Thirring Model Research Articles

Application of Reservoir Computers for Chaos Synchronization and Cracking Chaos-Based Cryptography in Fermionic 2D Thirring and 4D Gursey Models with Spinor Fields

Reservoir computers have recently become one of the most widely exploited model-free approaches to chaos synchronization due to their fast convergence speed and simple yet effective learning rules. In this study, reservoir computing has been utilized to emulate the dynamics of chaotic Thirring and Gursey systems. In the supervised learning of the reservoir, one-step ahead states of the dynamical systems have been employed as the teaching signals. After the training phase, the reservoir was first run autonomously and then weakly driven by chaotic systems. It has been shown that the trained reservoir computers can exhibit the same characteristics as the attractors of the learned chaotic systems, which enable their use in synchronization tasks of chaotic systems with possible application in cracking chaos-based cryptography. To investigate the effect of noise on the performance of reservoir computers, noise signals with different colors and amplitudes have been included in the transmitted signals. The obtained results indicate that the proposed scheme can provide lower error metrics for both models.

Evaluation of the Bilinear Condensate of the Planar Thirring Model in the Strongly Coupled Region

Evaluation of the Bilinear Condensate of the Planar Thirring Model in the Strongly Coupled Region

On the analysis and deeper properties of the fractional complex physical models pertaining to nonsingular kernels.

This study solves the coupled fractional differential equations defining the massive Thirring model and the Kundu Eckhaus equation using the Natural transform decomposition method. The massive Thirring model is a dynamic component of quantum field theory, consisting of a coupled nonlinear complex differential equations. Initially, we study the suggested equations under the fractional derivative of Caputo-Fabrizio. The Atangana-Baleanu derivative is then used to evaluate the comparable equations. The results are significant and necessary for exploring a range of physical processes. This paper uses modern approach and the fractional operators in this situation to develop satisfactory approximations to the offered problems. The proposed approach combines the natural transform technique with the efficient Adomian decomposition scheme. Obtaining numerical findings in the form of a fast-converge series significantly improves the scheme's accuracy. Some graphical plot distributions are presented to show that the present approach is very simple and straightforward. We performed a fractional order analysis of assumed phenomena to demonstrate and validate the effectiveness of the future technique. The behaviour of the approximate series solution for several fractional orders is shown visually. Additionally, the nature of the derived outcome has been observed for various fractional orders. The derived results demonstrate how simple and efficient the proposed method is to apply for analysing the behaviour of fractionally-order complex nonlinear differential equations that arise in related fields of engineering and science.

Testing the equivalence between the planar Gross-Neveu and Thirring models at N=1

It is known that the Fierz identities predict that the Gross-Neveu and Thirring models should be equivalent when describing systems composed of a single fermionic flavor, N=1. Here, we consider the planar version of both models within the framework of the optimized perturbation theory at the two-loop level, in order to verify if the predicted equivalence emerges explicitly when different temperature and density regimes are considered. At vanishing densities, our results indicate that both models indeed describe exactly the same thermodynamics, provided that N=1. However, at finite chemical potentials we find that the N=1 Fierz equivalence no longer holds. After examining the relevant thermodynamic potentials, we have identified the contributions which lead to this puzzling discrepancy. Finally, we discuss different frameworks in which this (so far open) problem could be further understood and eventually circumvented. Published by the American Physical Society 2024

Algebraic solitons in the massive Thirring model.

We present exact solutions describing dynamics of two algebraic solitons in the massive Thirring model. Each algebraic soliton corresponds to a simple embedded eigenvalue in the Kaup-Newell spectral problem and attains the maximal mass among the family of solitary waves traveling with the same speed. By coalescence of speeds of the two algebraic solitons, we find a new solution for an algebraic double-soliton which corresponds to a double embedded eigenvalue. We show that the double-soliton attains the double mass of a single soliton and describes a slow interaction of two identical algebraic solitons.

A Microlocal Investigation of Stochastic Partial Differential Equations for Spinors with an Application to the Thirring Model

On a d-dimensional Riemannian, spin manifold (M, g) we consider non-linear, stochastic partial differential equations for spinor fields, driven by a Dirac operator and coupled to an additive Gaussian, vector-valued white noise. We extend to the case in hand a procedure, introduced in Dappiaggi et al (Commun Contemp Math 27(07):2150075, 2022), for the scalar counterpart, which allows to compute at a perturbative level the expectation value of the solutions as well as the associated correlation functions accounting intrinsically for the underlying renormalization freedoms. This framework relies strongly on tools proper of microlocal analysis and it is inspired by the algebraic approach to quantum field theory. As a concrete example we apply it to a stochastic version of the Thirring model proving in particular that it lies in the subcritical regime if d≤2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$d\\le 2$$\\end{document}.

Fixed lines in four fermion models in two dimensions

Motivated by conjectures about near-horizon dynamics in quantum gravity, we search for lines of perturbatively accessible fixed points emanating from models of N free fermions. Through two loops we find a new class of models, apart from the well-known Abelian Thirring models. Further study is needed to see whether these can lead to true conformal manifolds, or perhaps a new class of large-N fixed points. Published by the American Physical Society 2024

A new approach for solving fractional Kundu-Eckhaus equation and fractional massive Thirring model using controlled Picard’s technique with [formula omitted]-Laplace transform

This paper introduces a novel approach called the Controlled Picard technique along with ρ-Laplace transform for simulating the solutions of the fractional Kundu–Eckhaus equation (FKEE) and time-fractional massive Thirring model (FMTM). Significant applications of these equations can be found in nonlinear optics, weakly dynamic dispersive waves of water, and the study of quantum fields. Our proposed approach merges the Controlled Picard technique with the ρ-Laplace transform, resulting in an elegant framework that effectively solves nonlinear fractional equations without the need for Lagrange multipliers or Adomian polynomials. The inclusion of a small parameter h enhances convergence and is particularly advantageous for nonlinear differential equations. We demonstrate the accuracy and convergence of our method by comparing it with other analytical approaches. In addition, we offer graphical depictions of the numerical results to demonstrate the effectiveness as well as reliability of our suggested methodology. Overall, our approach is highly efficient, accurate, and straightforward to implement.

On small black holes in string theory

We discuss the worldsheet sigma-model whose target space is the d+1 dimensional Euclidean Schwarzschild black hole. We argue that in the limit where the Hawking temperature of the black hole, T, approaches the Hagedorn temperature, TH, it can be described in terms of a generalized version of the Horowitz-Polchinski effective theory. For d ≥ 6, where the Horowitz-Polchinski EFT [1, 2] does not have suitable solutions, the modified effective Lagrangian allows one to study the black hole CFT in an expansion in powers of d − 6 and TH − T. At T = TH, the sigma model is non-trivial for all d > 6. It exhibits an enhanced SU(2) symmetry, and is described by a non-abelian Thirring model with a radially dependent coupling. The resulting picture connects naturally to the results of [3–5], that relate Schwarzschild black holes in flat spacetime at large d to the two dimensional black hole. We also discuss an analogous open string system, in which the black hole is replaced by a system of two separated D-branes connected by a throat. In this system, the asymptotic separation of the branes plays the role of the inverse temperature. At the critical separation, the system is described by a Kondo-type model, which again exhibits an enhanced SU(2) symmetry. At large d, the brane system gives rise to the hairpin brane [6].

Integrable coupled bosonic massive Thirring model and its nonlocal reductions

A coupled bosonic massive Thirring model (BMTM), involving an interaction between the two independent spinors, is introduced and shown to be integrable. By incorporating suitable reductions between the field components of the coupled BMTM, five novel integrable models with various type of nonlocal interactions are constructed. Lax pairs satisfying the zero curvature condition are obtained for the coupled BMTM and for each of the related nonlocal models. An infinite number of conserved quantities are derived for each of these models which confirms the integrability of the systems. It is shown that the coupled BMTM respects important symmetries of the original BMTM such as parity, time reversal, global U(1)-gauge and the proper Lorentz transformations. Similarly, all the nonlocal models obtained from the coupled BMTM remain invariant under combined operation of parity and time reversal transformations. However, it is found that only one of the nonlocal models is invariant under proper Lorentz transformation and two other models are invariant under global U(1)-gauge transformation. By using ultralocal Poisson bracket relations among the elements of the Lax operator, it is shown that the coupled BMTM and one of the nonlocal models are completely integrable in the Liouville sense.

Entanglement entropy of two disjoint intervals and spin structures in interacting chains in and out of equilibrium

We take the paradigm of interacting spin chains, the Heisenberg spin-12\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\frac{1}{2} $$\\end{document} XXZ model, as a reference system and consider interacting models that are related to it by Jordan-Wigner transformations and restrictions to sub-chains. An example is the fermionic analogue of the gapless XXZ Hamiltonian, which, in a continuum scaling limit, is described by the massless Thirring model. We work out the Rényi-α entropies of disjoint blocks in the ground state and extract the universal scaling functions describing the Rényi-α tripartite information in the limit of infinite lengths. We consider also the von Neumann entropy, but only in the limit of large distance. We show how to use the entropies of spin blocks to unveil the spin structures of the underlying massless Thirring model. Finally, we speculate about the tripartite information after global quenches and conjecture its asymptotic behaviour in the limit of infinite time and small quench. The resulting conjecture for the “residual tripartite information”, which corresponds to the limit in which the intervals’ lengths are infinitely larger than their (large) distance, supports the claim of universality recently made studying noninteracting spin chains. Our mild assumptions imply that the residual tripartite information after a small quench of the anisotropy in the gapless phase of XXZ is equal to − log 2.

On bosonic Thirring model in Minkowski signature

We present the way to continue the bosonic Thirring model or βγ-system with quartic interaction to Minkowski signature, based on the symmetries of this model. It is shown that the considered Minkowski version of the model is one-loop renormalizable. Based on this, we find the amplitudes of scattering of the excitations corresponding to the γ and γ¯ fields up to the one-loop order. In particular, it was computed that the 2→2 amplitudes of these excitations possess property of reflectionless scattering and thus the corresponding S-matrix of such excitations satisfies the Yang-Baxter equation. The obtained S-matrix elements for γ and γ¯ are shown to coincide with the corresponding S-matrix elements of the solitons in the complex sine-Gordon model proposed by Dorey and Hollowood.

Integrable coupled massive Thirring model with field values in a Grassmann algebra

A coupled massive Thirring model of two interacting Dirac spinors in 1 + 1 dimensions with fields taking values in a Grassmann algebra is introduced, which is closely related to a SU(1) version of the Grassmannian Thirring model also introduced in this work. The Lax pair for the system is constructed, and its equations of motion are obtained from a zero curvature condition. It is shown that the system possesses several infinite hierarchies of conserved quantities, which strongly confirms its integrability. The model admits a canonical formulation and is invariant under space-time translations, Lorentz boosts and global U(1) gauge transformations, as well as discrete symmetries like parity and time reversal. The conserved quantities associated to the continuous symmetries are derived using Noether’s theorem, and their relation to the lower-order integrals of motion is spelled out. New nonlocal integrable models are constructed through consistent nonlocal reductions between the field components of the general model. The Lagrangian, the Hamiltonian, the Lax pair and several infinite hierarchies of conserved quantities for each of these nonlocal models are obtained substituting its reduction in the expressions of the analogous quantities for the general model. It is shown that, although the Lorentz symmetry of the general model breaks down for its nonlocal reductions, these reductions remain invariant under parity, time reversal, global U(1) gauge transformations and space-time translations.

Data-driven localized waves and parameter discovery in the massive Thirring model via extended physics-informed neural networks with interface zones

In this paper, we study data-driven localized wave solutions and parameter discovery in the massive Thirring (MT) model via the deep learning in the framework of physics-informed neural networks (PINNs) algorithm. Abundant data-driven solutions including soliton of bright/dark type, breather and rogue wave are simulated accurately and analyzed contrastively with relative and absolute errors. For higher-order localized wave solutions, we employ the extended PINNs (XPINNs) with domain decomposition to capture the complete pictures of dynamic behaviors such as soliton collisions, breather oscillations and rogue-wave superposition. In particular, we modify the interface line in domain decomposition of XPINNs into a small interface zone and introduce the pseudo initial, residual and gradient conditions as interface conditions linked adjacently with individual neural networks. Then this modified approach is applied successfully to various solutions ranging from bright-bright soliton, dark-dark soliton, dark-antidark soliton, general breather, Kuznetsov-Ma breather and second-order rogue wave. Experimental results show that this improved version of XPINNs reduce the complexity of computation with faster convergence rate and keep the quality of learned solutions with smoother stitching performance as well. For the inverse problems, the unknown coefficient parameters of linear and nonlinear terms in the MT model are identified accurately with and without noise by using the classical PINNs algorithm.

Formation of Rogue Waves and Modulational Instability with Zero-Wavenumber Gain in Multicomponent Systems with Coherent Coupling.

It is known that rogue waves (RWs) are generated by the modulational instability (MI) of the baseband type. Starting with the Bers-Kaup-Reiman system for three-wave resonant interactions, we identify a specific RW-building mechanism based on MI which includes zero wavenumber in the gain band. An essential finding is that this mechanism works solely under a linear relation between the MI gain and a vanishingly small wavenumber of the modulational perturbation. The same mechanism leads to the creation of RWs by MI in other multicomponent systems-in particular, in the massive Thirring model.

Lifetime of locally stable states near a phase transition in the Thirring model

We study the lifetime of locally stable states in the Thirring model, which describes a system of particles whose interactions are long-range. The model exhibits first-order phase transitions in the canonical ensemble and, therefore, a free energy barrier separates two free energy minima. The energy of the system diffuses as a result of thermal fluctuations and we show that its dynamics can be described by means of a Fokker–Planck equation. Considering an initial state where the energy takes the value corresponding to one of the minima of the free energy, we can define the lifetime of the initial state as the mean first-passage time for the system to reach the top of the free energy barrier between the minima. We use an analytical formula for the mean first-passage time which is based on the knowledge of the exact free energy of the model, even at a finite number of particles. This formula shows that the lifetime of locally stable states increases exponentially in the number of particles, which is a typical feature of systems with long-range interactions. We also perform Monte Carlo simulations in the canonical ensemble in order to obtain the probability distribution of the first-passage time, which turns out to be exponential in time in a long time limit. The numerically obtained mean first-passage time agrees with the theoretical prediction. Combining theory and simulations, our work provides a new insight in the study of metastability in many-body systems with long-range interactions.

Bosonization, duality, and the C-theorem in the non-abelian Thirring model

We revisit the two dimensional non-Abelian Thirring model in order to investigate its fixed point structure and the corresponding renormalization group (RG) flow. For this purpose we discuss the bosonization of the model, and we present different, but of course equivalent, bosonic versions of the theory. The bosonic theories are illuminating in that they exhibit the fixed points in a manifest way, and also possess a remarkable strong/weak duality that sheds light on the fixed point structure of the theory. We study the RG flow through the computation of the Zamolodchikov C-function and of the β-function in the large-level limit. Within this framework, we discuss how close to the infrared fixed point the RG flow can reach, since this point is strictly unachievable due to an emergent gauge invariance.

Rogue waves in the massive Thirring model

AbstractIn this paper, general rogue wave solutions in the massive Thirring (MT) model are derived by using the Kadomtsev–Petviashvili (KP) hierarchy reduction method and these rational solutions are presented explicitly in terms of determinants whose matrix elements are elementary Schur polynomials. In the reduction process, three reduction conditions including one index‐ and two dimension‐ones are proved to be consistent by only one constraint relation on parameters of tau‐functions of the KP‐Toda hierarchy. It is found that the rogue wave solutions in the MT model depend on two background parameters, which influence their orientation and duration. Differing from many other coupled integrable systems, the MT model only admits the rogue waves of bright‐type, and the higher order rogue waves represent the superposition of fundamental ones in which the nonreducible parameters determine the arrangement patterns of fundamental rogue waves. Particularly, the super rogue wave at each order can be achieved simply by setting all internal parameters to be zero, resulting in the amplitude of the sole huge peak of order N being times the background. Finally, rogue wave patterns are discussed when one of the internal parameters is large. Similar to other integrable equations, the patterns are shown to be associated with the root structures of the Yablonskii–Vorob'ev polynomial hierarchy through a linear transformation.

Deep learning of fermion sign fluctuations

We describe a procedure for alleviating the fermion sign problem in which phase fluctuations are explicitly subtracted from the Boltzmann factor. Several ans\"atze for fluctuations are designed and compared. In the absence of a sufficiently high-quality ansatz, a neural network can be trained to parameterize the fluctuations. Demonstrating on the staggered Thirring model in $1+1$ dimensions, we examine the performance of this method as deeper neural networks are used, and in conjunction with the well-studied contour deformation methods.

The Coleman correspondence at the free fermion point

We prove that the truncated correlation functions of the charge and gradient fields associated with the massless sine-Gordon model on \mathbb{R}^2 with \beta=4\pi exist for all coupling constants and are equal to those of the chiral densities and vector current of free massive Dirac fermions. This is an instance of Coleman's prediction that the massless sine-Gordon model and the massive Thirring model are equivalent (in the above sense of correlation functions). Our main novelty is that we prove this correspondence starting from the Euclidean path integral in the nonperturbative regime of the infinite volume models. We use this correspondence to show that the correlation functions of the massless sine-Gordon model with \beta=4\pi decay exponentially and that the corresponding probabilistic field is localized.