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Time dependent solutions of the massless Gross-Neveu model

Christian Fitzner

Published Feb 22, 2021• 0 citations
OpenAlex

Abstract

The subject of this work time dependent solutions of the massless Gross-Neveu model. The Gross-Neveu model is a 1+1 dimensional field theory of N flavours of fermions with four-fermion interaction. In the limit of large flavour numbers N , this theory can be solved exactly by a mean field approximation equivalent to the Hartree-Fock method of relativistic many-particle physics. Nearly all solutions that have been found until now, like “kinks” and “DHN-baryons”, are time independent, since there exists a systematic way to generate such solutions. The only time dependent solution, the breather, has been found by guessing. In this thesis we present new time dependent solutions. Firstly, we show that a special case, the scattering of an arbitrary number of kinks, can be mapped onto an already solved problem. Secondly, we use the fact that Hartree-Fock potential and wave functions of the already known solutions can be written as fractions of polynomials of exponential functions. We make an ansatz of new polynomials, determine their coefficients via the Dirac equation and confirm the self consistency. That way we find analytic solutions for the scattering of up to four DHN-baryons or breathers and can even give a, yet unproven, formula for the scattering of an arbitrary number of such objects.

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Multiconfigurational time-dependent Hartree method for bosons: Many-body dynamics of bosonic systems

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Small oscillations of a chiral Gross-Neveu system

Paulo Laerte Natti, A. F. R. de Toledo Piza

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arXiv:hep-th/9704188•Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields•Mar 15, 1997•3 citations
The subject of this work time dependent solutions of the massless Gross-Neveu model. The Gross-Neveu model is a 1+1 dimensional field theory of N flavours of fermions with four-fermion interaction. In the limit of large flavour numbers N , this theory can be solved exactly by a mean field approximation equivalent to the Hartree-Fock method of relativistic many-particle physics. Nearly all solutions that have been found until now, like “kinks” and “DHN-baryons”, are time independent, since there exists a systematic way to generate such solutions. The only time dependent solution, the breather, has been found by guessing. In this thesis we present new time dependent solutions. Firstly, we show that a special case, the scattering of an arbitrary number of kinks, can be mapped onto an already solved problem. Secondly, we use the fact that Hartree-Fock potential and wave functions of the already known solutions can be written as fractions of polynomials of exponential functions. We make an ansatz of new polynomials, determine their coefficients via the Dirac equation and confirm the self consistency. That way we find analytic solutions for the scattering of up to four DHN-baryons or breathers and can even give a, yet unproven, formula for the scattering of an arbitrary number of such objects.
Due to integrability, baryon-baryon scattering in the massless Gross-Neveu model at large NNN features only forward elastic scattering. A bare mass term breaks integrability and is therefore expected to induce backward elastic scattering as well as inelastic reactions. We confirm these expectations by a study of baryon-baryon scattering in the massive Gross-Neveu model near the nonrelativistic limit. This restriction enables us to solve the time-dependent Hartree-Fock equations with controlled approximations, using a combination of analytical methods from an effective field theory and the numerical solution of partial differential equations.
Abstract This paper employs the Cauchy matrix approach to investigate the (1+1)-dimensional Gross-Neveu (GN) model. We establish a theoretical framework using Sylvester-type equations and master function matrices, and reveal the connection between the model’s time-dependent Hartree–Fock (TDHF) potential and the sine-Gordon equation. Through variable transformations, exact analytical N-soliton solutions are derived. The localized propagation characteristics of the soliton fundamental components and key physical quantities (condensate function and fermion density for N = 1 and N = 2) are validated via spatiotemporal evolution analysis and static slice visualizations. The results confirm the model’s full dynamical solvability and its parameter dependency. This work builds a concrete bridge between quantum field theory and classical physics, offering analytical evidence for the quantum particle behavior in the semiclassical GN model.
The evolution of Bose-Einstein condensates is amply described by the time-dependent Gross-Pitaevskii mean-field theory which assumes all bosons to reside in a single time-dependent one-particle state throughout the propagation process. In this work, we go beyond mean field and develop an essentially exact many-body theory for the propagation of the time-dependent Schr\"odinger equation of NNN interacting identical bosons. In our theory, the time-dependent many-boson wave function is written as a sum of permanents assembled from orthogonal one-particle functions, or orbitals, where both the expansion coefficients and the permanents (orbitals) themselves are time-dependent and fully determined according to a standard time-dependent variational principle. By employing either the usual Lagrangian formulation or the Dirac-Frenkel variational principle we arrive at two sets of coupled equations of motion, one for the orbitals and one for the expansion coefficients. The first set comprises of first-order differential equations in time and nonlinear integrodifferential equations in position space, whereas the second set consists of first-order differential equations with time-dependent coefficients. We call our theory multiconfigurational time-dependent Hartree for bosons, or MCTDHB(M)\text{MCTDHB}(M)MCTDHB(M), where MMM specifies the number of time-dependent orbitals used to construct the permanents. Numerical implementation of the theory is reported and illustrative numerical examples of many-body dynamics of trapped Bose-Einstein condensates are provided and discussed. The convergence of the method with a growing number MMM of orbitals is demonstrated in a specific example of four interacting bosons in a double well.
The (2+12+12+1)-dimensional generalized massless Thirring model with four-component Fermi fields is investigated by the Hartree-Fock method. The Lagrangian of this model is constructed from two different four-fermion structures. One of them takes into account the vector $\ifmmode\times\else\texttimes\fi{}$ vector channel of fermion interaction with coupling constant Gv{G}_{v}Gv​, the other, the scalar $\ifmmode\times\else\texttimes\fi{}$ scalar channel with coupling Gs{G}_{s}Gs​. At some relation between bare couplings Gs{G}_{s}Gs​ and Gv{G}_{v}Gv​, the Hartree-Fock equation for self-energy of fermions can be renormalized, and dynamical generation of the Dirac and Haldane fermion masses is possible. As a result, a phase portrait of the model consists of two nontrivial phases. In the first one the chiral symmetry is spontaneously broken due to dynamical appearance of the Dirac mass term, while in the second phase a spontaneous breaking of the spatial parity P\mathcal{P}P is induced by a Haldane mass term. It is shown that in the particular case of a pure Thirring model, i.e., at Gs=0{G}_{s}=0Gs​=0, the ground state of the system is indeed a mixture of these phases. Moreover, it was found that dynamical generation of fermion masses is possible for any finite number of Fermi fields.
In this paper we consider the problem of solving quantum field theories with time dependent interaction strengths. We show that the recently formulated framework [P. R. Pasnoori, Phys. Rev. B 112, L060409 (2025)], which is a generalization of the regular Bethe ansatz technique, provides the exact many-body wavefunction. In this framework, the time-dependent Schrodinger equation is reduced to a set of analytic difference equations and matrix difference equations, called the quantum Knizhnik-Zamolodchikov (qKZ) equations. The consistency of the solution gives rise to constraints on the time-dependent interaction strengths. For interaction strengths satisfying these constraints, the system is integrable, and the solution to the qKZ and the analytic difference equations provides the explicit form of the many-body wavefunction that satisfies the time-dependent Schrodinger equation. We provide a concrete example by considering the SU(2)SU(2)SU(2) Gross-Neveu model with time dependent interaction strength. Using this framework we solve the model with the most general time-dependent interaction strength and obtain the explicit form of the wave function.