The scattering of fermions in the background field of a topological soliton of the modified $(2 + 1)$-dimensional $\mathbb{CP}^{1}$ model is studied here both analytically and numerically. Unlike the original $\mathbb{CP}^{1}$ model, the Lagrangian of the modified model contains a potential term. Due to this, a dilatation zero mode of the topological soliton disappears, which results in stability of the fermion-soliton system. The symmetry properties of the fermion-soliton system are established, and the asymptotic forms of fermionic radial wave functions are studied. Questions related to the bound states of the fermion-soliton system are then discussed. General formulae describing the scattering of fermions are presented. The amplitudes of the fermion-soliton scattering are obtained in an analytical form within the framework of the Born approximation, and their symmetry properties and asymptotic forms are studied. The energy levels of the fermionic bound states and the partial phase shifts of fermionic scattering are obtained by numerical methods, and the ultrarelativistic limits of the partial phase shifts are found.
We consider the fermionic bound states associated with a soliton-antisoliton pair in 1+1 dimensions which have zero energy when the solitons are infinitely far apart. We calculate the energies of these states when the solitons are separated by a finite distance. The energies decay exponentially with the distance between the soliton and antisoliton. When the fermion mass is much larger than the boson mass, the energy simplifies substantially. These energies may be interpreted as a contribution to the effective potential between the soliton and antisoliton. The character of this contribution dep…
Necessary and sufficient conditions are found for any object in $3+1$ dimensions to have integer rather than fractional fermion number. Nontrivial examples include the Jackiw-Rebbi monopole and the already well studied Su-Schrieffer-Heeger soliton, both displaying integer multiples of elementary charges in combinations that normally are forbidden.
We study soliton solutions of a modified non-linear Schroedinger (MNLS) equation. Using an Ansatz for the time and azimuthal angle dependence previously considered in the studies of the spinning Q-balls, we construct multi-node solutions of MNLS as well as spinning generalisations.
In presence of string solitons, index theorems for the generalised Dirac operators have to be revisited. We show that in supersymmetric configurations the fermionic operators decouple, so that there are no mixing effects between different fermions in the index theorems. We extend the index theorems in presence of torsion to the generic case of manifolds with boundary, which naturally appear in string solutions and apply this result to the soliton solution by Callan, Harvey and Strominger.
We argue that having an odd number of Majorana fermion zero modes on a dynamical point-like soliton signifies an inconsistency in a theory with 3+1 and higher dimensions. We check this statement in a couple of examples in field theory and in string/M theory.
We study effects of backreaction of the fermionic modes localized by the baby Skyrmion in the (2+1)-dimensional Skyrme model. It is shown that there is a tower of fermionic modes of two different types, localized by the soliton, however there is only one fermionic level, which flows from positive to negative value as coupling increases. Considering the strong coupling regime we observe that the coupling of the bosonic field to the fermions may strongly deform the Skyrmion, in particular the regions of negative topological charge density appear.
We investigate a coupled system of a Dirac particle and a pseudoscalar field in the form of a soliton in (1+1) dimensions and find some of its exact solutions numerically. We solve the coupled set of equations self-consistently and non-perturbatively by the use of a numerical method and obtain the bound states of the fermion and the shape of the soliton. That is the shape of the static soliton in this problem is not prescribed and is determined by the equations themselves. This work goes beyond the perturbation theory in which the back reaction of the fermion on soliton is its first order corr…
The scattering of Dirac fermions in the background fields of topological solitons of the $(2+1)$-dimensional nonlinear $O(3)$ $σ$-model is studied using both analytical and numerical methods. General formulae describing fermion scattering are obtained and the symmetry properties of the partial scattering amplitudes and elements of the $S$-matrix are determined. Within the framework of the Born approximation, the scattering amplitudes, differential cross-sections, and total cross-sections of fermion-soliton scattering are obtained in analytical forms, and their symmetry properties and asymptoti…
Formation of fermion bag solitons is an important paradigm in the theory of hadron structure. We report here on our non-perturbative analysis of this phenomenon in the 1+1 dimensional massive Gross-Neveu model, in the large $N$ limit. Our main result is that the extremal static bag configurations are reflectionless, as in the massless Gross-Neveu model. Explicit formulas for the profiles and masses of these solitons are presented. We also present a particular type of self-consistent reflectionless solitons which arise in the massive Nambu-Jona-Lasinio models, in the large-N limit.
In this paper we show how to quantize Hopf solitons using the Finkelstein-Rubinstein approach. Hopf solitons can be quantized as fermions if their Hopf charge is odd. Symmetries of classical minimal energy configurations induce loops in configuration space which give rise to constraints on the wave function. These constraints depend on whether the given loop is contractible. Our method is to exploit the relationship between the configuration spaces of the Faddeev-Hopf and Skyrme models provided by the Hopf fibration. We then use recent results in the Skyrme model to determine whether loops are…
In this paper we study Jackiw-Rebbi model, in which a massless fermion is coupled to the kink of $λϕ^4$ theory through a Yukawa interaction. In the original Jackiw-Rebbi model the soliton is prescribed. However, we are interested in the back-reaction of the fermion on the soliton besides the effect of the soliton on the fermion. Also, as a particular example, we consider a minimal supersymmetric kink model, $\mathcal{N}=1$, in ($1+1$) dimensions. In this case, the bosonic self-coupling, $λ$, and the Yukawa coupling between fermion and soliton, $g$, have specific relation, $g=\sqrt{λ/2}$. As th…
The large N limit of fermionic vectors models is studied using bilocal variables, in the framework of a collective field theory approach. The large N configuration is determined completely using only classical solutions of the model. Further, the Bethe-Salpeter equations of the model are cast as a Green's function problem. One of the main results of this work is to show that this Green's function is in fact the large N bilocal itself.
We analyze the model of topological fermions (MTF), where charged fermions are treated as soliton solutions of the field equations. In the region far from the sources we find plane waves solutions with the properties of electro-magnetic waves.
We investigate $4+d$ dimensional fermionic models in which the system in codimension-$d$ supports a topologically stable solution, and in which the fermion may be localised to the brane, with power law in 'instanton' backgrounds and exponentially in 'soliton' backgrounds. When the fermions are isoscalars, the mechanism fails, while for isospinor fermions it is successful. As backgrounds we consider instantons of Yang--Mills and sigma models in even codimensions, solitons of sigma models in odd codimensions, as well as solitons of Higgs and Goldstone models in all codimensions.
We study the stability of Hopfions embedded in a certain modification Ginzburg-Landau model of two equally charged condensates. It has been shown by Ward [Phys. Rev. D66, 041701(R) (2002)] that certain modification of the ordinary model results in system which supports stable topological solitons (Hopfions) for some values of the parameters of the model. We expand the search for stability into previously uninvestigated region of the parameter space, charting an approximate shape for the stable/unstable boundary and find that, within the accuracy of the numerical methods used, the energy of the…
We consider the coupling of a single Dirac fermion to the three component unit vector field which appears as an order parameter in the Faddeev model. Classically, the coupling is determined by requiring that it preserves a certain local frame independence. But quantum mechanically the separate left and right chiral fermion number currents suffer from a frame anomaly. We employ this anomaly to compute the fermion number of a knotted soliton. The result coincides with the self-linking number of the soliton. In particular, the anomaly structure of the fermions relates directly to the inherent chi…
Formation of fermion bag solitons is an important paradigm in the theory of hadron structure. We study this phenomenon non-perturbatively in the 1+1 dimensional Massive Gross-Neveu model, in the large $N$ limit. We find, applying inverse scattering techniques, that the extremal static bag configurations are reflectionless, as in the massless Gross-Neveu model. This adds to existing results of variational calculations, which used reflectionless bag profiles as trial configurations. Only reflectionless trial configurations which support a single pair of charge-conjugate bound states of the assoc…
In the present paper we discuss arguments, favouring the view that massive fermions represent dislocations (i.e. topological solitons) in discrete space-time with Burgers vectors, parallel to an axis of time. If to put symmetrical parts of tensors of distortions (i.e. derivatives of atomic displacements on coordinates) and mechanical stresses equal zero, the equations of the field theory of dislocations get the form of the Maxwell equations. If to consider these tensors as symmetrical, we shall receive the equations of the theory of gravitation, and it turns out that the sum of tensor of disto…
The generalized massive Thirring model (GMT) with three fermion species is bosonized in the context of the functional integral and operator formulations and shown to be equivalent to a generalized sine-Gordon model (GSG) with three interacting soliton species. The generalized Mandelstam soliton operators are constructed and the fermion-boson mapping is established through a set of generalized bosonization rules in a quotient positive definite Hilbert space of states. Each fermion species is mapped to its corresponding soliton in the spirit of particle/soliton duality of Abelian bosonization. I…
The generalized massive Thirring model (GMT) with $N_{f}$[=number of positive roots of $su(n)$] fermion species is bosonized in the context of the functional integral and operator formulations and shown to be equivalent to a generalized sine-Gordon model (GSG) with $N_{f}$ interacting soliton species. The generalized Mandelstam-Halpern soliton operators are constructed and the fermion-boson mapping is established through a set of generalized bosonization rules in a quotient positive definite Hilbert space of states. Each fermion species is mapped to its corresponding soliton in the spirit of p…
We examine solitons in theories with heavy fermions. These ``quantum'' solitons differ dramatically from semi-classical (perturbative) solitons because fermion loop effects are important when the Yukawa coupling is strong. We focus on kinks in a $(1+1)$--dimensional $ϕ^4$ theory coupled to fermions; a large-$N$ expansion is employed to treat the Yukawa coupling $g$ nonperturbatively. A local expression for the fermion vacuum energy is derived using the WKB approximation for the Dirac eigenvalues. We find that fermion loop corrections increase the energy of the kink and (for large $g$) decrease…
I consider a class of one-dimensional models where Majorana fermions interact with bosonic fields. Contrary to a more familiar situation where bosonic degrees of freedom are phonons and as such form a slow subsystem, I consider fast bosons. Such situation exists when the bosonic modes appear as collective excitations of interacting electrons as, for instance, in superconductors or carbon nanotubes. It is shown that an entire new class of excitations emerge, namely bound states of solitons and Majorana fermions. The latter bound states are not topological and their existence and number depend o…
We construct the N=2, D=9 supergravity theory up to the quartic fermionic terms and derive the supersymmetry transformation rules for the fields modulo cubic fermions. We consider a class of p-brane solutions of this theory, the stainless p-branes, which cannot be isotropically oxidized into higher dimensions. The new stainless elementary membrane and elementary particle solutions are found. It is explicitly verified that these solutions preserve half of the supersymmetry.
We study a modified version of the Ginzburg-Landau model suggested by Ward and show that Hopfions exist in it as stable static solutions, for values of the Hopf invariant up to at least 7. We also find that their properties closely follow those of their counterparts in the Faddeev-Skyrme model. Finally, we lend support to Babaev's conjecture that longer core lengths yield more stable solitons and propose a possible mechanism for constructing Hopfions in pure Ginzburg-Landau model.
This paper presents work done in collaboration with E. Farhi, J. Goldstone, and A. Lue which is described in full in Ref. [1]. We consider a variant of the standard electroweak theory in which the Higgs sector has been modified so that there is a classically stable weak scale soliton. We explore fermion number violating processes which involve soliton decay. A soliton can decay by tunnelling under the sphaleron barrier or the decay can be collision induced if the energy is sufficient for the barrier to be traversed. We discuss classical solutions to the Minkowski space equations of motion in w…
The index theorem implies that there are fermionic states localized on a soliton. Presence of these modes may significantly alter the pattern of interaction between the solitons. As a particular example we investigate the chiral magnetic Skyrmions coupled to spin-isospin fermions. It is shown that there are sequences of fermionic modes localized on the Skyrmions. We investigate the pattern of interaction between the soltions with localized modes and proved the existence of stable system of magnetic Skyrmions bounded by the strong attractive dipole interaction mediated by the chargeless fermion…
Solitons of a nonlinear field interacting with fermions often acquire a fermionic number or an electric charge if fermions carry a charge. We show how the same mechanism (chiral anomaly) gives solitons statistical and rotational properties of fermions. These properties are encoded in a geometrical phase, i.e., an imaginary part of a Euclidian action for a nonlinear sigma-model. In the most interesting cases the geometrical phase is non-perturbative and has a form of an integer-valued theta-term.
Bosonization in curved spacetime maps massive Thirring model (self-interacting Dirac fermions) to a generalized sine-Gordon model, both living in $1+1$-dimensional curved spacetime. Applying this duality we have shown that the Thirring model fermion, in non-relativistic limit, gets identified with the soliton of non-linear Scrodinger model with Kerr form of non-linearity. We discuss one particular optical soliton in the latter model and relate it with the Thirring model fermion.
The generalized massive Thirring model (GMT) with $N_{f}(=$number of positive roots of $su(n)$) fermion species is bosonized in the context of the functional integral and operator formulations and shown to be equivalent to a generalized sine-Gordon model (GSG) with $N_{f}$ interacting soliton species. The generalized Mandelstam-Halpern soliton operators are constructed and the fermion-boson mapping is established through a set of generalized bosonization rules in a quotient positive definite Hilbert space of states. Each fermion species is mapped to its corresponding soliton in the spirit of p…
We consider $λϕ^{4}$ kink and sine-Gordon soliton in the presence of a minimal length uncertainty proportional to the Planck length. The modified Hamiltonian contains an extra term proportional to $p^4$ and the generalized Schrödinger equation is expressed as a forth-order differential equation in quasiposition space. We obtain the modified energy spectrum for the discrete states and compare our results with 1-loop resummed and Hartree approximations for the quantum fluctuations. We finally find some lower bounds for the deformations parameter so that the effects of the minimal length have the…
The Skyrme-Faddeev system, a modified O(3) sigma model in three space dimensions, admits topological solitons with nonzero Hopf number. One may learn something about these solitons by considering the system on the 3-sphere of radius R. In particular, the Hopf map is a solution which is unstable for R > \sqrt{2}.
We present and study new mechanism of interaction between the solitons based on the exchange interaction mediated by the localized fermion states. As particular examples, we consider solutions of simple 1+1 dimensional scalar field theories with self-interaction potentials, including sine-Gordon model and the polynomial $ϕ^4$, $ϕ^6$ models, coupled to the Dirac fermions with back-reaction. We discover that there is an additional fermion exchange interaction between the solitons, it leads to the formation of static multi-soliton bound states. Further, we argue that similar mechanisms of formati…
We analyze gravitationally localized states of multiple fermions with high angular momenta, in the formalism introduced by Finster, Smoller, and Yau [Phys Rev. D 59, 104020 (1999)]. We show that the resulting soliton-like wave functions can be naturally interpreted in terms of a form of self-trapping, where the fermions become localized on shells the locations of which correspond to those of `bulges' in the optical geometry created by their own energy density.
The dynamics of a four dimensional generally covariant modified SU(N) Yang-Mills action, which depends on the complex structure of spacetime and not its metric, is studied. A general solution of the complex structure integrability conditions is found in the context of the G{2,2) Grassmannian manifold, which admits a global SL(4,C) symmetry group. A convenient definition of the physical energy and momentum permits the study of the vacuum and soliton sectors. The model has a set of conformally SU(2,2) invariant vacua and a set of Poincare invariant vacua. An algebraic integrability condition of …
We consider a variant of the standard electroweak theory in which the Higgs sector has been modified so that there is a classically stable weak scale soliton. We explore fermion number violating processes which involve soliton decay. A soliton can decay by tunnelling under the sphaleron barrier, or the decay can be collision induced if the energy is sufficient for the barrier to be traversed. We present a classical solution to the Minkowski space equations of motion in which a soliton is kicked over the barrier by an incoming pulse. This pulse corresponds to a quantum coherent state with mean …
We consider different methods of calculating the (fractional) fermion number of solitons based on the heat kernel expansion. We derive a formula for the localized eta function a more systematic version of the derivative expansion for spectral assymmetry and that provides a more systematic version of the derivative expansion for spectral asymmetry and compute the fermion number in a multiflavour extension of the Goldstone-Wilczek model.We also propose an improved expansionof the heat kernelthat allows the tackling ofthe convergence issues and permits an automated computation of the coefficients
We show that quantum effects can stabilize a soliton in a model with no soliton at the classical level. The model has a scalar field chirally coupled to a fermion in 1+1 dimensions. We use a formalism that allows us to calculate the exact one loop fermion contribution to the effective energy for a spatially varying scalar background. This energy includes the contribution from counterterms fixed in the perturbative sector of the theory. The resulting energy is therefore finite and unambiguous. A variational search then yields a fermion number one configuration whose energy is below that of a si…
Topological objects resulting from symmetry breakdown may be either stable or metastable depending on the pattern of symmetry breaking. However, if they acquire zero-energy modes of fermions, and in the process acquire non-integer fermionic charge, the metastable configurations also get stabilized. In the case of Dirac fermions the spectrum of the number operator shifts by 1/2. In the case of majorana fermions it becomes useful to assign negative values of fermion number to a finite number of states occupying the zero-energy level, constituting a \textit{majorana pond}. We determine the pariti…
We find static solitons stabilized by quantum corrections in a (1+1)-dimensional model with a scalar field chirally coupled to fermions. This model does not support classical solitons. We compute the renormalized energy functional including one-loop quantum corrections. We carry out a variational search for a configuration that minimizes the energy functional. We find a nontrivial configuration with fermion number whose energy is lower than the same number of free fermions quantized about the translationally invariant vacuum. In order to compute the quantum corrections for a given background f…
We demonstrate an unambiguous and robust method for computing fermionic corrections to the energies of classical background field configurations. We consider the particular case of a sequence of background field configurations that interpolates continuously between the trivial vacuum and a widely separated soliton/antisoliton pair in 1+1 dimensions. Working in the continuum, we use phase shifts, the Born approximation, and Levinson's theorem to avoid ambiguities of renormalization procedure and boundary conditions. We carry out the calculation analytically at both ends of the interpolation and…
We investigate the validity of collective coordinate approaximations to the scattering of solitons in several classes of models in (1+1) dimensional field theory models. We look at models which are deformations of the sine-Gordon (SG) or the nonlinear Schrödinger (NLS) model as they posses solitons which are topological (SG) or non-topological (NLS). Our deformations preserve their topology (SG), but changes their integrability properties, either completely or partially (models become `quasi-integrable'). As our collective coordinate approximation does not allow for the radiation of energy out…
We consider a fermion chirally coupled to a prescribed pseudoscalar field in the form of the soliton of the sine-Gordon model and calculate and investigate the Casimir energy and all of the relevant quantities for each parity channel, separately. We present and use a simple prescription to construct the simultaneous eigenstates of the Hamiltonian and parity in the continua from the scattering states. We also use a prescription we had introduced earlier to calculate unique expressions for the phase shifts and check their consistency with both the weak and strong forms of the Levinson theorem. I…
Some modified (defocusing) non-linear Schrödinger models (MNLS) possess infinite towers of anomalous conservation laws with asymptotically conserved charges. The so-called anomalies of the quasi-conservation laws vanish upon space-time integration for a special ${\cal C}{\cal P}_s{\cal T}_d$ symmetric field configurations. We verify numerically the degree of modifications of the charges around the dark-soliton interaction regions by computing numerically some representative anomalies related to lowest order quasi-conservation laws of the non-integrable cubic-quintic NLS model as a modified (de…
We show that space-time evolution of one-dimensional fermionic systems is described by nonlinear equations of soliton theory. We identify a space-time dependence of a matrix element of fermionic systems related to the {\it Orthogonality Catastrophe} or {boundary states} with the $τ$-function of the modified KP-hierarchy. The established relation allows to apply the apparatus of soliton theory to the study of non-linear aspects of quantum dynamics. We also describe a {\it bosonization in momentum space} - a representation of a fermion operator by a Bose field in the presence of a boundary state…
We present a simple algorithm to implement the generalized derivative expansion introduced previously by L-H. Chan, and apply it to the calculation of the one-loop mass correction to the classical soliton mass in the 1+1 dimensional Jacobi model. We then show how this derivative expansion approach implies that the total (bosonic plus fermionic) mass correction in an N=1 supersymmetric soliton model is determined solely by the asymptotic values (and derivatives) of the fermionic background potential. For a static soliton the total mass correction is $-m/(2π)$, in agreement with recent analyses …
We present an analysis of excited-state solutions for a gravitationally localized system consisting of a filled shell of high-angular-momentum fermions, using the Einstein-Dirac formalism introduced by Finster, Smoller, and Yau [Phys. Rev. D 59, 104020 (1999)]. We show that, even when the particle number is relatively low ($N_f\ge 6$), the increased nonlinearity in the system causes a significant deviation in behavior from the two-fermion case. Excited-state solutions can no longer be uniquely identified by the value of their central redshift, with this multiplicity producing distortions in th…
Fullerenes and nanotubes consist of a large number of carbon atoms sitting on the sites of a regular lattice. For pratical reasons it is often useful to approximate the equations on this lattice in terms of the continuous equation. At the moment, the best candidate for such an equation is the modified non-linear Schroedinger equation. In this paper, we study the modified non-linear Schroedinger equation, which arises as continuous equation in a system describing an excitation on a hexagonal lattice. This latter system can e.g. describe electron-phonon interaction on fullerene related structure…
An integrable two-dimensional system related to certain fermion-soliton systems is studied. The self-consistent solutions of a static version of the system are obtained by using the tau function approach. The self-consistent solutions appear as an infinite number of topological sectors labeled by $n \in \IZ_{+}$, such that in each sector the scalar field would evolve continuously from a trivial configuration to the one with half integer topological charge. The spinor bound states are found analytically for each topological configuration of the background scalar field. The bound state energy sa…
We construct a fermion system in a multi-kink-antikink soliton background, and present in an explicit form all its trapped configurations (bound state solutions) as well as scattering states. This is achieved by exploiting an exotic N=4 centrally extended nonlinear supersymmetry of completely isospectral pairs of reflectionless Schrodinger systems with potentials to be n-soliton solutions for the Korteweg-de Vries equation. The obtained reflectionless Dirac system with a position-dependent mass is shown to possess its own exotic nonlinear supersymmetry associated with the matrix Lax-Novikov op…
In this paper, we study a holographic dual of a confined fermi liquid state by putting a charged fluid of fermions in the AdS soliton geometry. This can be regarded as a confined analogue of electron stars. Depending on the parameters such as the mass and charge of the bulk fermion field, we found three different phase structures when we change the values of total charge density at zero temperature. In one of the three cases, our confined solution (called soliton star) is always stable and this solution approaches to the electron star away from the tip. In both the second and third case, we fi…
