In a recent work we studied the phase structure of the Gross-Neveu (GN) model in $1+1$ dimensions at finite number of fermion flavors $N_\mathrm{f} = 2, 8, 16$, finite temperature and finite chemical potential using lattice field theory. Most importantly, we found an inhomogeneous phase at low temperature and large chemical potential, quite similar to the analytically solvable $N_\mathrm{f} \rightarrow \infty$ limit. In the present work we continue our lattice field theory investigation of the finite-$N_\mathrm{f}$ GN model by studying the formation of baryons, their spatial distribution and their relation to the chiral condensate. As a preparatory step we also discuss a linear coupling of lattice fermions to the 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.
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 discuss the relevance of the Lee-Yang edge singularity to the finite-temperature Z_2-symmetry restoration transition of the Gross-Neveu model in three dimensions. We present an explicit result for its large-N free energy density in terms of Zeta(3) and the absolute maximum of Clausen's function.
Using results from the theory of non-degenerate conducting polymers like cis-polyacetylene, we generalize our previous work on dense baryonic matter and the soliton crystal in the massless Gross-Neveu model to finite bare fermion mass. In the large N limit, the exact crystal ground state can be constructed analytically, in close analogy to the bipolaron lattice in polymers. These findings are contrasted to the standard scenario with homogeneous phases only and a first order phase transition at a critical chemical potential.
We investigate the Gross-Neveu model in $2+1$ dimensions in a constant and homogeneous magnetic field using one reducible flavor of overlap fermions. Our lattice simulations suggest that the magnetic catalysis phenomenon, i.e., an increase of the chiral condensate with the magnetic field, is present for all temperatures below the chiral phase transition, in accordance with predictions from mean-field and beyond-mean-field calculations.
The Gross-Neveu model in 1+1 dimensions is generalized to the case of different scalar and pseudoscalar coupling constants. This enables us to interpolate smoothly between the standard massless Gross-Neveu models with either discrete or continuous chiral symmetry. We present the solution of the generalized model in the large N limit including the vacuum, fermion-antifermion scattering and bound states, solitonic baryons with fractional baryon number and the full phase diagram at finite temperature and chemical potential.
We study the $(1+1)$-dimensional chiral Gross-Neveu model on the lattice. At finite density, analytic mean-field results predict the existence of inhomogeneous condensates breaking both chiral symmetry and spacetime symmetries spontaneously. We investigate the fate of these inhomogeneities for two flavors and find remnant structural order, albeit with a decaying amplitude. We also map out phase diagrams in the plane spanned by the chemical potential and temperature for different lattice spacings and physical volumes. Finally, we comment on the interpretation of our results in the light of vari…
We study the phase diagram of the 1+1 dimensional Gross-Neveu model at finite number of fermion flavors using lattice field theory. Numerical results are presented, which indicate the existence of an inhomogeneous phase, where the chiral condensate is a spatially oscillating function.
We investigate the discrete chiral transformation of a Majorana fermion on a torus. Depending on the boundary conditions the integration measure can change sign. Taking this anomalous behavior into account we define a chiral order parameter as a ratio of partition functions with differing boundary conditions. Then the lattice realization of the Gross-Neveu model with Wilson fermions is simulated using the recent `worm' technique on the loop gas or all-order hopping representation of the fermions. An algorithm is formulated that includes the Gross-Neveu interaction for N fermion species. The cr…
We investigate the chiral phase structure of the Gross-Neveu model on a 2-D lattice using the Borici-Creutz fermion action. We present a strong coupling analysis of the Gross-Neveu model and perform a hybrid Monte Carlo simulation of the model with Borici-Creutz fermions. Both analytic and lattice results show a second order chiral phase transition.
We consider universal finite size effects in the large-N limit of the continuum Gross-Neveu model as well as in its discretized versions with Wilson and with staggered fermions. After extrapolation to zero lattice spacing the lattice results are compared to the continuum values.
We present a numerical simulation of the Gross-Neveu model on the lattice using a new representation in terms of fermion loops. In the loop representation all signs due to Pauli statistics are eliminated completely and the partition function is a sum over closed loops with only positive weights. We demonstrate that the new formulation allows to simulate volumes which are two orders of magnitude larger than those accessible with standard methods.
Guided by a study of kink-antikink scattering in the Gross-Neveu model and other known solutions of the Hartree-Fock approach of a particularly simple type, we demonstrate a quantitative relationship between three different problems: Quantized 1+1-dimensional fermions in the large N limit, solitons of the classical sinh-Gordon equation and classical strings moving in 3-dimensional anti de Sitter space. Aside from throwing light on the relationship between quantum field theory and classical physics, this points to the full solvability of the dynamical N-kink-antikink problem in the Gross-Neveu …
There is substantial evidence that the ground state of the 2d chiral Gross-Neveu model, in the presence of a $U(1)$ fermion number chemical potential $μ$ and in the large $N$ limit, is given by a {\it chiral spiral} phase, namely an inhomogeneous phase with a chiral condensate having a spatially periodic phase. We show that the chiral spiral configuration persists at finite $N$ and $T=0$ for any $μ>0$. Our analysis is based on non-abelian bosonization, that relates the model to a $U(N)_1$ WZW model deformed by current-current interactions. In this description the appearance of the inhomogeneou…
The divergence of the axial current is used to relate the spatial derivative of the fermion density to the bare fermion mass and scalar/pseudoscalar condensates in 1+1 dimensional Gross-Neveu models. This serves as a novel test of known results, to explain simple features of the continuous chiral model and to resolve a conflict concerning the assignment of baryon number to certain multi-fermion bound states.
The phase diagram of the massive chiral Gross-Neveu model (the 1+1-dimensional Nambu-Jona-Lasinio model at large N) is investigated in the vicinity of the tricritical point. Using the derivative expansion, the grand canonical potential is cast into the form of a Ginzburg-Landau effective action. Minimization of this action by variational and numerical methods reveals both 1st and 2nd order phase transitions to a chiral crystal phase, separated by a tricritical line. These findings are contrasted to the massive Gross-Neveu model with discrete chiral symmetry where only 2nd order transitions hav…
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.
We consider a lattice version of the bosonized Gross-Neveu model. It is explicitely chiral symmetric and its numerical simulation does not involve any anticommuting field. We study its non trivial $1/N$ expansion up to the next-to-leading term comparing the results with explicit numerical simulations.
The effectiveness of the Glasgow algorithm is explored via implementation in the 3d U(1) Gross-Neveu model and the realisation of the Goldstone mechanism in this model is compared and contrasted with its realisation in QCD.
We consider the lattice formulation of bosonized quantum field theories. As a non trivial example, we study the two dimensional Gross-Neveu model. Analytical investigations and direct numerical simulation strongly suggest that the lattice model reproduces the continuum physics. Anticommuting fields are not required and there are not related doubling problems. (3 PostScript figures available from the author)
We investigate the two flavor chiral Gross-Neveu model in the Schroedinger functional on the lattice. The procedure necessary to recover chiral symmetry in the continuum limit of this model with Wilson fermions is discussed. We introduce several useful observables and present a first demonstration of the feasibility of Monte-Carlo simulations in this model.
We apply the Grassmann tensor renormalization group (GTRG) to the one-flavor lattice Gross-Neveu model in the presence of chemical potential. We compute the fermion number density and its susceptibility and confirm the validity of GTRG for the finite density system. We introduce a method analogous to the reweighting method for Monte Carlo method and test it for some parameters.
We present numerical evidence that the 2d Yukawa models with strong quartic selfcoupling of the scalar field have the same phase structure and are asymptotically free in the Yukawa coupling like the Gross-Neveu models.
The phase diagram of the massive chiral Gross-Neveu model (the massive Nambu-Jona-Lasinio model in 1+1 dimensions) is constructed. In the large N limit, the Hartree-Fock approach can be used. We find numerically a chiral crystal phase separated from a massive Fermi gas phase by a 1st order transition. Using perturbation theory, we also construct the critical sheet where the homogeneous phase becomes unstable in a 2nd order transition. A tricritical curve is located. The phase diagram is mapped out as a function of fermion mass, chemical potential and temperature and compared with the one of th…
We present preliminary results of an excited state spectroscopy calculation in the 2-d lattice Gross-Neveu model. We address the construction of suitable interpolators for the variational method and their overlap with excitations. We comment on the role of the eigenvectors as a tool for matching scattering states on lattices with different volumes.
Two flavour massless QCD has a second order chiral transition which has been argued to belong to the universality class of the $3d$ O(4) spin model. The arguments have been questioned recently, and the transition was claimed to be mean field behaved. We discuss this issue at the example of the $3d$ Gross-Neveu model. A solution is obtained by applying various well established analytical methods.
We investigate the phase structure of the 2+1-dimensional Gross-Neveu model in the large-Nf limit, where Nf denotes the number of fermion flavors. We discuss two different fermion representations and their implication on the interpretation of a discrete symmetry of the action. We present numerical results, which indicate the existence of an inhomogeneous phase similar as in the 1+1-dimensional Gross-Neveu model.
The nature of the phase transition in the lattice Gross-Neveu model with Wilson fermions is investigated using a new analytical technique. This involves a new type of weak coupling expansion which focuses on the partition function zeroes of the model. Its application to the single flavour Gross-Neveu model yields a phase diagram whose structure is consistent with that predicted from a saddle point approach. The existence of an Aoki phase is confirmed and its width in the weakly coupled region is determined. Parity, rather than chiral symmetry breaking naturally emerges as the driving mechanism…
We study quantum critical behavior in three dimensional lattice Gross-Neveu models containing two massless Dirac fermions. We focus on two models with SU(2) flavor symmetry and either a $Z_2$ or a U(1) chiral symmetry. Both models could not be studied earlier due to sign problems. We use the fermion bag approach which is free of sign problems and compute critical exponents at the phase transitions. We estimate $ν= 0.83(1)$, $η= 0.62(1)$, $η_ψ= 0.38(1)$ in the $Z_2$ and $ν= 0.849(8)$, $η= 0.633(8)$, $η_ψ= 0.373(3)$ in the U(1) model.
Encouraged by the successful applications of Lüscher's method to boson--boson scattering we discuss the possibility of extracting scattering phase shifts from finite--volume energies for fermion--fermion scattering in the Gross--Neveu model.
We study the two-dimensional lattice Gross--Neveu model with Wilson twisted mass fermions in order to explore the phase structure in this setup. In particular, we investigate the behaviour of the phase transitions found earlier with standard Wilson fermions as a function of the twisted mass parameter $μ$. We find that qualitatively the dependence of the phase transitions on $μ$ is very similar to the case of lattice QCD.
We investigate the phase structure of the two-dimensional lattice Gross-Neveu model formulated with the Wilson fermion action to leading order of 1/N expansion. Structural change of the parity-broken phase under the influence of finite temperature and chemical potential is studied. The connection between the lattice phase structure and the chiral phase transition of the continuum theory is clarified.
We derive a new exact self-consistent crystalline condensate in the 1+1 dimensional chiral Gross-Neveu model. This also yields a new exact crystalline solution for the one dimensional Bogoliubov-de Gennes equations and the Eilenberger equation of semiclassical superconductivity. We show that the functional gap equation can be reduced to a solvable nonlinear equation, and discuss implications for the temperature-chemical potential phase diagram.
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 Lüscher's method, which exploits the volume dependence of the energies of two-particle states. The results are compared with the analytical predictions.
We investigate the two-dimensional lattice Gross-Neveu model, using the domain-wall fermion formulation, as a toy model of lattice QCD. We study features of the phase diagram related to the mechanism of chiral symmetry restoration, and find that the parity-broken phase (Aoki phase) exists for finite extent in the extra dimension ($N_s$). We also find that O(a) scaling violation terms vanishes in the limit of $N_s\to\infty$.
We present results from Monte Carlo simulations of the three-dimensional Gross-Neveu model with a U(1) chiral symmetry at nonzero temperature. We provide evidence that the model undergoes a Berezinskii-Kosterlitz-Thouless transition in accordance with the dimensional reduction scenario. We also identify a regime in the high temperature phase in which the fermions acquire nonzero dynamical mass, analogous to the pseudogap behaviour observed in cuprate superconductors.
Baryons in the large N limit of two-dimensional Gross-Neveu models are reconsidered. The time-dependent Dirac-Hartree-Fock approach is used to boost a baryon to any inertial frame and shown to yield the covariant energy-momentum relation. Momentum distributions are computed exactly in arbitrary frames and used to interpolate between the rest frame and the infinite momentum frame, where they are related to structure functions. Effects from the Dirac sea depend sensitively on the occupation fraction of the valence level and the bare fermion mass and do not vanish at infinite momentum. In the cas…
Chiral symmetry of the 2-dimensional chiral Gross-Neveu model is broken explicitly by a bare mass term as well as a splitting of scalar and pseudo-scalar coupling constants. The vacuum and light hadrons - mesons and baryons which become massless in the chiral limit - are explored analytically in leading order of the derivative expansion by means of a double sine-Gordon equation. Depending on the parameters, this model features new phenomena as compared to previously investigated 4-fermion models: spontaneous breaking of parity, a non-trivial chiral vacuum angle, twisted kink-like baryons whose…
We introduce a finite volume renormalization scheme for the N-Majorana-component O(N) invariant Gross-Neveu model. Universal observables are defined that are accessible to precise numerical simulation in various discretizations and allow for an extrapolation to the continuum limit. Here first numerical results with Wilson fermions are reported. For N=2 they reproduce exact finite volume continuum results in the massless Thirring model. Our N=8 data are ready for comparison for instance with staggered results in the future.
We study the lattice O(2N) Gross-Neveu model with Wilson fermions in the fermion loop formulation. Employing a worm algorithm for an open fermionic string, we simulate fluctuating topological boundary conditions and use them to tune the system to the critical point. We show how the worm algorithm can be extended to sample correlation functions of bound states involving an arbitrary number of Majorana fermions and present first results.
We study the chiral Gross-Neveu model with Wilson fermions. In the framework of the Schroedinger functional we show that in general not only the bare mass has to be tuned to achieve chiral symmetry in the continuum, but also coupling constants multiplying chirally non-invariant four fermion terms. The strategy for fixing the parameters of the theory is explained in perturbation theory and the results of a first order calculation are presented. The results are shown to agree with the infinite flavour limit.
Baryons in the large N limit of (1+1)-dimensional Gross-Neveu models with either discrete or continuous chiral symmetry have long been known. We generalize their construction to the case where the symmetry is explicitly broken by a bare mass term in the Lagrangian. In the discrete symmetry case, the exact solution is found for arbitrary bare fermion mass, using the Hartree-Fock approach. It is mathematically closely related to polarons and bipolarons in conducting polymers. In the continuous symmetry case, a derivative expansion allows us to rederive a formerly proposed Skyrme-type model and t…
We apply the method of Hasenfratz and Niedermayer to analytically construct perfect lattice actions for the Gross--Neveu model. In the large $N$ limit these actions display an exactly perfect scaling, i.e. cut-off artifacts are completely eliminated even at arbitrarily short correlation length. Also the energy spectrum coincides with the spectrum in the continuum and continuous translation and rotation symmetries are restored in physical observables. This is the first (analytic) construction of an exactly perfect lattice action at finite correlation length.
We study numerically on the lattice the 2D Yukawa model with the U(1) chiral symmetry and $N_F$ = 16 at infinite scalar field self-coupling. The scaling behaviour of the fermion mass, as the Yukawa coupling approaches zero, is analysed using the mean field method. It is found to agree with that of the Gross-Neveu model with the same symmetry and $N_F$. The results suggest that the 2D Yukawa models belong to the universality class of the Gross-Neveu models not only at weak scalar field self-coupling but also for a broad range of the bare parameters which is not accessible to the $1/N_F$ expansi…
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.
Fixed point actions for free and interacting staggered lattice fermions are constructed by iterating renormalization group transformations. At large N the fixed point action for the Gross-Neveu model is a perfect action in the sense of Hasenfratz and Niedermayer, i.e. cut-off effects are completely eliminated. In particular, the fermionic 1-particle energy spectrum of the lattice theory is identical with the one of the continuum even for arbitrarily small correlation lengths. The cut-off effects of the chiral condensate are eliminated using a perfect operator. (The paper is stored as a ps-file…
A new method to analytically determine the partition function zeroes of weakly coupled theories on finite-size lattices is developed. Applied to the lattice Schwinger model, this reveals the possible absence of a phase transition at fixed weak coupling. We show how finite-size scaling techniques on small or moderate lattice sizes may mimic the presence of a spurious phase transition. Application of our method to the Gross-Neveu model yields a phase diagram consistent with that coming from a saddle point analysis.
The 2+1d Gross-Neveu model with finite density and finite temperature are studied by the staggered fermion discretization. The kinetic part of this staggered fermion in momentum space is used to build the relation between the staggered fermion and Wilson-like fermion. In the large Nf limit (the number Nf of staggered fermion flavors), the chiral condensate and fermion density are solved from the gap equation in momentum space, and thus the phase diagram of fermion coupling, temperature and chemical potential are obtained. Moreover, an analytic formula for the inverse of the staggered fermion m…
We measure the critical exponents of the three dimensional Gross-Neveu model with two four-component fermions. The exponents are inferred from the scaling behaviour of observables on different lattice sizes. We also calculate the exponents, through a second order epsilon-expansion around 4d, for the three dimensional Higgs-Yukawa model, which is expected to be in the same universality class and we find that the exponents agree. We conclude that the equivalence of the two models remains valid in 3d at fixed small N_f values.
The recently proposed construction of chiral fermions on lattices with boundaries is tested in an interacting theory up to first order of perturbation theory. We confirm that, in the bulk of the lattice, the chiral Ward identities take their continuum value up to cutoff effects without any tuning. Universal quantities are defined that have an expansion in the renormalised couplings with coefficients that are functions of the physical size and the periodicity in the spatial direction. These coefficient functions have to be identical for different discretisations. We find agreement with the stan…
We present the detailed properties of a self-consistent crystalline chiral condensate in the massless chiral Gross-Neveu model. We show that a suitable ansatz for the Gorkov resolvent reduces the functional gap equation, for the inhomogeneous condensate, to a nonlinear Schrödinger equation, which is exactly soluble. The general crystalline solution includes as special cases all previously known real and complex condensate solutions to the gap equation. Furthermore, the associated Bogoliubov-de Gennes equation is also soluble with this inhomogeneous chiral condensate, and the exact spectral pro…
We investigate chiral properties of the overlap lattice fermion by using solvable model in two dimensions, the gauged Gross-Neveu model. In this model, the chiral symmetry is spontaneously broken in the presence of small but finite fermion mass. We calculate the quasi-Nambu-Goldstone(NG) boson mass as a function of the bare fermion mass and two parameters in the overlap formula. We find that the quasi-NG boson mass has desired properties as a result of the extended chiral symmetry found by Lüscher. We also examine the PCAC relation and find that it is satisfied in the continuum limit. Comparis…
We investigate the parity-broken phase structure for staggered and naive fermions in the Gross-Neveu model as a toy model of QCD. We consider a generalized staggered Gross-Neveu model including two types of four-point interactions. We use generalized mass terms to split the doublers for both staggered and naive fermions. The phase boundaries derived from the gap equations show that the mass splitting of tastes results in an Aoki phase both in the staggered and naive cases. We also discuss the continuum limit of these models and explore taking the chirally-symmetric limit by fine-tuning a mass …
We investigate the two dimensional lattice Gross--Neveu model in large flavor number limit using the domain-wall fermion formulation, as a toy model of lattice QCD. We study nonperturbative behaviorn of the restoration of chiral symmetry of the domain-wall fermions as the extent of the extra dimension $(N_s)$ is increased to infinity. We find the the parity broken phase (Aoki phase) for finite $N_s$, and study the phase diagram, which is related to the mechanism of the chiral restoration in $N_s\to\infty$ limit. The continuum limit is taken and $O(a)$ scaling violation of observables vanishes …
We present results from numerical simulations of the (2+1)-dimensional Gross-Neveu model with a U(1) chiral symmetry and N_f=4 fermion species at non-zero temperature. We provide evidence that there are two different chirally symmetric phases, one critical and one with finite correlation length, separated by a Berezinskii-Kosterlitz-Thouless transition. We have also identified a regime above the critical temperature in which the fermions acquire a screening mass even in the absence of chiral symmetry breaking, analogous to the pseudogap behaviour observed in cuprate superconductors.
We present results from a numerical study of bound state wavefunctions in the (2+1)-dimensional Gross-Neveu model with staggered lattice fermions at both zero and nonzero temperature. Mesonic channels with varying quantum numbers are identified and analysed. In the strongly coupled chirally broken phase at T=0 the wavefunctions expose effects due to varying the interaction strength more effectively than straightforward spectroscopy. In the weakly coupled chirally restored phase information on fermion - antifermion scattering is recovered. In the hot chirally restored phase we find evidence for…
