Gross-Neveu Research Articles

Gross-Neveu model with O(2)L×O(2)R chiral symmetry: Duality with Zakharov-Mikhailov model and large N solution

The two-flavor Gross-Neveu model with ${\mathrm{U}(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{U}(2{)}_{R}$ chiral symmetry in $1+1$ dimensions is used to construct a novel variant of four-fermion theories with ${\mathrm{O}(2)}_{L}\ifmmode\times\else\texttimes\fi{}\mathrm{O}(2{)}_{R}$ chiral symmetry. The spontaneous breaking of the group O(2), a continuous group with two connected components (rotations and reflections), gives rise to new phenomena. It is ideally suited to describe a situation where two distinct kinds of condensation compete, in particular chiral symmetry breaking (particle-hole condensation) and Cooper pairing (particle-particle condensation). After solving the O(2) chiral Gross-Neveu model in detail, we demonstrate that it is dual to another classically integrable model due to Zakharov and Mikhailov. The duality enables us to solve the quantum version of this model in the large $N$ limit with semiclassical methods, supporting its integrability at the quantum level. The resulting model is the unique four-fermion theory sharing the full Pauli-G\"ursey symmetry with free, massless fermions (``perfect Gross-Neveu model'') and provides us with a solvable model for competing chiral and Cooper pair condensates, including explicit soliton dynamics and the phase diagram.

Integrable sigma models on Riemann surfaces

We consider quantum aspects of a class of generalized Gross-Neveu models, which in special cases reduce to sigma models. We show that, in the case of gauged models, an admissible gauge is $A_\mu=0$, which is a direct analogue of the conformal gauge in string models. Chiral anomalies are a gauge counterpart of the Weyl anomaly, and are required to vanish. Topological effects on the worldsheet lead to an integration over moduli spaces of connections on a Riemann surface. This is an initial step in studying the effects of worldsheet geometry and topology in integrable sigma models.

Chiral critical behavior of 3D lattice fermionic models with quartic interactions

We study the critical behavior of the three-dimensional Gross-Neveu (GN) model with ${N}_{f}$ four-component Dirac fermionic flavors and quartic interactions, at the chiral ${\mathbb{Z}}_{2}$ transition in the massless ${\mathbb{Z}}_{2}$-symmetric limit. For this purpose, we consider a lattice GN model with staggered Kogut-Susskind fermions and a scalar field coupled to the scalar bilinear fermionic operator, which effectively realizes the attractive four-fermion interaction. We perform Monte Carlo simulations for ${N}_{f}=4$, 8, 12, 16. By means of finite-size scaling analyses of the numerical data, we obtain estimates of the critical exponents, which are compared with the large-${N}_{f}$ predictions obtained using the continuum GN field theory. We observe a substantial agreement. This confirms that lattice GN models with staggered fermions provide a nonperturbative realization of the GN quantum field theory, even though the lattice interactions explicitly break the flavor $\mathrm{U}({N}_{f})\ensuremath{\bigotimes}\mathrm{U}({N}_{f})$ symmetry of the GN field theory, which is only recovered in the critical limit.

Improved coarse-graining methods for two dimensional tensor networks including fermions

We show how to apply renormalization group algorithms incorporating entanglement filtering methods and a loop optimization to a tensor network which includes Grassmann variables which represent fermions in an underlying lattice field theory. As a numerical test a variety of quantities are calculated for two dimensional Wilson-Majorana fermions and for the two flavor Gross-Neveu model. The improved algorithms show much better accuracy for quantities such as the free energy and the determination of Fisher’s zeros.

Spectroscopy in the 2+1D Thirring model with N=1 domain wall fermions

We employ the domain wall fermion (DWF) formulation of the Thirring model on a lattice in $2+1+1$ dimensions and perform $N=1$ flavor Monte Carlo simulations. At a critical interaction strength the model features a spontaneous $\mathrm{U}(2)\ensuremath{\rightarrow}\mathrm{U}(1)\ensuremath{\bigotimes}\mathrm{U}(1)$ symmetry breaking; we analyze the induced spin-0 mesons, both Goldstone and non-Goldstone, as well as the correlator of the fermion quasiparticles, in both resulting phases. Crucially, we determine the anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\psi}}\ensuremath{\approx}3$ at the critical point, in stark contrast with the Gross-Neveu model in 3D and with results obtained with staggered fermions. Our numerical simulations are complemented by an analytical treatment of the free fermion correlator, which exhibits large early-time artifacts due to branch cuts in the propagator stemming from unbound interactions of the fermion with its heavy doublers. These artifacts are generalizable beyond the Thirring model, being an intrinsic property of DWF, or more generally Ginsparg-Wilson fermions.

Generative learning for the problem of critical slowing down in lattice Gross-Neveu model

In lattice field theory, Monte Carlo simulation algorithms get highly affected by critical slowing down in the critical region, where autocorrelation time increases rapidly. Hence the cost of generation of lattice configurations near the critical region increases sharply. In this paper, we use a Conditional Generative Adversarial Network (C-GAN) for sampling lattice configurations. We train the C-GAN on the dataset consisting of Hybrid Monte Carlo (HMC) samples in regions away from the critical region, i.e., in the regions where the HMC simulation cost is not so high. Then we use the trained C-GAN model to generate independent samples in the critical region. We perform both interpolation and extrapolation to the critical region. Thus, the overall computational cost is reduced. We test our approach for Gross-Neveu model in 1+1 dimension. We find that the observable distributions obtained from the proposed C-GAN model match with those obtained from HMC simulations, while circumventing the problem of critical slowing down.

Infinite variance in Monte Carlo sampling of lattice field theories

In Monte Carlo calculations of expectation values in lattice quantum field theories, the stochastic variance of the sampling procedure that is used defines the precision of the calculation for a fixed number of samples. If the variance of an estimator of a particular quantity is formally infinite, or in practice very large compared to the square of the mean, then that quantity can not be reliably estimated using the given sampling procedure. There are multiple scenarios in which this occurs, including in Lattice Quantum Chromodynamics, and a particularly simple example is given by the Gross-Neveu model where Monte Carlo calculations involve the introduction of auxiliary bosonic variables through a Hubbard-Stratonovich (HS) transformation. Here, it is shown that the variances of HS estimators for classes of operators involving fermion fields are divergent in this model and an even simpler zero-dimensional analogue. To correctly estimate these observables, two alternative sampling methods are proposed and numerically investigated.

Correlated Chern insulators in two-dimensional Raman lattices: A cold-atom regularization of strongly coupled four-Fermi field theories

We show that synthetic spin-orbit coupling for ultracold atoms in optical Raman potentials can be exploited to build versatile quantum simulators of correlated Chern insulators connected to strongly coupled four-Fermi field theories similar to the Gross-Neveu model in (2+1) dimensions. Exploiting this multidisciplinary perspective, we identify a large-N quantum anomalous Hall (QAH) effect in absence of any external magnetic field, and use it to delimit regions in parameter space where these correlated topological phases appear, the boundaries of which are controlled by strongly coupled fixed points of these four-Fermi relativistic field theories. We further show how, for strong interactions, the QAH effect gives way to magnetic phases described by a two-dimensional quantum compass model in a transverse field. We present a detailed description of the phase diagram using the large-N effective potential, and variational techniques such as projected entangled pairs.Received 22 December 2020Revised 8 July 2022Accepted 25 July 2022DOI:https://doi.org/10.1103/PhysRevResearch.4.L042012Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasChern insulatorsLattice field theoryOptical lattices & trapsQuantum simulationAtomic, Molecular & OpticalQuantum InformationCondensed Matter, Materials & Applied Physics

Tricritical curve of massive chiral Gross-Neveu model with isospin

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.

Renormalization group improvement of the effective potential in a (1 + 1) dimensional Gross-Neveu model

In this work, we investigate the consequences of the Renormalization Group Equation (RGE) in the determination of the effective potential and the study of Dynamical Symmetry Breaking (DSB) in an Gross-Neveu (GN) model with N fermions fields in (1+1) dimensional space-time, which can be applied as a model to describe certain properties of the polyacetylene. The classical Lagrangian of the model is scale invariant, but radiative corrections to the effective potential can lead to dimensional transmutation, when a dimensionless parameter (coupling constant) of the classical Lagrangian is exchanged for a dimensionful one, a dynamically generated mass for the fermion fields. For the model we are considering, perturbative calculations of the effective potential and renormalization group functions up to three loops are available, but we use the RGE and the leading logs approximation to calculate an improved effective potential, including contributions up to six loops orders. We then perform a systematic study of the general aspects of DSB in the GN model with finite N, comparing the results we obtain with the ones derived from the original unimproved effective potential we started with.

Inhomogeneous Phase of the Chiral Gross-Neveu Model.

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 "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 nonabelian bosonization, that relates the model to a U(N)_{1} Wess-Zumino-Witten model deformed by current-current interactions. In this description, the appearance of the inhomogeneous phase is surprisingly simple. We also rederive the phase diagram of the large N chiral Gross-Neveu model via a direct diagrammatic computation, finding agreement with previous results in the literature.

Topological chiral currents in the Gross-Neveu model extension

We unveil an interesting connection of Lorentz-violating quantum field theories, studied in the context of the standard model extension, and Hubbard-type models of topological crystalline phases. These models can be interpreted as a regularisation of the former and, as hereby discussed, explored with current quantum simulators based on ultra-cold atoms in optical Raman lattices. In particular, we present a complete analysis of the Creutz-Hubbard ladder under a generic magnetic flux, which regularises a Gross-Neveu model extension, and presents a characteristic circulating chiral current whose non-zero value arises from a specific violation of Lorentz invariance. We present a complete phase diagram with trivial insulators, ferromagnetic and anti-ferromagnetic phases, and current-carrying topological crystalline phases. These predictions are benchmarked using tools from condensed matter and quantum-information science, showing that self-consistent Hartree-Fock and strong-coupling Dzyaloshinskii-Moriya methods capture the essence of the phase diagram in different regimes, which is further explored using extensive numerical simulations based on matrix-product states.

Exact tricritical point from next-to-leading-order stability analysis

In the massive chiral Gross-Neveu model, a phase boundary separates a homogeneous from an inhomogeneous phase. It consists of two parts, a second order line and a first order line, joined at a tricritical point. Whereas the first order phase boundary requires a full, numerical Hartree-Fock calculation, the second order phase boundary can be determined exactly and with less effort by a perturbative stability analysis. We extend this stability analysis to higher order perturbation theory. This enables us to locate the tricritical point exactly, without need to perform a Hartree-Fock calculation. Divergencies due to the emergence of spectral gaps in a spatially periodic perturbation are handled using well established tools from many body theory.

The chiral Gross-Neveu model on the lattice via a Landau-forbidden phase transition

We study the phase diagram of the (1 + 1)-dimensional Gross-Neveu model with both {g}_x^2{left(overline{psi}psi right)}^2 and {g}_y^2{left(overline{psi}i{gamma}_5psi right)}^2 interaction terms on a spatial lattice. The continuous chiral symmetry, which is present in the continuum model when {g}_x^2={g}_y^2 , has a mixed ’t Hooft anomaly with the charge conservation symmetry, which guarantees the existence of a massless mode. However, the same ’t Hooft anomaly also implies that the continuous chiral symmetry is broken explicitly in our lattice model. Nevertheless, from numerical matrix product state simulations we find that for certain parameters of the lattice model, the continuous chiral symmetry reemerges in the infrared fixed point theory, even at strong coupling. We argue that, in order to understand this phenomenon, it is crucial to go beyond mean-field theory (or, equivalently, beyond the leading order term in a 1/N expansion). Interestingly, on the lattice, the chiral Gross-Neveu model appears at a Landau-forbidden second order phase transition separating two distinct and unrelated symmetry-breaking orders. We point out the crucial role of two different ’t Hooft anomalies or Lieb-Schultz-Mattis obstructions for this Landau-forbidden phase transition to occur.

Excitonic insulators and Gross-Neveu models

We introduce a generalized Gross-Neveu (GN) model to describe the excitonic instabilities in two different systems: a small overlap semi-metal (SM) and a small gap semi-conductor (SMC), both in two (2d) and three-dimensions (3d). We identify the excitonic order parameter (EOP) and obtain the effective potential within the Large $N$ limit approach where the GN model can be exactly solved. We obtain the excitonic insulator (EI) phase diagrams as a function of temperature, chemical potential, overlap between bands and gaps of the system. We show that the EI may undergo first- or second-order thermal transitions depending on the regime whereupon this phase is approached. We also investigate the expected thermodynamic signatures for the specific heat above the fine-tuned excitonic quantum critical point (EQCP), in both 2d and 3d, in the SMC regime. We show that the EQCP is a different kind of critical point since although the EOP vanishes at the EQCP, there is always a finite gap in the SMC regime. We find that for high temperatures, the specific heat might exhibit a scaling behavior in the form $C_V/T \propto T^{(d-z)/z}$, where $d$ is the dimension of the system and $z$ is the dynamical critical exponent. The very low temperature behavior has a dominant exponential thermally activated term due to the presence of a gap that does not vanish at the excitonic transition.

Black-hole-like saturons in Gross-Neveu

It has been argued recently that objects of maximal microstate entropy permitted by unitarity, the so-called saturons, have properties similar to black holes. We demonstrate the existence of such objects in Gross-Neveu model. From the large-$N$ scaling of $S$-matrix, we deduce the connection between the entropy of the bound-state and the unitarity of scattering. We observe that upon saturation of unitarity, the bound state exhibits a remarkable correspondence with a black hole. The scaling of its entropy is identical to Bekenstein-Hawking entropy. The saturon decays via Hawking's thermal rate of temperature given by the inverse size. The information retrieval time from the Gross-Neveu saturon is isomorphic to Page's time. Our observations indicate that black hole properties are exhibited by saturated states in simple calculable models.

Inhomogeneities in the two-flavor chiral Gross-Neveu model

We investigate the finite-temperature and -density chiral Gross-Neveu model with an axial U$_A$(1) symmetry in $1+1$ dimensions on the lattice. In the limit where the number of flavors $N_\mathrm{f}$ tends to infinity the continuum model has been solved analytically and shows two phases: a symmetric high-temperature phase with a vanishing condensate and a low-temperature phase in which the complex condensate forms a chiral spiral which breaks translation invariance. In the lattice simulations we employ chiral SLAC fermions with exact axial symmetry. Similarly to $N_\mathrm{f}\to\infty$, we find for $8$ flavors, where quantum and thermal fluctuations are suppressed, two distinct regimes in the $(T,\mu)$ phase diagram, characterized by qualitatively different behavior of the two-point functions of the condensate fields. More surprisingly, at $N_\mathrm{f}=2$, where fluctuations are no longer suppressed, the model still behaves similarly to the $N_\mathrm{f}\to\infty$ model and we conclude that the chiral spiral leaves its footprints even on systems with a small number of flavors. For example, at low temperature the two-point functions are still dominated by chiral spirals with pitches proportional to the inverse chemical potential, although in contrast to large-$N_\mathrm{f}$ their amplitudes decrease with distance. We argue that these results should not be interpreted as the spontaneous breaking of a continuous symmetry, which is forbidden in two dimensions. Finally, using Dyson-Schwinger equations we calculate the decay of the U$_A$(1)-invariant fermion four-point function in search for a BKT phase at zero temperature.

Inhomogeneous Phases in the Chirally Imbalanced 2 + 1-Dimensional Gross-Neveu Model and Their Absence in the Continuum Limit

We studied 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 was found for one of the two regularizations. Our results suggest that there is no inhomogeneous phase in the continuum limit. We showed 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.

Spontaneous non-Hermiticity in the ( 2+1 )-dimensional Gross-Neveu model

Using a nonperturbative approach based on the Cornwall-Jackiw-Tomboulis (CJT) effective action $\mathrm{\ensuremath{\Gamma}}(S)$ for composite operators ($S$ is the full fermion propagator), the phase structure of the simplest massless ($2+1$)-dimensional Gross-Neveu model is investigated. We have calculated $\mathrm{\ensuremath{\Gamma}}(S)$ and its stationary (or Dyson-Schwinger) equation in the first order of the bare coupling constant $G$ and have shown that there exist a well-defined dependence of $G\ensuremath{\equiv}G(\mathrm{\ensuremath{\Lambda}})$ on the cutoff parameter $\mathrm{\ensuremath{\Lambda}}$, such that the Dyson-Schwinger equation is renormalized. It has three different solutions for fermion propagator $S$ corresponding to possible dynamical appearance of three different mass terms in the model. One is a Hermitian, but two others are non-Hermitian and $\mathcal{P}\mathcal{T}$ even or odd. It means that two phases with spontaneous non-Hermiticity can be emerged in the system. Moreover, mass spectrum of quasiparticles is real in these non-Hermitian and $\mathcal{P}\mathcal{T}$ even/odd phases.

The $\mathsf{CP}^{n-1}$-model with fermions: a new look

We elaborate the formulation of the $\mathsf{CP^{n-1}}$ sigma model with fermions as a gauged Gross-Neveu model. This approach allows to identify the super phase space of the model as a supersymplectic quotient. Potential chiral gauge anomalies are shown to receive contributions from bosons and fermions alike and are related to properties of this phase space. Along the way we demonstrate that the worldsheet supersymmetric model is a supersymplectic quotient of a model with target space supersymmetry. Possible generalizations to other quiver supervarieties are briefly discussed.