3D Gross-Neveu Model Research Articles

Quark-Antiquark and Diquark Condensates in Vacuum in Two-Flavor Four-Fermion Interaction Models with Any Color Number Nc

The color number Nc-dependence of the interplay between quark-antiquark condensates ⟨q̄q⟩ and diquark condensates ⟨qq⟩ in vacuum in two-flavor four-fermion interaction models is researched. The results show that the GS-HS (the coupling constant of scalar (q̄q)2-scalar (qq)2 channel) phase diagrams will be qualitatively consistent with the case of Nc = 3 as Nc varies in 4D Nambu–Jona–Lasinio model and 2D Gross–Neveu (GN) model. However, in 3D GN model, the behavior of the GS-HP (the coupling constant of pseudoscalar (qq)2 channel) phase diagram will obviously depend on Nc. The known characteristic that a 3D GN model does not have the coexistence phase of the condensates ⟨q̄q⟩ and ⟨qq⟩ is proven to appear only in the case of Nc ≤ 4. In all the models, the regions occupied by the phases containing the diquark condensates hqqi in corresponding phase diagrams will gradually decrease as Nc grows up and finally go to zero if Nc → ∞, i.e. in this limit only the pure ⟨q̄q⟩ phase could exist.

Phase diagram of quark-antiquark and diquark condensates in the 3-dimensional Gross-Neveu model with the 4-component spinor representation

We construct the phase diagram of the quark-antiquark and diquark condensates at finite temperature and density in the 2+1 dimensional (3D) two flavor massless Gross-Neveu (GN) model with the 4-component quarks. In contrast to the case of the 2-component quarks, there appears the coexisting phase of the quark-antiquark and diquark condensates. This is the crucial difference between the 2-component and 4-component quark cases in the 3D GN model. The coexisting phase is also seen in the 4D Nambu Jona-Lasinio model. Then we see that the 3D GN model with the 4-component quarks bears closer resemblance to the 4D Nambu Jona-Lasinio model.

High energy behavior of gravity at largeN

A first step in the analysis of the renormalizability of gravity at large $\mathbf{N}$ is carried out. Suitable resummations of planar diagrams give rise to a theory in which there is only a finite number of primitive, superficially divergent, Feynman diagrams. The mechanism is similar to the one which makes the 3D Gross-Neveu model renormalizable at large $\mathbf{N}$. The connections with gravitational confinement and Kawai-Lewellen-Tye relations are briefly analyzed. Some potential problems in fulfilling the Zinn-Justin equations are pointed out.

Holography in 4D (super) higher spin theories and a test via cubic scalar couplings

The correspondences proposed previously between higher spin gauge theories and free singleton field theories were recently extended into a more complete picture by Klebanov and Polyakov in the case of the minimal bosonic theory in D=4 to include the strongly coupled fixed point of the 3d O(N) vector model. Here we propose an N=1 supersymmetric version of this picture. We also elaborate on the role of parity in constraining the bulk interactions, and in distinguishing two minimal bosonic models obtained as two different consistent truncations of the minimal N=1 model that retain the scalar or the pseudo-scalar field. We refer to these models as the Type A and Type B models, respectively, and conjecture that the latter is holographically dual to the 3d Gross-Neveu model. In the case of the Type A model, we show the vanishing of the three-scalar amplitude with regular boundary conditions. This agrees with the O(N) vector model computation of Petkou, thereby providing a non-trivial test of the Klebanov-Polyakov conjecture.

Particle Density in Zero Temperature Symmetry Restoring Phase Transitions in Four-Fermion Interaction Models**The project supported by National Natural Science Foundation of China

By means of critical behaviors of the dynamical fermion mass in four-fermion interaction models, we show by explicit calculations that when T = 0 the particle density will have a discontinuous jumping across the critical chemical potential μc in 2D and 3D Gross-Neveu (GN) model and these physically explain the first-order feature of the corresponding symmetry restoring phase transitions. For the second-order phase transitions in the 3D GN model when T → 0 and in 4D Nambu–Jona–Lasinio (NJL) model when T = 0, it is proven that the particle density itself will be continuous across μc but its derivative over the chemical potential μ will have a discontinuous jumping. The results give a physical explanation of implications of the tricritical point in the 3D GN model. The discussions also show effectiveness of the critical analysis approach of phase transitions.

Phase Transitions in the Gross–Neveu Model on a Sphere for High-Tc Materials

Phase transitions were studied in the 3D Gross–Neveu model on a sphere at T ≠ 0. The zeta function of the Dirac operator of the model was represented with a double sum, and its derivative was calculated using recurrent formulas for generalized Epstein–Hurwitz functions. The effective potential Vwas determined as a function of the σ field, temperature T, and inverse radius of curvature r. Phase transitions first-order in temperature at R= 0 (Tc = 1/2 ln 2) and second-order in curvature R= 2/r2 at T= 0 (rc = 1/1.5) were identified. For the general case, T ≠ 0 and R ≠ 0, the critical phase dependence Tc(rc) was obtained, and the V(σ, T, r) surface was constructed. External stimuli such as temperature and curvature were shown to restore the symmetry of the system in the 3DGross–Neveu model.

Discrete symmetry breaking and restoration at finite temperature in 3D Gross-Neveu model

Dynamical spontaneous breaking of some discrete symmetries including special parities and time reversal and their restoration at finite temperature T are researched in 3D Gross-Neveu model by means of Schwinger-Dyson equation in the real-time thermal field theory in the fermion bubble diagram approximation. When the momentum cut-off Λ is large enough, the equation of critical chemical potential μ c and critical temperature T c will be Λ-independent and identical to the one obtained by auxiliary scalar field approach. The dynamical fermion mass m≡ m( T, μ), as the order parameter of symmetry breaking, has the same ( T c − T) 1/2 behavior at T≲ T c as one in 4D NJL-model and this shows the second-order phase transition feature of the symmetry restoration at T> T c . It is also proven that no scalar bound state could exist in this model.

Three-dimensional Gross-Neveu model on curved spaces

The large N limit of the 3D Gross-Neveu model is here studied on manifolds with positive and negative constant curvature. Using the ζ-function regularization we analyze the critical properties of this model on the spaces S 2 × S 1 and H 2 × S 1. We evaluate the free energy density, the spontaneous magnetization and the correlation length at the ultraviolet fixed point. The limit S 1 → R , which is interpreted as the zero temperature limit, is also studied.

Interplay of universality classes in a three-dimensional Yukawa model.

We investigate numerically on the lattice the interplay of universality classes of the three-dimensional Yukawa model with U(1) chiral symmetry, using the Binder method of finite size scaling. At zero Yukawa coupling the scaling related to the magnetic Wilson-Fisher fixed point is confirmed. At sufficiently strong Yukawa coupling the dominance of the chiral fixed point associated with the 3D Gross-Neveu model is observed for various values of the coupling parameters, including infinite scalar self-coupling. In both cases the Binder method works consistently in a broad range of lattice sizes. However, when the Yukawa coupling is decreased the finite size behavior gets complicated and the Binder method gives inconsistent results for different lattice sizes. This signals a crossover between the universality classes of the two fixed points. \textcopyright{} 1996 The American Physical Society.

Phase transitions at finite temperature and dimensional reduction for fermions and bosons

In a recent letter we discussed the fact that large- N expansions and computer simulations indicate that the universality class of the finite temperature chiral symmetry restoration transition in the 3D Gross-Neveu model is mean field theory. This was seen to be a counterexample to the standard ‘sigma model’ scenario which predicts the 2D Ising model universality class. In this article we present more evidence, both theoretical and numerical, that this result is correct. We develop a physical picture for our results and discuss the width of the scaling region (Ginzburg criterion), 1/N corrections, and differences between the dynamics of BCS superconductors and Gross-Neveu models. Lattices as large as 12 x 72 2 are simulated for both the N = 12 and N = 4 cases and the numerical evidence for mean field scaling is quite compelling. We point out that the amplitude ratio for the model's susceptibility is a particularly good observable for distinguishing between the dimensional reduction and the mean field scenarios, because this universal quantity differs by almost a factor of 20 in the two cases. The simulations are done close to the critical point in both the symmetric and broken phases, and correlation lengths of order 10 are measured. The critical indices β mag and S also pick out mean field behavior. We trace the breakdown of the standard scenario (dimensional reduction and universality) to the composite character of the mesons in the model. We point out that our results should be generic for theories with dynamical symmetry breaking, such as Quantum Chromodynamics. We also simulated the O(2) model on 8 x 16 3 lattices to establish that our methods give the results of dimensional reduction in purely bosonic cases where its theoretical basis is firm. We also show that Z 2 Nambu-Jona-Lasinio models simulated on 8 x 16 3 lattices give mean field rather than three-dimensional Ising model indices.

Can sigma models describe finite temperature chiral transitions?

Large-N expansions and computer simulations indicate that the universality class of the finite temperature chiral symmetry restoration transition in the 3D Gross-Neveu model is mean field theory. This is a counterexample to the standard 'sigma model' scenario which predicts the 2D Ising model universality class. We trace the breakdown of the standard scenario (dimensional reduction and universality) to the absence of canonical scalar fields in the model. We point out that our results could be generic for theories with dynamical symmetry breaking, such as Quantum Chromodynamics.