Two-dimensional Gross-Neveu Model Research Articles

Peierls Transition in Gross-Neveu Model from Bethe Ansatz

The two-dimensional Gross-Neveu model is anticipated to undergo a crystalline phase transition at high baryon charge densities. This conclusion is drawn from the mean-field approximation, which closely resembles models of Peierls instability. We demonstrate that this transition indeed occurs when both the rank of the symmetry group and the dimension of the particle representation contributing to the baryon density are large (the large N limit). We derive this result through the exact solution of the model, developing the large N limit of the Bethe ansatz. Our analytical construction of the large-N solution of the Bethe ansatz equations aligns perfectly with the periodic (finite-gap) solution of the Korteweg–de Vries (KdV) of the mean-field analysis. Published by the American Physical Society 2024

The effect of chiral density waves on the superconducting phase in the two-dimensional Gross-Neveu model

The phase structure of a 2-dimensional Gross-Neveu model with a difermion channel of interaction is studied in the variables of the temperature and chemical potentials in the limit of a large number of field components N → ∞.

Minimal doubling fermions and hermiticity

We analyze the lattice fermion kinetic term using PT symmetry, R-hermiticity, and $\gamma_{5}$-hermiticity. R-hermiticity is a condition for Hermite action and it is related to $\gamma_{5}$-hermiticity and PT symmetry. Assuming that a translation-invariant kinetic term with continuum and periodic function does not have PT symmetry, it can have R-hermiticity or $\gamma_{5}$-hermiticity. We prove that a kinetic term with continuum and periodic function that is PT symmetric does not reduce doublers. As a simple example, we analyze the two-dimensional two-flavor Gross-Neveu model with minimal doubling fermions. The minimal doubling fermions break PT symmetry and R-hermiticity, hence complex or non-Hermite coupling constants are caused by quantum correction.

The Gross-Neveu Model under the conditions of violated Lorentz invariance

The aim of this work is to calculate the effective potential in the two-dimensional Gross-Neveu model with violated Lorentz invariance, to study the properties of the obtained potential, and to find extreme points by solving the gap equation and to compare of the results with the theory in the case of nonviolated Lorentz invariance. It is shown that, in the model under study, the effects that make it crucially different from the well-studied model without violation are observed.

Critical and tricritical points for the massless 2D Gross-Neveu model beyond largeN

Using optimized perturbation theory, we evaluate the effective potential for the massless two-dimensional Gross-Neveu model at finite temperature and density containing corrections beyond the leading large-N contribution. For large N, our results exactly reproduce the well-known 1/N leading order results for the critical temperature, chemical potential, and tricritical points. For finite N, our critical values are smaller than the ones predicted by the large-N approximation and seem to observe Landau's theorem for phase transitions in one space dimension. New analytical results are presented for the tricritical points that include 1/N corrections. The easiness with which the calculations and renormalization are carried out allied to the seemingly convergent optimized results displayed, in this particular application, show the robustness of this method and allows us to obtain neat analytical expressions for the critical as well as tricritical values beyond the results currently known.

Two-dimensional Gross-Neveu model with Wilson fermion action at finite temperature and density

We analytically investigate the 2-dimensional Gross-Neveu model at finite temperature and density using Wilson fermion action. The relation between the phase structure on the lattice and that in the continuum is clarified.

Perturbative expansion around the Gaussian effective potential of the fermion field theory

We have extended the perturbative expansion method around the Gaussian effective action to fermionic field theory, by taking the two-dimensional Gross-Neveu model as an example. We have computed both the zero-temperature and the finite-temperature effective potentials of the Gross-Neveu model up to the first perturbative correction terms, and have found that the critical temperature, at which dynamically broken symmetry is restored, is significantly improved for the small value of the flavor number.

Oscillation phenomena in polyacetylene: R 1×S 1 Gross-Neveu model with a chemical potential

The phase structure of the two-dimensional Gross-Neveu model in a spacetime of the form R 1×S 1 and in the presence of a chemical potential µ is investigated. The phase portrait of the model is constructed in the parameter plane (µ, γ), where γ=1/L and L is the length of the circle S 1. In the portrait there exist two massive phases, with spontaneously broken chiral invariance, as well as infinitely many massless symmetric phases. Such a vacuum structure leads to oscillations of the critical curve µ c (γ) of chiral phase transitions. In addition, the particle number density in the vacuum state of the model oscillates if µ<µ c (γl).

Scattering phases for fermion-fermion scattering in the Gross-Neveu model

We extract scattering phases for fermion-fermion scattering from Monte Carlo simulations of the two-dimensional Gross-Neveu model. This is done by means of L\"uscher's method, which exploits the volume dependence of the energies of two-particle states. The results are compared with the analytical predictions.

Kinks and bound states in the Gross-Neveu model.

We investigate static space-dependent \ensuremath{\sigma}(x)=〈\ensuremath{\psi}\ifmmode\bar\else\textasciimacron\fi{}\ensuremath{\psi}〉 saddle point configurations in the two-dimensional Gross-Neveu model in the large N limit. We solve the saddle point condition for \ensuremath{\sigma}(x) explicitly by employing supersymmetric quantum mechanics and using simple properties of the diagonal resolvent of one-dimensional Schr\"odinger operators rather than inverse scattering techniques. The resulting solutions in the sector of unbroken supersymmetry are the Callan-Coleman-Gross-Zee kink configurations. We thus provide a direct and clean construction of these kinks. In the sector of broken supersymmetry we derive the DHN saddle point configurations. Our method of finding such nontrivial static configurations may be applied also in other two-dimensional field theories.

Vacuum instability as the origin of asymptotic freedom

By using non-perturbative approaches to the β-function, an analysis of the two-dimensional Gross-Neveu model and four-dimensional Yang-Mills and massless λØ 4 theory is presented. Asymptotic freedom is a consequence of vacuum instability occuring for arbitrarily small values of the coupling constant.

Three-loop β function(s) and effective potential in the Gross-Neveu model

We extend the perturbative analysis of the two-dimensional Gross-Neveu model by cimputing within the MS scheme the three-loop renormalization functions β,γ, and γ m in a two-parameter version of the model appropriate to the discussion of the RG properties of the effective potential, whose perturbative expansion is performed to the same order.

The Gross-Neveu chiral condensate in finite volume

The two-dimensional Gross-Neveu model exhibits spontaneous breaking of its discrete chiral symmetry. In finite volume the zero momentum fermionic modes condense and a perturbative condensate is formed. Previous calculations of the chiral condensate, are extended to two-loop order and an estimate for the infinite volume limit is given.

A comparison of numerical algorithms for dynamical fermions

Abstract The Langevin algorithm with bilinear noise and the hybrid Monte Carlo algorithm for dynamical fermions are applied to the simulation of the two-dimensional Gross-Neveu model with Wilson fermions. The hybrid Monte Carlo algorithm is implemented both with the classical “leap-frog” and with a new improved discretization scheme. The results are compared to theoretical predictions obtained by 1/N expansion and to previous simulations based on the pseudo-fermion algorithm. In the determination of the chiral transition point the three algorithms behave quite differently, and only the hybrid Monte Carlo method gives satisfactory results. The three methods are essentially equivalent in the evaluation of vacuum expectation values in a definite physical phase.

Exact method for nonlinear fermions on finite lattice: Two dimensions.

A new method is proposed to solve exactly the problem of a nonlinear fermion on a finite lattice. A lattice version of the two-dimensional chiral Gross-Neveu model with broken SU(2) is examined as a simple example. We found a phase transition through analyses of zeros of the partition function and of the specific heat.

Monte Carlo study of chiral structure: The Gross-Neveu model

The two-dimensional Gross-Neveu model is studied by Monte Carlo integration of Fermi fields. A clear transition is seen from a strong-coupling, massive, chiral symmetric phase to the weak-coupling chiral broken phase found in the continuum model. The weak-coupling behaviour of the dynamically generated scale agrees well with asymptotic freedom results.