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Journal of Physics A Mathematical and General

The (n=0)-component Gross-Neveu model as a description of polymers in two dimensions

K. Ziegler

Published Jun 21, 1988• 6 citations
Publisher

Abstract

A modified Gross-Neveu model with n=0 (two fermion and two boson) components is considered in two space dimensions. This model is equivalent, in terms of perturbation theory, to the Gross-Neveu model with a 2n-component fermion field, combined with the replica trick n to 0. Using the universality hypothesis, the author demonstrates that the model is related to the statistics of polymer chains. This result extends earlier investigations, obtained in a strong coupling expansion.

DOI
10.1088/0305-4470/21/12/008

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A modified Gross-Neveu model with n=0 (two fermion and two boson) components is considered in two space dimensions. This model is equivalent, in terms of perturbation theory, to the Gross-Neveu model with a 2n-component fermion field, combined with the replica trick n to 0. Using the universality hypothesis, the author demonstrates that the model is related to the statistics of polymer chains. This result extends earlier investigations, obtained in a strong coupling expansion.
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During the last few years, the phase diagram of the large N Gross–Neveu model in 1 + 1 dimensions at finite temperature and chemical potential has undergone a major revision. Here, we present a streamlined account of this development, collecting the most important results. Quasi-one-dimensional condensed matter systems like conducting polymers provide real physical systems which can be approximately described by the Gross–Neveu model and have played some role in establishing its phase structure. The kink–antikink phase found at low temperatures is closely related to inhomogeneous superconductors in the Larkin–Ovchinnikov–Fulde–Ferrell phase. With the complete phase diagram at hand, the Gross–Neveu model can now serve as a firm testing ground for new algorithms and theoretical ideas.
The quantum critical behavior of the (2 + 1)-dimensional Gross-Neveu model in the vicinity of its zero-temperature critical point is considered.The model is known to be renormalisable in the large-N limit, which offers the possibility to obtain expressions for various thermodynamic functions in closed form.We have used the concept of finite-size scaling to extract information about the leading temperature behavior of the free energy and the mass term, defined by the fermionic condensate and determined the crossover lines in the coupling (g)-temperature (T ) plane.These are given by T ∼ |g -gc|, where gc denotes the critical coupling at zero temperature.According to our analysis no spontaneous symmetry breaking survives at finite temperature.We have found that the leading temperature behavior of the fermionic condensate is proportional to the temperature with the critical amplitude √ 5 3 π.The scaling function of the singular part of the free energy is found to exhibit a maximum at ln 2 2π corresponding to one of the crossover lines.The critical amplitude of the singular part of the free energy is given by the universal number 1 3 1 2π ζ(3) -Cl2 π 3 = -0.274543 . .., where ζ(z) and Cl2(z) are the Riemann zeta and Clausen's functions, respectively.Interpreted in terms of the thermodynamic Casimir effect, this result implies an attractive Casimir "force".This study is expected to be useful in shedding light on a broader class of four fermionic models.
We consider the Gross-Neveu model with a continuous chiral symmetry in two and three spacetime dimensions at zero and finite temperature. In order to study long-range phase coherence, we construct an effective low-energy Lagrangian for the phase θθθ. This effective Lagrangian is used to show that the fermionic two-particle correlation function at finite temperature decays algebraically in 2+1 dimensions and exponentially in 1+1 dimensions.
We define and study various tensorial generalizations of the Gross-Neveu model in two dimensions, that is, models with four-fermion interactions and G3 symmetry, where we take either G = U(N) or G = O(N). Such models can also be viewed as two-dimensional generalizations of the Sachdev-Ye-Kitaev model, or more precisely of its tensorial counterpart introduced by Klebanov and Tarnopolsky, which is in part our motivation for studying them. Using the Schwinger-Dyson equations at large-N, we discuss the phenomenon of dynamical mass generation and possible combinations of couplings to avoid it. For the case G = U(N),we introduce an intermediate field representation and perform a stability analysis of the vacua. It turns out that the only apparently viable combination of couplings that avoids mass generation corresponds to an unstable vacuum. The stable vacuum breaks U(N)3 invariance, in contradiction with the Coleman-Mermin-Wagner theorem, but this is an artifact of the large-N expansion, similar to the breaking of continuous chiral symmetry in the chiral Gross-Neveu model.