Nonlinear Field Theory Research Articles

Quantum electronic circuit simulation of generalized sine-Gordon models

Investigation of strongly interacting, nonlinear quantum field theories (QFT-s) remains one of the outstanding challenges of modern physics. Here, we describe analog quantum simulators for nonlinear QFT-s using mesoscopic superconducting circuit lattices. Using the Josephson effect as the source of nonlinear interaction, we investigate generalizations of the quantum sine-Gordon model. In particular, we consider a two-field generalization, the double sine-Gordon model. In contrast to the sine-Gordon model, this model can be purely quantum integrable, when it does not admit a semi-classical description - a property that is generic to many multi-field QFT-s. The primary goal of this work is to investigate different thermodynamic properties of the double sine-Gordon model and propose experiments that can capture its subtle quantum integrability. First, we analytically compute the mass-spectrum and the ground state energy in the presence of an external `magnetic' field using Bethe ansatz and conformal perturbation theory. Second, we calculate the thermodynamic Bethe ansatz equations for the model and analyze its finite temperature properties. Third, we propose experiments to verify the theoretical predictions.

Scientific Life and Works of Walter Noll

Walter Noll (1925–2017) was an American mathematician of German birth who made lasting contributions to the foundations of continuum physics and the classical non-linear field theory. This essay is an attempt to put in a broader perspective Noll’s methods and achievements in the hope that young generations of researchers may find their inspiration in the talent and depth of the old. By no means should this be considered as a historical account on the development of continuum mechanics through the second half of the twentieth century. We are content to illuminate Noll’s precious legacy.

Particle level screening of scalar forces in 1+1 dimensions

We investigate how non-linear scalar field theories respond to point sources. Taking the symmetron as a specific example of such a theory, we solve the non-linear equation of motion in one spatial dimension for (i) an isolated point source and (ii) two identical point sources with arbitrary separation. We find that the mass of a single point source can be screened by the symmetron field, provided that its mass is above a critical value. We find that two point sources behave as independent, isolated sources when the separation between them is large, but, when their separation is smaller than the symmetron's Compton wavelength, they behave much like a single point source with the same total mass. Finally, we explore closely related behavior in a toy Higgs-Yukawa model, and find indications that the maximum fermion mass that can be generated consistently via a Yukawa coupling to the Higgs in 1+1 dimensions is roughly the mass of the Higgs itself, with potentially intriguing implications for the hierarchy problem.

Klein–Gordon field from the XXZ Heisenberg model

We examine the recently introduced idea of Spin-Field Correspondence (SFC) focusing on the example of the spin system described by the XXZ Heisenberg model with external magnetic field. The Hamiltonian of the resulting nonlinear scalar field theory is derived for arbitrary value of the anisotropy parameter [Formula: see text]. We show that the linear scalar field theory is reconstructed in the large spin limit. For [Formula: see text], a nonrelativistic scalar field theory satisfying the Born reciprocity principle is recovered. As expected, for the vanishing anisotropy parameter [Formula: see text], the standard relativistic Klein–Gordon field is obtained. Various aspects of the obtained class of the scalar fields are studied, including the fate of the relativistic symmetries and the properties of the emerging interaction terms. We show that, in a certain limit, the so-called polymer quantization of the field variables is recovered. This and other discussed properties suggest a possible relevance of the considered framework in the context of quantum gravity.

Mechanics of hard-magnetic soft materials

Soft materials that can undergo rapid and large deformation through the remote and wireless action of external stimuli offer a range of tantalizing applications such as soft robots, flexible electronics, and biomedical devices. A natural and simple embodiment of such materials is to embed magnetic particles in soft polymers. Unfortunately, existing magnetically responsive soft materials such as magnetorheological elastomers and ferrogels typically use magnetically-soft particles such as iron and iron oxides, which are characterized by the low coercivity and hence lack the capability to retain remnant magnetism. Accordingly, their deformation is limited to simple elongation or shortening, rendering these materials substantially unsuited for the complex transformations required in many applications. To introduce shape-programmability, magnetically-hard particles with high coercivity have been incorporated in mechanically soft materials. In addition, recent works aimed at ameliorating this situation have developed fabrication techniques and facile routes to engineer rapid and complex transformations in a programmable manner by introducing intricate patterns of magnetic polarities in soft materials. The resulting structures, when properly designed, have been shown to exhibit a diverse and rich array of actuation behavior. In this work, we develop a suitable theoretical framework to analyze these so-called hard-magnetic soft materials to facilitate the rational design of magnetically activated functional structures and devices based on a quantitative prediction of complex shape changes. We adopt a nonlinear field theory to describe the finite deformation coupled with magnetic fields and argue that the macroscopic behavior of the fabricated materials requires a new constitutive classification — ideal hard-magnetic soft material — which assumes that (i) the material has a residual magnetic flux density, and (ii) the induced magnetic flux density exhibits a linear relation with the applied actuating magnetic field. We implement the theory and constitutive law in a finite-element framework and find remarkable agreement between the simulation and experimental results on various deformation modes of hard-magnetic soft materials. Using the developed (and validated) model, we present a set of illustrative examples to highlight the use of our model-based simulation to guide the design of experimentally realizable complex shape-morphing structures based on hard-magnetic soft materials.

Higgs boson pair production in non-linear Effective Field Theory with full mt-dependence at NLO QCD

We present a calculation of the NLO QCD corrections to Higgs boson pair production within the framework of a non-linearly realised Effective Field Theory in the Higgs sector, described by the electroweak chiral Lagrangian. We analyse how the NLO corrections affect distributions in the Higgs boson pair invariant mass and the transverse momentum of one of the Higgs bosons. We find that these corrections lead to significant and non-homogeneous K-factors in certain regions of the parameter space. We also provide an analytical parametrisation for the total cross-section and the mhh distribution as a function of the anomalous Higgs couplings that includes NLO corrections. Such a parametrisation can be useful for phenomenological studies.

Spectrum of connectivity fluctuations including the effect of activity-dependent feedback

The spatiotemporal spectrum of feedback-driven fluctuations of brain connectivity is investigated using nonlinear neural field theory of the corticothalamic system. Weakly nonlinear dynamics of neural feedbacks are expanded in terms of first order perturbations of neural activity relative to a fixed point. Susceptibilities are used to quantify the change in connectivity per unit change in presynaptic or postsynaptic activity caused by nonlinear feedbacks such as facilitation, depression, sensitization, potentiation, and the effects of discrete eigenmode structure are included for a spherical brain geometry. Spectral signatures such as resonances are identified that allow the presence of particular presynaptic and postsynaptic feedback effects to be inferred. These include additional resonances at high frequencies and shifts of existing spectral peaks, mostly visible in the lowest spatial modes of the response.

Dependence of absence seizure dynamics on physiological parameter evolution

A neural field model of the corticothalamic system is applied to investigate the temporal and spectral characteristics of absence seizures in the presence of a temporally varying connection strength between the cerebral cortex and thalamus. Increasing connection strength drives the system into an absence seizure-like state once a threshold is passed and a supercritical Hopf bifurcation occurs. The dynamics and spectral characteristics of the resulting model seizures are explored as functions of maximum connection strength, time above threshold, and the rate at which the connection strength increases (ramp rate). Our results enable spectral and temporal characteristics of seizures to be related to changes in the underlying physiological evolution of connections via nonlinear dynamics and neural field theory. Spectral analysis reveals that the power of the harmonics and the duration of the oscillations increase as the maximum connection strength and the time above threshold increase. It is also found that the time to reach the stable limit-cycle seizure oscillation from the instability threshold decreases with the square root of the ramp rate.

Supersymmetry and fermionic modes in an oscillon background

The excitations referred to as oscillons are long-lived time-dependent field configurations which emerge dynamically from non-linear field theories. Such long-lived solutions are of interest in applications that include systems of Condensed Matter Physics, the Standard Model of Particle Physics, Lorentz-symmetry violating scenarios and Cosmology. In this work, we show how oscillons may be accommodated in a supersymmetric scenario. We adopt as our framework simple (N=1) supersymmetry in D=1+1 dimensions. We focus on the bosonic sector with oscillon configurations and their (classical) effects on the corresponding fermionic modes, (supersymmetric) partners of the oscillons. The particular model we adopt to pursue our investigation displays cubic superfield which, in the physical scalar sector, corresponds to the usual quartic self-coupling.

Puncture mechanics of soft elastomeric membrane with large deformation by rigid cylindrical indenter

Soft elastomeric membrane structures are widely used and commonly found in engineering and biological applications. Puncture is one of the primary failure modes of soft elastomeric membrane at large deformation when indented by rigid objects. In order to investigate the puncture failure mechanism of soft elastomeric membrane with large deformation, we study the deformation and puncture failure of silicone rubber membrane that results from the continuous axisymmetric indentation by cylindrical steel indenters experimentally and analytically. In the experiment, effects of indenter size and the friction between the indenter and the membrane on the deformation and puncture failure of the membrane are investigated. In the analytical study, a model within the framework of nonlinear field theory is developed to describe the large local deformation around the punctured area, as well as to predict the puncture failure of the membrane. The deformed membrane is divided into three parts and the friction contact between the membrane and indenter is modeled by Coulomb friction law. The first invariant of the right Cauchy-Green deformation tensor I1 is adopted to predict the puncture failure of the membrane. The experimental and analytical results agree well. This work provides a guideline in designing reliable soft devices featured with membrane structures, which are present in a wide variety of applications.

Tackling non-linearities with the effective field theory of dark energy and modified gravity

We present the extension of the effective field theory framework to the mildly non-linear scales. The effective field theory approach has been successfully applied to the late time cosmic acceleration phenomenon and it has been shown to be a powerful method to obtain predictions about cosmological observables on linear scales. However, mildly non-linear scales need to be consistently considered when testing gravity theories because a large part of the data comes from those scales. Thus, non-linear corrections to predictions on observables coming from the linear analysis can help in discriminating among different gravity theories. We proceed firstly by identifying the necessary operators which need to be included in the effective field theory Lagrangian in order to go beyond the linear order in perturbations and then we construct the corresponding non-linear action. Moreover, we present the complete recipe to map any single field dark energy and modified gravity models into the non-linear effective field theory framework by considering a general action in the Arnowitt-Deser-Misner formalism. In order to illustrate this recipe we proceed to map the beyond-Horndeski theory and low-energy Hořava gravity into the effective field theory formalism. As a final step we derived the 4th order action in term of the curvature perturbation. This allowed us to identify the non-linear contributions coming from the linear order perturbations which at the next order act like source terms. Moreover, we confirm that the stability requirements, ensuring the positivity of the kinetic term and the speed of propagation for scalar mode, are automatically satisfied once the viability of the theory is demanded at linear level. The approach we present here will allow to construct, in a model independent way, all the relevant predictions on observables at mildly non-linear scales.

An ignored source in the foundations of continuum physics “Die Allgemeinen Ansätze der Mechanik der Kontinua” by E. Hellinger

AbstractIn 1913, Ernst Hellinger finished his remarkable article DIE ALLGEMEINEN ANSÄTZE DER MECHANIK DER KONTINUA, which appeared in German language in the “Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen”. Following his predecessors J.‐L. Lagrange, G. Piola and the Cosserat brothers, Hellinger presents in this paper a nonlinear field theory of continuum physics entirely based on a variational principle “the principle of virtual displacements”. Even though Hellinger focuses on the fundamentals of continuum mechanics, he presents within the very same variational framework the physics of optics, electrodynamics, thermodynamics and the theory of relativity. Accordingly, Hellinger's paper can be understood as a contribution to continuum physics in general. However, due to the establishment of English as the upcoming scientific language and due to the refusal of a variational formulation of continuum mechanics in the subsequent period, Hellinger's contribution to the foundations of continuum physics has been ignored for decades and has almost fallen into oblivion. (© 2017 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Self-gravitating oscillons and new critical behavior

The dynamical evolution of self-interacting scalars is of paramount importance in cosmological settings, and can teach us about the content of Einstein's equations. In flat space, nonlinear scalar field theories can give rise to localized, non-singular, time-dependent, long-lived solutions called {\it oscillons}. Here, we discuss the effects of gravity on the properties and formation of these structures, described by a scalar field with a double well potential. We show that oscillons continue to exist even when gravity is turned on, and we conjecture that there exists a sequence of critical solutions with infinite lifetime. Our results suggest that a new type of critical behavior appears in this theory, characterized by modulations of the lifetime of the oscillon around the scaling law and the modulations of the amplitude of the critical solutions.

Alignment as an Indicator of Changes to Modal Structure within the Roberts Flow

AbstractAlignment of the magnetic and velocity fields has previously been shown to play a role within nonlinear dynamo theory (Cameron and Galloway 2006), MHD turbulence (Matthaeus et al. 1980) and mean field theory (Yokoi 2013). What has not been previously examined is whether it is beneficial to examine alignment within kinematic dynamo theory. I show how measurements of alignment within kinematic dynamo theory for the Roberts flow can indicate a change in the structure of the magnetic field.

Methods to improve harvested energy and conversion efficiency of viscoelastic dielectric elastomer generators

As a new transduction technology, dielectric elastomer generators (DEGs) are capable of converting mechanical energy from diverse sources into electrical energy. However, their energy harvesting performance is strongly affected by the material viscoelasticity. Based on the finite-deformation viscoelasticity theory and the nonlinear coupled field theory for dielectric elastomers, this work presents a theoretical framework to model the performance of DEGs. Motivated by the recent experiments of DEGs with a triangular harvesting scheme, we propose a method to optimize the harvesting cycle, which could significantly improve the conversion efficiency of viscoelastic DEGs. From our simulation results, choosing a higher voltage power source appears to be an effective way to improve the performance of DEGs. In addition, optimizing the period of the discharging process of DEG can markedly increase its efficiency. Also, we have uncovered that the triangular harvesting scheme for DEGs, which is expected to harvest energy close to the maximum achievable energy, could be actually realized by choosing dielectric elastomers with a higher fraction of time-independent polymer networks. The theoretical framework and simulation results presented in this work are expected to benefit the optimal design of DEGs for different applications.

A nonlinear acoustomechanical field theory of polymeric gels

A nonlinear acoustomechanical field theory is presented for polymeric gels undergoing large deformation coupled with diffusion mass transport of solvent molecules in and out of the gel. The theory is developed by combining the acoustic radiation stress theory with the nonlinear elasticity theory of polymeric gels. Explicit velocity and acoustic fields are determined by solving the elastodynamical equations of wave propagation in Eulerian coordinates, which are then employed to determine the distribution of acoustic radiation stresses inside the gel. The nonlinear elasticity of gels is modeled by adopting the Flory–Rehner free energy functions for network stretching and molecules-polymer mixing. For illustration, the developed theory is applied to a layer of polymeric gel immersed in external solvent subjected to two counterpropagating acoustic waves. The acoustically actuated large deformation of the gel is analyzed under three different constraint conditions. Unique acoustomechanical behaviors of polymeric gels are revealed, such as periodical response and nonlinear chaos. This work is expected to enable novel design of ultrasound-triggered sensors and actuators made of polymeric gels, and can also enlighten the application of ultrasonic waves in biomedical engineering.

Kink-antikink asymmetry and impurity interactions in topological mechanical chains.

We study the dynamical response of a diatomic periodic chain of rotors coupled by springs, whose unit cell breaks spatial inversion symmetry. In the continuum description, we derive a nonlinear field theory which admits topological kinks and antikinks as nonlinear excitations but where a topological boundary term breaks the symmetry between the two and energetically favors the kink configuration. Using a cobweb plot, we develop a fixed-point analysis for the kink motion and demonstrate that kinks propagate without the Peierls-Nabarro potential energy barrier typically associated with lattice models. Using continuum elasticity theory, we trace the absence of the Peierls-Nabarro barrier for the kink motion to the topological boundary term which ensures that only the kink configuration, and not the antikink, costs zero potential energy. Further, we study the eigenmodes around the kink and antikink configurations using a tangent stiffness matrix approach appropriate for prestressed structures to explicitly show how the usual energy degeneracy between the two no longer holds. We show how the kink-antikink asymmetry also manifests in the way these nonlinear excitations interact with impurities introduced in the chain as disorder in the spring stiffness. Finally, we discuss the effect of impurities in the (bond) spring length and build prototypes based on simple linkages that verify our predictions.

Nonequilibrium quantum mechanics: A “hot quantum soup” of paramagnons

Motivated by recent measurements of the lifetime (decay width) of paramagnons in quantum antiferromagnet TlCuCl3, we investigate paramagnon decay in a heat bath and formulate an appropriate quantum theory. Our formulation can be split into two regimes: (i) a non-perturbative, 'hot quantum soup' regime where paramagnon width is comparable to its energy; (ii) usual perturbative regime where paramagnon width is significantly lower than its energy. Close to the Neel tempera- ture the paramagnon width becomes comparable to its energy and falls into the hot quantum soup regime. To describe this regime we develop a new finite frequency, finite temperature technique for a nonlinear quantum field theory; the 'golden rule of quantum kinetics'. The formulation is generic and applicable to any three dimensional quantum antiferromagnet in the vicinity of a quantum critical point. Specifically we apply our results to TlCuCl3 and find agreement with experimental data. Additionally, we show that logarithmic running of the coupling constant in the upper critical dimension changes the commonly accepted picture of the quantum disordered and quantum critical regimes.

Coupling oftt̄andγγwith a strongly interacting Electroweak Symmetry Breaking Sector

We report the coupling of an external $\gamma\gamma$ or $t\bar t$ state to a strongly interacting EWSBS satisfying unitarity. We exploit perturbation theory for those coupling of the external state, whereas the EWSBS is taken as strongly interacting. We use a modified version of the IAM unitarization procedure to model such a strongly interacting regime. The matrix elements $V_LV_L\to V_LV_L$, $V_LV_L\leftrightarrow hh$, $hh\to hh$, $V_LV_L\leftrightarrow\{\gamma\gamma,t\bar t\}$, and $hh\leftrightarrow\{\gamma\gamma,t\bar t\}$ are all computed to NLO in perturbation theory with the Nonlinear Effective Field Theory of the EWSBS, within the Equivalence Theorem. This allows us to describe resonances of the electroweak sector that may be found at the LHC and their effect on other channels such as $\gamma\gamma$ or $t\bar t$ where they may be discovered.

The first-order Euler-Lagrange equations and some of their uses

In many nonlinear field theories, relevant solutions may be found by reducing the order of the original Euler-Lagrange equations, e.g., to first order equations (Bogomolnyi equations, self-duality equations, etc.). Here we generalise, further develop and apply one particular method for the order reduction of nonlinear field equations which, despite its systematic and versatile character, is not widely known.