in which an additional (non-thermal) vortex is present, and which is restricted to move along a closed curve enclosing the bulk fluid. By generalizing the well-known connection between the Coulomb-gas partition function and the two-dimensional sine-Gordon model, to include adiabatic phases, we derive a series of renormalization-group equations, the asymptotic solutions of which determined the macroscopic superfluid density. Numerical solutions to these equations are given, from which it is concluded that although the density is discontinuous along a line of critical temperatures T c, the size of the discontinuity is reduced by a factor of 1/ n compared to the case where adiabatic phases are absent.

Scattering in soliton models and boson-exchange descriptions.

We analyze the relationship between the conventional one-boson-exchange model (OBE) for nuclear forces with the description of the nucleon as a soliton within the context of the two-dimensional sine-Gordon model. We find that the soliton-soliton $S$ matrix contains poles and residues compatible with the ones used in the OBE. Implications of this result for the four-dimensional case are briefly discussed.

Dynamics of quasi-one-dimensional kinks in the two-dimensional sine-Gordon model

New results on the old-standing problem stated in the title of the paper are reported. Emission of radiation by a plane quasi-one-dimensional (Q1D) kink coming across a local defect is investigated in detail. In the same situation, excitation of flexural waves on a “crest” of the Q1D kink is investigated in the long-wave approximation. Next, stable stationary shapes of the kink's “crest” that admit its motion with a constant velocity V are found in the framework of the homogeneous damped dc-driven model. For any V < 1, there is a single v-like profile of the crest. For any “tachyonic” value V > 1, an x-like profile exists (in the absence of the dissipation and the drive terms, it corresponds to an exact solution of the two-dimensional sine-Gordon equation). For a densely packed periodic array of the Q1D kinks, the tachyonic profile is Λ-like.

THE HIERARCHICAL SINE GORDON MODEL IN THE HIGH TEMPERATURE PHASE

The two-dimensional hierarchical Sine Gordon model in the high temperature phase is investigated both by analytical and numerical methods. We show that the “continuous hierarchical renormalization group” excludes double well fixed points in two dimensions. This might be interpreted as a hint that truncating continuous flow equations can lead to a qualitative change in the renormalization group flow.

On The Cut-off Parameters in the Bosonization Technique

Abstract Jordan's boson representation and cut-off repolarization for the one-dimensional two-fermion model (TFT) is used to get the equivalence of the model with the two-dimensional Coulomb gas and sine-Gordon model. The scaling equations for the coupling constants are thereby obtained up to the third order.

Commensurate-Incommensurate Transition in the Two-Dimensional Sine-Gordon System: the Bethe Ansatz Calculation

Recently Haldane has derived the Bethe ansatz equations describing the ground state properties of the massive Thirring model with finite `hole' density (the quantum sine-Gordon model with finite soliton density). His derivation is explicitly confirmed here using a particular cutoff scheme, and his solutions of the equations in various limits are reviewed with complementary results. The commensurate-incommensurate transition in the two-dimensional sine-Gordon model with a uniaxial misfit is examined at finite temperatures using the results obtained in the massive Thirring model.