Massive Thirring Model Research Articles

L2 orbital stability of Dirac solitons in the massive Thirring model

ABSTRACTWe prove L2 orbital stability of Dirac solitons in the massive Thirring model. Our method uses local well posedness of the massive Thirring model in L2, conservation of the charge functional, and the auto–Bäcklund transformation. The latter transformation exists because the massive Thirring model is integrable via the inverse scattering transform method.

Transverse Instability of Line Solitary Waves in Massive Dirac Equations

Working in the context of localized modes in periodic potentials, we consider two systems of the massive Dirac equations in two spatial dimensions. The first system, a generalized massive Thirring model, is derived for the periodic stripe potentials. The second one, a generalized massive Gross--Neveu equation, is derived for the hexagonal potentials. In both cases, we prove analytically that the line solitons suffer from instability with respect to periodic transverse perturbations of large periods. The instability is induced by the spatial translation for the massive Thirring model and by the gauge rotation for the massive Gross--Neveu model. We also observe numerically that the instability holds for the transverse perturbations of any period in the massive Thirring model and exhibits a finite threshold on the period of the transverse perturbations in the massive Gross--Neveu model.

Lefschetz thimble structure in one-dimensional lattice Thirring model at finite density

We investigate Lefschetz thimble structure of the complexified path-integration in the one-dimensional lattice massive Thirring model with finite chemical potential. The lattice model is formulated with staggered fermions and a compact auxiliary vector boson (a link field), and the whole set of the critical points (the complex saddle points) are sorted out, where each critical point turns out to be in a one-to-one correspondence with a singular point of the effective action (or a zero point of the fermion determinant). For a subset of critical point solutions in the uniform-field subspace, we examine the upward and downward cycles and the Stokes phenomenon with varying the chemical potential, and we identify the intersection numbers to determine the thimbles contributing to the path-integration of the partition function. We show that the original integration path becomes equivalent to a single Lefschetz thimble at small and large chemical potentials, while in the crossover region multi thimbles must contribute to the path integration. Finally, reducing the model to a uniform field space, we study the relative importance of multiple thimble contributions and their behavior toward continuum and low-temperature limits quantitatively, and see how the rapid crossover behavior is recovered by adding the multi thimble contributions at low temperatures. Those findings will be useful for performing Monte-Carlo simulations on the Lefschetz thimbles.

Darboux polynomial matrices: the classical massive Thirring model as a study case

One way of constructing explicit expressions of solutions of integrable systems of partial differential equations goes via the Darboux method. This requires the construction of Darboux matrices. Here we introduce a novel algorithm to obtain such matrices in polynomial form. Our method is illustrated by applying it to the classical massive Thirring model, and by combining it with the Dihedral group of symmetries possessed by this model.

Global solution to nonlinear Dirac equation for Gross–Neveu model in [formula omitted] dimensions

This paper studies a class of nonlinear Dirac equations with cubic terms in R1+1, which include the equations for the massive Thirring model and the massive Gross–Neveu model. Under the assumption that the initial data has bounded L2 norm, the global existence and the uniqueness of the strong solution in C([0,∞),L2(R1)) are proved.

Vakhitov–Kolokolov and energy vanishing conditions for linear instability of solitary waves in models of classical self-interacting spinor fields

We study the linear stability of localized modes in self-interacting spinor fields, analysing the spectrum of the operator corresponding to linearization at solitary waves. Following the generalization of the Vakhitov–Kolokolov approach, we show that the bifurcation of real eigenvalues from the origin is completely characterized by the Vakhitov–Kolokolov condition dQ/dω = 0 and by the vanishing of the energy functional. We give the numerical data on the linear stability in the generalized Gross–Neveu model and the generalized massive Thirring model in the charge-subcritical, charge-critical and charge-supercritical cases, illustrating the agreement with the Vakhitov–Kolokolov and the energy vanishing conditions.

Bragg grating rogue wave

Abstract We derive the rogue wave solution of the classical massive Thirring model, that describes nonlinear optical pulse propagation in Bragg gratings. Combining electromagnetically induced transparency with Bragg scattering four-wave mixing may lead to extreme waves at extremely low powers.

Orbital Stability of Dirac Solitons

We prove H 1 orbital stability of Dirac solitons in the integrable massive Thirring model by working with an additional conserved quantity which complements Hamiltonian, momentum and charge functionals of the general nonlinear Dirac equations. We also derive a global bound on the H 1 norm of the L 2-small solutions of the massive Thirring model.

The algebraic structure behind the derivative nonlinear Schrödinger equation

The Kaup–Newell (KN) hierarchy contains the derivative nonlinear Schrödinger equation (DNLSE) amongst others interesting and important nonlinear integrable equations. In this paper, a general higher grading affine algebraic construction of integrable hierarchies is proposed and the KN hierarchy is established in terms of an Kac–Moody algebra and principal gradation. In this form, our spectral problem is linear in the spectral parameter. The positive and negative flows are derived, showing that some interesting physical models arise from the same algebraic structure. For instance, the DNLSE is obtained as the second positive, while the Mikhailov model as the first negative flows. The equivalence between the latter and the massive Thirring model is also explicitly demonstrated. The algebraic dressing method is employed to construct soliton solutions in a systematic manner for all members of the hierarchy. Finally, the equivalence of the spectral problem introduced in this paper with the usual one, which is quadratic in the spectral parameter, is achieved by setting a particular automorphism of the affine algebra, which maps the homogeneous into principal gradation.

Bragg solitons in nonlinearPT-symmetric periodic potentials

It is shown that slow Bragg soliton solutions are possible in nonlinear complex parity-time (PT) symmetric periodic structures. Analysis indicates that the PT-symmetric component of the periodic optical refractive index can modify the grating band structure and hence the effective coupling between the forward and backward waves. Starting from a classical modified massive Thirring model, solitary wave solutions are obtained in closed form. The basic properties of these slow solitary waves and their dependence on their respective PT-symmetric gain/loss profile are then explored via numerical simulations.

TWISTED BOSONIZATION IN TWO-DIMENSIONAL NONCOMMUTATIVE SPACETIME

We study the twisted bosonization of massive Thirring model to relate to sine-Gordon model in Moyal spacetime using twisted commutation relations. We obtain the relevant twisted bosonization rules. We show that there exists dual relationship between twisted bosonic and fermionic operators. The strong–weak duality is also observed to be preserved as its commutative counterpart.

Inverse scattering approach for massive Thirring models with integrable type-II defects

We discuss the integrability of the bosonic and Grassmannian massive Thirring models in the presence of defects through the inverse scattering approach. We present a general method to compute the generating functions of modified conserved quantities for any integrable field equation associated with the m × m spectral linear problem. We apply the method to derive in particular the defect contributions for the number of occupation, energy and momentum of the massive Thirring models.

Study of quasi-elastic Bragg soliton collisions in uniform fiber Bragg gratings by using the solution of the massive Thirring model

Bragg soliton is a solution of the nonlinear coupled mode equations (NLCMEs) that are a non-integrable system. Nevertheless, we show that for a broad region of soliton parameters the interaction between two Bragg solitons in a uniform fiber Bragg grating can be accurately described by using a trial function based on the known solution for two-soliton interaction in the massive Thirring model (MTM). In this region the similar behavior of Bragg solitons and solitons of the Thirring model enables one to calculate explicitly the approximate asymptotic properties of the interaction between two Bragg solitons such as the shifts in locations and phases of the solitons as a result of the interaction. We have validated that the similar behavior of Bragg solitons and solitons of the MTM is obtained for a broad range of parameters of the interacting solitons that can also be realized experimentally. Since the NLCMEs are not an integrable system, there is a parameter regime in which the interaction between Bragg solitons does not resemble the elastic interaction between MTM solitons. We describe an interaction between two co-propagating Bragg solitons that causes the inversion of the propagation direction of one of the solitons.

New reductions of integrable matrix partial differential equations: Sp(m)-invariant systems

We propose a new type of reduction for integrable systems of coupled matrix partial differential equations; this reduction equates one matrix variable with the transposition of another multiplied by an antisymmetric constant matrix. Via this reduction, we obtain a new integrable system of coupled derivative modified KdV equations and a new integrable variant of the massive Thirring model, in addition to the already known systems. We also discuss integrable semidiscretizations of the obtained systems and present new soliton solutions to both continuous and semidiscrete systems. As a by-product, a new integrable semidiscretization of the Manakov model (self-focusing vector nonlinear Schrödinger equation) is obtained.

Chiral Confinement in Quasirelativistic Bose-Einstein Condensates

In the presence of a laser-induced spin-orbit coupling an interacting ultracold spinor Bose-Einstein condensate may acquire a quasirelativistic character described by a nonlinear Dirac-like equation. We show that as a result of the spin-orbit coupling and the nonlinearity the condensate may become self-trapped, resembling the so-called chiral confinement, previously studied in the context of the massive Thirring model. We first consider 1D geometries where the self-confined condensates present an intriguing sinusoidal dependence on the interparticle interactions. We further show that multidimensional chiral confinement is also possible under appropriate feasible laser arrangements, and discuss the properties of 2D and 3D condensates, which differ significantly from the 1D case.

Massless Sine-Gordon and Massive Thirring Models: Proof of Coleman’s Equivalence

We prove the Coleman's conjecture on the equivalence between the massless Sine-Gordon model with finite volume interaction and the Thirring model with a finite volume mass term.

DUAL BOSONIC THERMAL GREEN FUNCTION AND FERMION CORRELATORS OF THE MASSIVE THIRRING MODEL AT FINITE TEMPERATURE

The Euclidean thermal Green function of the two-dimensional (2D) free massless scalar field in coordinate space is written as the real part of a complex analytic function of a variable that conformally maps the infinite strip -∞ < x < ∞ (0 < τ < β) of the z = x + iτ (τ: imaginary time) plane into the upper-half-plane. Using this fact and the Cauchy–Riemann conditions, we identify the dual thermal Green function as the imaginary part of that function. Using both the thermal Green function and its dual, we obtain an explicit series expression for the fermionic correlation functions of the massive Thirring model (MTM) at a finite temperature.

Two-dimensional thermofield bosonization II: Massive fermions

We consider the perturbative computation of the N-point function of chiral densities of massive free fermions at finite temperature within the thermofield dynamics approach. The infinite series in the mass parameter for the N-point functions are computed in the fermionic formulation and compared with the corresponding perturbative series in the interaction parameter in the bosonized thermofield formulation. Thereby we establish in thermofield dynamics the formal equivalence of the massive free fermion theory with the sine-Gordon thermofield model for a particular value of the sine-Gordon parameter. We extend the thermofield bosonization to include the massive Thirring model.

Gap solitons in spatiotemporal photonic crystals

We generalize the concept of nonlinear periodic structures to systems that show arbitrary spacetime variations of the refractive index. Nonlinear pulse propagation through these spatiotemporal photonic crystals can be described, for shallow nonstationary gratings, by coupled mode equations which are a generalization of the traditional equations used for stationary photonic crystals. Novel gap soliton solutions are found by solving a modified massive Thirring model. They represent the missing link between the gap solitons in static photonic crystals and resonance solitons found in dynamic gratings.

Light front field theory: An advanced Primer

Light front field theory: An advanced PrimerWe present an elementary introduction to quantum field theory formulated in terms of Dirac's light front variables. In addition to general principles and methods, a few more specific topics and approaches based on the author's work will be discussed. Most of the discussion deals with massive two-dimensional models formulated in a finite spatial volume starting with a detailed comparison between quantization of massive free fields in the usual field theory and the light front (LF) quantization. We discuss basic properties such as relativistic invariance and causality. After the LF treatment of the soluble Federbush model, a LF approach to spontaneous symmetry breaking is explained and a simple gauge theory - the massive Schwinger model in various gauges is studied. A LF version of bosonization and the massive Thirring model are also discussed. A special chapter is devoted to the method of discretized light cone quantization and its application to calculations of the properties of quantum solitons. The problem of LF zero modes is illustrated with the example of the two-dimensional Yukawa model. Hamiltonian perturbation theory in the LF formulation is derived and applied to a few simple processes to demonstrate its advantages. As a byproduct, it is shown that the LF theory cannot be obtained as a "light-like" limit of the usual field theory quantized on an initial space-like surface. A simple LF formulation of the Higgs mechanism is then given. Since our intention was to provide a treatment of the light front quantization accessible to postgradual students, an effort was made to discuss most of the topics pedagogically and a number of technical details and derivations are contained in the appendices.