We use the critical point large $N$ formalism to calculate the critical exponents corresponding to the fermion mass operator and flavour non-singlet fermion bilinear operator in the universality class of Quantum Electrodynamics (QED) coupled to the Gross-Neveu model for an $SU(N)$ flavour symmetry in $d$-dimensions. The $ε$ expansion of the exponents in $d$ $=$ $4$ $-$ $2ε$ dimensions are in agreement with recent three and four loop perturbative evaluations of both renormalization group functions of these operators. Estimates of the value of the non-singlet operator exponent in three dimensions are provided.
The critcal exponent $ω$ is evaluated at $O(1/N)$ in $d$-dimensions in the Gross-Neveu model using the large $N$ critical point formalism. It is shown to be in agreement with the recently determined three loop $β$-functions of the Gross-Neveu-Yukawa model in four dimensions. The same exponent is computed for the chiral Gross-Neveu and non-abelian Nambu-Jona-Lasinio universality classes.
We calculate CFT data for the Gross-Neveu model in $2<d<4$ dimensions at the next-to-leading order in the $1/N$ expansion. In particular, we make use of the background field method to derive various conformal triangles involving the composite operator $s^2$, for the Hubbard-Stratonovich field $s$. We then apply these conformal triangles to obtain the corresponding OPE coefficients.
We study a special classical current in the O(3) Gross-Neveu model that becomes supersymmetric when quantum anomalies are included. Following its definition, we generalize the current for the general case, the O(N) Gross-Neveu models. We compute its algebra and discuss the possibility of supersymmetry can be established for these models.
The free field representation of the Zamolodchikov-Faddeev algebra for the chiral Gross-Neveu model is analysed in detail, and used to construct an integral representation for form factors of the model.
A class of 1+1 dimensional supersymmetric theories with four-fermionic interaction will be built from scratch. The vacua of selected examples will be examined in the 't Hooft limit and compared to the Gross-Neveu model.
We summarize recent work showing how the Thermodynamic Bethe Ansatz may be used to study the finite-density first-order phase transition in the Gross-Neveu model. The application to trans-polyacetylene is discussed, and the significance of the results is addressed.
We show that it is possible to obtain the Gross-Neveu model in 1+1 dimensions from gauge fields only. This is reminiscent of the fact that in 1+1 dimensions the gauge field tensor is essentially a pseudo-scalar. We also show that it is possible in this context to combine the Gross-Neveu model with the massive Schwinger model in the limit where the fermion mass is larger than the electric charge.
We compute the anomalous dimensions of a class of operators of the form $(\barψψ)^p$ and $(\barψψ)^pψ$ to leading order in $ε$ in the Gross-Neveu model in $2+ε$ dimensions. We use the techniques developed in arXiv: 1505.00963.
The isomorphism $SU(4) \simeq O(6)$ is used to construct the form factors of the O(6) Gross-Neveu model as bound state form factors of the SU(4) chiral Gross-Neveu model. This technique is generalized and is then applied to use the O(6) as the starting point of the nesting procedure to obtain the O(N) form factors for general even N.
We study the scattering theory for the Gross-Neveu model on the half-line. We find the reflection matrices for the elementary fermions, and by fusion we compute the ones for the two-particle bound-states, showing that they satisfy non-trivial bootstrap consistency conditions. We also compute more general reflection matrices for the Gross-Neveu model and the nonlinear sigma model, and argue that they correspond to the integrable boundary conditions we identified in our previous paper hep-th/9809178.
We use the recently developed CFT techniques of Rychkov and Tan to compute anomalous dimensions in the $O(N)$ Gross-Neveu model in $d=2+ε$ dimensions. To do this, we extend the "cowpie contraction" algorithm of arXiv:1506.06616 to theories with fermions. Our results match perfectly with Feynman diagram computations.
We introduce the 2PPI (2-point-particle-irreducible) expansion, which sums bubble graphs to all orders. We prove the renormalizibility of this summation. We use it on the Gross-Neveu model to calculate the mass gap and vacuum energy. After an optimization of the expansion, the final results are qualitatively good.
We use the large $N$ critical point formalism to compute $d$-dimensional critical exponents at several orders in $1/N$ in an Ising Gross-Neveu universality class where the core interaction includes a Lie group generator. Specifying a particular symmetry group or taking the abelian limit of the final exponents recovers known results but also provides expressions for any Lie group or fermion representation.
Recently it was shown that kink baryons and kink-antikink scattering in the massless Gross-Neveu model are closely related to one- and two-soliton solutions of the sinh-Gordon equation. Here we generalize these findings to the case of n kinks and antikinks. Using the known n soliton solution of the sinh-Gordon equation, we solve the general n kink-antikink scattering problem in the large N Gross-Neveu model analytically, mapping the time-dependent Hartree-Fock approach onto inverse scattering theory.
We consider a version of the Gross-Neveu model in 1+1 dimensions with discrete chiral and continuous flavor symmetry (isospin). In 2+1 dimensions, this model is known as chiral Heisenberg Gross-Neveu model. Spontaneous symmetry breaking and the emergence of two massless and one massive scalar bosons are shown. A duality to the Nambu--Jona-Lasinio model with isospin is exhibited, provided that the isovector pseudoscalar mean field is constrained to a plane in isospin space. This enables us to find the phase diagram as a function of temperature, chemical potential and isospin chemical potential …
The massive Gross-Neveu model is solved in the large N limit at finite temperature and chemical potential. The phase diagram features a kink-antikink crystal phase which was missed in previous works. Translated into the framework of condensed matter physics our results generalize the bipolaron lattice in non-degenerate conducting polymers to finite temperature.
We reconsider the two-flavor version of the massive, chiral Gross-Neveu model in 1+1 dimensions. Its phase diagram as a function of baryon chemical potential, isospin chemical potential and temperature has previously been explored. We recapitulate the results, adding the missing tricritical curves. They can be determined exactly by extending the standard stability analysis, using fourth order almost degenerate perturbation theory. Results for three different bare masses are presented and discussed.
This paper presents a case study of the effects of increasing the order of a Ginzburg-Landau type expansion, by using the well known Gross-Neveu model in 1+1 dimensions as a test case. It is found that as the order of expansion increases, the predicted phase diagram increasingly resembles the known exact phase diagram. Finally, some properties of arbitrary large order phase diagrams are examined.
The massless Gross-Neveu and chiral Gross-Neveu models are well known examples of integrable quantum field theories in 1+1 dimensions. We address the question whether integrability is preserved if one either replaces the four-fermion interaction in fermion-antifermion channels by a dual interaction in fermion-fermion channels, or if one adds such a dual interaction to an existing integrable model. The relativistic Hartree-Fock-Bogoliubov approach is adequate to deal with the large N limit of such models. In this way, we construct and solve three integrable models with Cooper pairing. We also i…
The Gross-Neveu model with ${\rm U}_L(N_f)\times{\rm U}_R(N_f)$ chiral symmetry is reconsidered in the large $N_c$ limit. The known analytical solution for the time dependent interaction of any number of twisted kinks and breathers is cast into a more revealing form. The ($x,t$)-dependent factors are isolated from constant coefficients and twist matrices. These latter generalize the twist phases of the single flavor model. The crucial tool is an identity for the inverse of a sum of two square matrices, derived from the known formula for the determinant of such a sum.
The existence of non-local charges, generating a Yangian symmetry is discussed in generalized chiral Gross-Neveu models. Their conservation can be proven by a finite-loop perturbative computation, the order of which is determined from group theoretic constants and is independent of the number of flavors. Examples, where the 1-loop calculation is sufficient, include the SO(n)-models and other more exotic groups and representations.
The scattering of two Dashen-Hasslacher-Neveu (DHN) baryons in the large N Gross-Neveu model is solved exactly using the relativistic, time-dependent Hartree-Fock approach. Unlike the special case of kink-antikink scattering, the scattering of DHN baryons is sensitive to the back-reaction of the fermions bound inside the baryons. Correspondingly the solution is much more complicated than the kink-antikink scattering solutions, which can be expressed in terms of sinh-Gordon solitons. Nevertheless, we present a simple ansatz form that gives closed analytic expressions for both the space-time dep…
We illustrate the phase structure of a deformed two-dimensional Gross-Neveu model which is defined by undeformed field contents plus deformed Pauli matrices. This deformation is based on two motives to find a more general polymer model and to estimate how $q$-deformed field theory affects on its effective potential. There found some regions where chiral symmetry breaking and restoration take place repeatedly as temperature increasing.
We study scalar and chiral fermionic models in next-to-leading order with the help of the functional renormalisation group. Their critical behaviour is of special interest in condensed matter systems, in particular graphene. To derive the beta functions, we make extensive use of computer algebra. The resulting flow equations were solved with pseudo-spectral methods to guarantee high accuracy. New estimates on critical quantities for both the Ising and the Gross-Neveu model are provided. For the Ising model, the estimates agree with earlier renormalisation group studies of the same level of app…
The phase transition of the Gross-Neveu model with N fermions is investigated by means of a non-perturbative evolution equation for the scale dependence of the effective average action. The critical exponents and scaling amplitudes are calculated for various values of N in d=3. It is also explicitely verified that the Neveu-Yukawa model belongs to the same universality class as the Gross-Neveu model.
We calculate the critical exponents $ω_\pm$ in the $d$-dimensional Gross-Neveu model in $1/N$ expansion with $1/N^2$ accuracy. These exponents are related to the slopes of the $β$-functions at the critical point in the Gross - Neveu - Yukawa model. They have been computed recently to four loops accuracy. We checked that our results are in complete agreement with the results of the perturbative calculations.
We study the spectrum of stable static fermion bags in the 1+1 dimensional Gross-Neveu model with $N$ flavors of Dirac fermions, in the large $N$ limit. In the process, we discover a new kink, heavier than the Callan-Coleman-Gross-Zee (CCGZ) kink, which is marginally stable (at least in the large $N$ limit). The connection of this new kink and the conjectured $S$ matrix of the Gross-Neveu model is obscured at this point. After identifying all stable static fermion bags, we arrange them into a periodic table, according to their $O (2N)$ and topological quantum numbers.
Multi-critical behavior of interacting fermions in graphene's honeycomb lattice is presented. In particular, we considered the spin triplet insulating orders, where the spin rotational symmetry of the order parameter is explicitly broken. By casting the problem in terms of Gross-Neveu-Yukawa theory we show that such symmetry-breaking terms are irrelevant near the metal-insulator critical point. A finite Yukawa coupling among bosons and fermions improves the stability of such critical point against the symmetry-breaking perturbations. Physical sources of such symmetry-breaking terms are pointed…
We consider intersecting D-brane models which have two dimensional chiral fermions localized at the intersections. At weak coupling, the interactions of these fermions are described by generalized Gross-Neveu models. At strong coupling, these configurations are described by the dynamics of probe D-branes in a curved background spacetime. We study patterns of dynamical chiral symmetry breaking in these models at weak and strong coupling, and also discuss relationships between these two descriptions.
The QED$_3$-Gross-Neveu model is a (2+1)-dimensional U(1) gauge theory involving Dirac fermions and a critical real scalar field. This theory has recently been argued to represent a dual description of the deconfined quantum critical point between Neel and valence bond solid orders in frustrated quantum magnets. We study the critical behavior of the QED$_3$-Gross-Neveu model by means of an epsilon expansion around the upper critical space-time dimension of $D_c^+=4$ up to the three-loop order. Estimates for critical exponents in 2+1 dimensions are obtained by evaluating the different Pade appr…
We compute at the one-loop order the beta-functions for a renormalisable non-commutative analog of the Gross Neveu model defined on the Moyal plane. The calculation is performed within the so called x-space formalism. We find that this non-commutative field theory exhibits asymptotic freedom for any number of colors. The beta-function for the non-commutative counterpart of the Thirring model is found to be non vanishing.
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 …
We calculate the anomalous dimensions of higher-spin currents, both singlet and non-singlet, in the Gross-Neveu model at the $1/n^2$ order. It was conjectured that in the critical regime this model is dual to a higher-spin gauge theory on $AdS_4$. The $AdS/CFT$ correspondence predicts that the masses of higher-spin fields correspond to the scaling dimensions of the singlet currents in the Gross-Neveu model.
We renormalize the SU(N) Gross-Neveu model in the modified minimal subtraction (MSbar) scheme at four loops and determine the beta-function at this order. The theory ceases to be multiplicatively renormalizable when dimensionally regularized due to the generation of evanescent 4-fermi operators. The first of these appears at three loops and we correctly take their effect into account in deriving the renormalization group functions. We use the results to provide estimates of critical exponents relevant to phase transitions in graphene.
Formation of fermion bag solitons is an important paradigm in the theory of hadron structure. We report here on our non-perturbative analysis of this phenomenon in the 1+1 dimensional massive Gross-Neveu model, in the large $N$ limit. Our main result is that the extremal static bag configurations are reflectionless, as in the massless Gross-Neveu model. Explicit formulas for the profiles and masses of these solitons are presented. We also present a particular type of self-consistent reflectionless solitons which arise in the massive Nambu-Jona-Lasinio models, in the large-N limit.
In 2+1 dimensions, for low momenta, using dimensional renormalization we study the effect of a Chern-Simons field on the perturbative expansion of fermions self interacting through a Gross Neveu coupling. For the case of just one fermion field, we verify that the dimension of operators of canonical dimension lower than three decreases as a function of the Chern-Simons coupling.
Dirac and Weyl fermions appear as quasi-particle excitations in many different condensed-matter systems. They display various quantum transitions which represent unconventional universality classes related to the variants of the Gross-Neveu model. In this work we study the bosonized version of the standard Gross-Neveu model -- the Gross-Neveu-Yukawa theory -- at three-loop order, and compute critical exponents in $4-ε$ dimensions for general number of fermion flavors. Our results fully encompass the previously known two-loop calculations, and agree with the known three-loop results in the pure…
We construct an explicit example of unitarity violation in fermionic quantum field theories in non-integer dimensions. We study the two-point correlation function of four-fermion operators. We compute the one-loop anomalous dimensions of these operators in the Gross-Neveu-Yukawa Model. We find that at one loop order, the four-fermion operators split into three classes with one class having negative norms. This implies that the theory violates unitarity following the definition in arXiv:1512.00013 [hep-th].
In the search for QFT's that admit boundstates, we reinvestigate the two dimensional Gross-Neveu model, but with massive fermions. By computing the self-energy for the auxiliary boundstate field and the effective potential, we show that there are no bound states around the lowest minimum, but there is a meta-stable bound state around the other minimum, a local one. The latter decays by tunneling. We determine the dependence of its lifetime on the fermion mass and coupling constant.
The general SU(N) form factor formula is constructed. Exact form factors for the field, the energy momentum and the current operators are derived and compared with the 1/N-expansion of the chiral Gross-Neveu model and full agreement is found. As an application of the form factor approach the equal time commutation rules of arbitrary local fields are derived and in general anyonic behavior is found.
The large N limit of the 3-d 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 $S^2 \times S^1$ and $H^2\times S^1$. We evaluate the free energy density, the spontaneous magnetization and the correlation length at the ultraviolet fixed point. The limit $S^1\to R$, which is interpreted as the zero temperature limit, is also studied.
This work is dedicated to the study of the noncommutative Gross-Neveu model. As it is known, in the canonical Weyl-Moyal approach the model is inconsistent, basically due to the separation of the amplitudes into planar and nonplanar parts. We prove that if instead a coherent basis representation is used, the model becomes renormalizable and free of the aforementioned difficulty. We also show that, although the coherent states procedure breaks Lorentz symmetry in odd dimensions, in the Gross-Neveu model this breaking can be kept under control by assuming the noncommutativity parameters to be sm…
We find the general solution to the time-dependent Hartree-Fock problem for scattering solutions of the Gross-Neveu models, with both discrete (GN_2) and continuous (NJL_2) chiral symmetry. We find new multi-breather solutions both for the GN_2 model, generalizing the Dashen-Hasslacher-Neveu breather solution, and also new twisted breathers for the NJL_2 model. These solutions satisfy the full TDHF consistency conditions, and only in the special cases of GN_2 kink scattering do these conditions reduce to the integrable Sinh-Gordon equation. We also show that all baryons and breathers are compo…
Magnetic catalysis describes the enhancement of symmetry breaking quantum fluctuations in chirally symmetric quantum field theories by the coupling of fermionic degrees of freedom to a magnetic background configuration. We use the functional renormalization group to investigate this phenomenon for interacting Dirac fermions propagating in (2+1)-dimensional spacetime, described by the Gross-Neveu model. We identify pointlike operators up to quartic fermionic terms that can be generated in the renormalization group flow by the presence of an external magnetic field. We employ the beta function f…
We analyse the spinor action on a curved noncommutative space, the so-called truncated Heisenberg algebra, and in particular, the nonminimal coupling of spinors to the torsion. We find that dimensional reduction of the Dirac action gives the noncommutative extension of the Gross-Neveu model, the model which is, as shown by Vignes-Tourneret, fully renormalisable.
We study the critical behavior of the D (2<D<4) dimensional Gross-Neveu model with a Thirring interaction, where a vector-vector type four-fermi interaction is on equal terms with a scalar-scalar type one. By using inversion method up to the next-to-leading order of 1/N expansion, we construct a gauge invariant effective potential. We show the existence of the chiral order phase transition, and determine explicitly the critical surface. It is observed that the critical behavior is mainly controlled by the Gross-Neveu coupling g. The critical surface can be divided into two parts by the surface…
We investigate whether the beta function of the finite-$N$ Gross-Neveu model, as calculated up to the four-loop level, exhibits evidence for an infrared zero. As part of our analysis, we calculate and analyze Padé approximants to this beta function and evaluate effects of scheme dependence. From our study, we find that in the range of coupling where the perturbative calculation of the four-loop beta function is reliable, it does not exhibit robust evidence for an infrared zero.
