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We study the μ-μ45-T phase diagram of the 2+1-dimensional Gross-Neveu model, where μ denotes the ordinary chemical potential, μ45 the chiral chemical potential and T the temperature. We use the mean-field approximation and two different lattice regularizations with naive chiral fermions. An inhomogeneous phase at finite lattice spacing is found for one of the two regularizations. Our results suggest that there is no inhomogeneous phase in the continuum limit. We show that a chiral chemical potential is equivalent to an isospin chemical potential. Thus, all results presented in this work can also be interpreted in the context of isospin imbalance.
The phase diagram of the (1+1)-dimensional Gross-Neveu model is reanalyzed for (non-)zero chemical potential and (non-)zero temperature within the mean-field approximation. By investigating the momentum dependence of the bosonic two-point function, the well-known second-order phase transition from the Z2 symmetric phase to the so-called inhomogeneous phase is detected. In the latter phase the chiral condensate is periodically varying in space and translational invariance is broken. This work is a proof of concept study that confirms that it is possible to correctly localize second-order phase transition lines between phases without condensation and phases of spatially inhomogeneous condensation via a stability analysis of the homogeneous phase. To complement other works relying on this technique, the stability analysis is explained in detail and its limitations and successes are discussed in context of the Gross-Neveu model. Additionally, we present explicit results for the bosonic wave-function renormalization in the mean-field approximation, which is extracted analytically from the bosonic two-point function. We find regions -- a so-called moat regime -- where the wave function renormalization is negative accompanying the inhomogeneous phase as expected.
The phase diagram of the (1+1)-dimensional Gross-Neveu model is reanalyzed for (non-)zero chemical potential and (non-)zero temperature within the mean-field approximation. By investigating the momentum dependence of the bosonic two-point function, the well-known second-order phase transition from the Z2 symmetric phase to the so-called inhomogeneous phase is detected. In the latter phase the chiral condensate is periodically varying in space and translational invariance is broken. This work is a proof of concept study that confirms that it is possible to correctly localize second-order phase transition lines between phases without condensation and phases of spatially inhomogeneous condensation via a stability analysis of the homogeneous phase. To complement other works relying on this technique, the stability analysis is explained in detail and its limitations and successes are discussed in context of the Gross-Neveu model. Additionally, we present explicit results for the bosonic wave-function renormalization in the mean-field approximation, which is extracted analytically from the bosonic two-point function. We find regions -- a so-called moat regime -- where the wave function renormalization is negative accompanying the inhomogeneous phase as expected.
The phase diagram of the (1+1)-dimensional Gross-Neveu model is reanalyzed for (non-)zero chemical potential and (non-)zero temperature within the mean-field approximation. By investigating the momentum dependence of the bosonic two-point function, the well-known second-order phase transition from the Z2 symmetric phase to the so-called inhomogeneous phase is detected. In the latter phase the chiral condensate is periodically varying in space and translational invariance is broken. This work is a proof of concept study that confirms that it is possible to correctly localize second-order phase transition lines between phases without condensation and phases of spatially inhomogeneous condensation via a stability analysis of the homogeneous phase. To complement other works relying on this technique, the stability analysis is explained in detail and its limitations and successes are discussed in context of the Gross-Neveu model. Additionally, we present explicit results for the bosonic wave-function renormalization in the mean-field approximation, which is extracted analytically from the bosonic two-point function. We find regions -- a so-called moat regime -- where the wave function renormalization is negative accompanying the inhomogeneous phase as expected.
We study the μ-μ45-T phase diagram of the 2+1-dimensional Gross-Neveu model, where μ denotes the ordinary chemical potential, μ45 the chiral chemical potential and T the temperature. We use the mean-field approximation and two different lattice regularizations with naive chiral fermions. An inhomogeneous phase at finite lattice spacing is found for one of the two regularizations. Our results suggest that there is no inhomogeneous phase in the continuum limit. We show that a chiral chemical potential is equivalent to an isospin chemical potential. Thus, all results presented in this work can also be interpreted in the context of isospin imbalance.
We explore the phase structure of the 1+1 dimensional Gross-Neveu model at finite number of fermion flavors using lattice field theory. Besides a chirally symmetric phase and a homogeneously broken phase we find evidence for the existence of an inhomogeneous phase, where the condensate is a spatially oscillating function. Our numerical results include a crude μ-T phase diagram.
In this work, the phase diagram of the 2+1-dimensional Gross-Neveu model is investigated with baryon chemical potential as well as chiral chemical potential in the mean-field approximation. We study the theory using two lattice discretizations, which are both based on naive fermions. An inhomogeneous chiral phase is observed only for one of the two discretizations. Our results suggest that this phase disappears in the continuum limit.
We continue previous investigations of the (inhomogeneous) phase structure of the Gross-Neveu model in a noninteger number of spatial dimensions (1≤d<3) in the limit of an infinite number of fermion species (N→∞) at (non)zero chemical potential μ. In this work, we extend the analysis from zero to nonzero temperature T.
The phase diagram of the Gross-Neveu model in 1≤d<3 spatial dimensions is well known under the assumption of spatially homogeneous condensation with both a symmetry broken and a symmetric phase present for all spatial dimensions. In d=1 one additionally finds an inhomogeneous phase, where the order parameter, the condensate, is varying in space. Similarly, phases of spatially varying condensates are also found in the Gross-Neveu model in d=2 and d=3, as long as the theory is not fully renormalized, i.e., in the presence of a regulator. For d=2, one observes that the inhomogeneous phase vanishes, when the regulator is properly removed (which is not possible for d=3 without introducing additional parameters).
In the present work, we use the stability analysis of the symmetric phase to study the presence (for 1≤d<2) and absence (for 2≤d<3) of these inhomogeneous phases and the related moat regimes in the fully renormalized Gross-Neveu model in the μ,T-plane. We also discuss the relation between "the number of spatial dimensions" and "studying the model with a finite regulator" as well as the possible consequences for the limit d→3.
Inhomogeneous condensation in the Gross-Neveu model in noninteger spatial dimensions 1 ≤ d < 3
(2023)
The Gross-Neveu model in the N→∞ approximation in d=1 spatial dimensions exhibits a chiral inhomogeneous phase (IP), where the chiral condensate has a spatial dependence that spontaneously breaks translational invariance and the Z2 chiral symmetry. This phase is absent in d=2, while in d=3 its existence and extent strongly depends on the regularization and the value of the finite regulator. This work connects these three results smoothly by extending the analysis to non-integer spatial dimensions 1≤d<3, where the model is fully renormalizable. To this end, we adapt the stability analysis, which probes the stability of the homogeneous ground state under inhomogeneous perturbations, to non-integer spatial dimensions. We find that the IP is present for all d<2 and vanishes exactly at d=2. Moreover, we find no instability towards an IP for 2≤d<3, which suggests that the IP in d=3 is solely generated by the presence of a regulator.
We show the absence of an instability of homogeneous (chiral) condensates against spatially inhomogeneous perturbations for various 2+1-dimensional four-fermion and Yukawa models. All models are studied at non-zero baryon chemical potential, while some of them are also subjected to chiral and isospin chemical potential. The considered theories contain up to 16 Lorentz-(pseudo)scalar fermionic interaction channels. We prove the stability of homogeneous condensates by analyzing the bosonic two-point function, which can be expressed in a purely analytical form at zero temperature. Our analysis is presented in a general manner for all of the different discussed models. We argue that the absence of an inhomogeneous chiral phase (where the chiral condensate is spatially non-uniform) follows from this lack of instability. Furthermore, the existence of a moat regime, where the bosonic wave function renormalization is negative, in these models is ruled out.