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Physica Scripta

Cauchy matrix approach to the Gross-Neveu model: soliton solutions and spatiotemporal evolution

Jiaxi Peng, Gang Chen, Hong-juan Tian

Published Apr 23, 2026• 0 citations
Publisher

Abstract

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.

DOI
10.1088/1402-4896/ae6424

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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.
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We study the quantum relaxation dynamics for a lattice version of the one-dimensional (1D) NNN-flavor Gross-Neveu (GN) model after a Hamiltonian parameter quench. Allowing for a system-reservoir coupling γγγ, we numerically describe the system dynamics through a time-dependent self-consistent Lindblad master equation. For a closed (γ=0γ=0γ=0) finite-size system subjected to an interaction parameter quench, the order parameter dynamics exhibits oscillations and revivals. In the thermodynamic limit, our results imply that the order parameter reaches its post-quench stationary value in accordance with the eigenstate thermalization hypothesis (ETH). However, time-dependent finite-momentum correlation matrix elements equilibrate only if γ>0γ>0γ>0. Our findings are consistent with the system being described by a pertinent Generalized Gibbs Ensemble (GGE) and, accordingly, highlight subtle yet important aspects of the post-quench relaxation dynamics of quantum many-body systems.
With the view of investigating quantum effects in (1 + 1)-dimensional scalar-field solitons, coupled equations for a soliton profile function and a background boson field are derived by the use of a squeezed coherent state and the time-dependent variational principle. These equations take the form of canonical equations of motion with the aid of canonicity conditions developed in the time-dependent Hartree-Fock theory. They also incorporate self-consistent higher order quantum corrections in the sense of the mean field theory. A method of heat-kernel expansion is developed to deal with the background field of static solitons. At the semi-classical level, this method can reproduce the mass corrections obtained by WKB method with respect to the sine-Gordon system and φ4-model.
We present a large family of exact solitary wave solutions of the one-dimensional Gross-Pitaevskii equation, with time-varying scattering length and gain or loss, in both expulsive and regular parabolic confinement regimes. The consistency condition governing the soliton profiles is shown to map onto a linear Schrödinger eigenvalue problem, thereby enabling one to find analytically the effect of a wide variety of temporal variations in the control parameters, which are experimentally realizable. Corresponding to each solvable quantum mechanical system, one can identify a soliton configuration. These include soliton trains in close analogy to experimental observations of Streckeret al. [Nature (London) 417, 150 (2002)], spatiotemporal dynamics, solitons undergoing rapid amplification, collapse and revival of condensates, and analytical expression of two-soliton bound states, to name a few.
A quantum algorithm to simulate the real time dynamics of two-flavor massive Gross-Neveu model is presented in Schrodinger picture. We implement the simulation on a classic computer by applying the matrix product state representation. The real time evolutions of up to four particles on a site in initial state are figured out in space-time coordinate. The state evolutions are effectively affected by fermion mass and coupling constant of the model. Especially when the mass of fermion is small enough and the coupling is strong enough, the fundamental fermions evolve synchronistically in space from the two-fermion and four-fermion initial states. These are also the conditions on which the bound states made up of fundamental fermion pairs were found to arise automatically in the literatures.