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We apply the dressing method on the Non Linear Sigma Model (NLSM), which describes the propagation of strings on $\mathbb{R}\times \mathrm{S}^2$, for an arbitrary seed. We obtain a formal solution of the corresponding auxiliary system, which is expressed in terms of the solutions of the NLSM that have the same Pohlmeyer counterpart as the seed. Accordingly, we show that the dressing method can be applied without solving any differential equations. In this context a superposition principle emerges: The dressed solution is expressed as a non-linear superposition of the seed with solutions of the NLSM with the same Pohlmeyer counterpart as the seed., 31 pages, 3 appendices

We suggest a generalization of the nonlinear σ model for diffusive superconducting systems to account for magnetoelectric effects due to spin-orbit scattering. In the leading orders of spin-orbit strength and gradient expansion, it includes two additional terms responsible for the spin-Hall effect and the spin-current swapping. First, assuming a delta-correlated disorder, we derive the terms from the Keldysh path integral representation of the generating functional. Then we argue phenomenologically that they exhaust all invariants allowed in the effective action to the leading order in the spin-orbit coupling (SOC). Finally, the results are confirmed by a direct derivation of the saddle-point (Usadel) equation from the quantum kinetic equations in the presence of randomly distributed impurities with SOC. At this point, we correct a recent derivation of the Usadel equation that includes magnetoelectric effects and does not resort to the Born approximation., P.V. and F.S.B. acknowledge funding from EU’s Horizon 2020 research and innovation program under Grant Agreement No. 800923 (SUPERTED). P.V. acknowledges funding from Academy of Finland Project 317118. I.V.T. acknowledges support by Grupos Consolidados UPV/EHU del Gobierno Vasco (Grant No. IT1249-19). F.S.B. acknowledges funding by the Spanish Ministerio de Ciencia, Innovacion y Universidades (MICINN) (Project No. FIS2017-82804-P).

We extend the recent one loop analysis of the ultraviolet completion of the $CP(N)$ nonlinear $\sigma$ model in six dimensions to two loop order in the MSbar scheme for an arbitrary covariant gauge. In particular we compute the anomalous dimensions of the fields and $\beta$-functions of the four coupling constants. We note that like Quantum Electrodynamics (QED) in four dimensions the matter field anomalous dimension only depends on the gauge parameter at one loop. As a non-trivial check we verify that the critical exponents derived from these renormalization group functions at the Wilson-Fisher fixed point are consistent with the $\epsilon$ expansion of the respective large $N$ exponents of the underlying universal theory. Using the Ward-Takahashi identity we deduce the three loop MSbar renormalization group functions for the six dimensional ultraviolet completeness of scalar QED., Comment: 11 latex pages, minor changes and several references added

We consider four-dimensional gravity coupled to a non-linear sigma model whose scalar manifold is a non-compact geometrically finite surface $\Sigma$ endowed with a Riemannian metric of constant negative curvature. When the space-time is an FLRW universe, such theories produce a very wide generalization of two-field $\alpha$-attractor models, being parameterized by a positive constant $\alpha$, by the choice of a finitely-generated surface group $\Gamma\subset \mathrm{PSL}(2,\mathbb{R})$ (which is isomorphic with the fundamental group of $\Sigma$) and by the choice of a scalar potential defined on $\Sigma$. The traditional two-field $\alpha$-attractor models arise when $\Gamma$ is the trivial group, in which case $\Sigma$ is the Poincar\'e disk. We give a general prescription for the study of such models through uniformization in the so-called "non-elementary" case and discuss some of their qualitative features in the gradient flow approximation, which we relate to Morse theory. We also discuss some aspects of the SRST approximation in these models, showing that it is generally not well-suited for studying dynamics near cusp ends. When $\Sigma$ is non-compact and the scalar potential is "well-behaved" at the ends, we show that, in the {\em naive} local one-field truncation, our generalized models have the same universal behavior as ordinary one-field $\alpha$-attractors if inflation happens near any of the ends of $\Sigma$ where the extended potential has a local maximum, for trajectories which are well approximated by non-canonically parameterized geodesics near the ends, we also discuss spiral trajectories near the ends., Comment: 60 pages

An exact hairy asymptotically locally AdS black hole solution with a flat horizon in the Einstein-nonlinear sigma model system in (3+1) dimensions is constructed. The ansatz for the nonlinear $SU(2)$ field is regular everywhere and depends explicitly on Killing coordinates, but in such a way that its energy-momentum tensor is compatible with a metric with Killing fields. The solution is characterized by a discrete parameter which has neither topological nor Noether charge associated with it and therefore represents a hair. A $U(1)$ gauge field interacting with Einstein gravity can also be included. The thermodynamics is analyzed. Interestingly, the hairy black hole is always thermodynamically favored with respect to the corresponding black hole with vanishing Pionic field., 15 pages, 1 figure. Accepted for publication in Physics Letters B

We consider a class of ans\"atze for the construction of exact solutions of the Einstein-nonlinear $\sigma$-model system with an arbitrary cosmological constant in (3+1) dimensions. Exploiting a geometric interplay between the $SU(2)$ field and Killing vectors of the spacetime reduces the matter field equations to a single scalar equation (identically satisfied in some cases) and simultaneously simplifies Einstein's equations. This is then exemplified over various classes of spacetimes, which allows us to construct stationary black holes with a NUT parameter and uniform black strings, as well as time-dependent solutions such as Robinson-Trautman and Kundt spacetimes, Vaidya-type radiating black holes and certain Bianchi~IX cosmologies. In addition to new solutions, some previously known ones are rederived in a more systematic way., Comment: 27 pages. v2: added 3 footnotes and brief comments, 2 new references

A discrete nonlinear $\ensuremath{\sigma}$-model is obtained by triangulate both the space-time ${M}^{d+1}$ and the target space $K$. If the path integral is given by the sum of all the simplicial homomorphisms $\ensuremath{\phi}:{M}^{d+1}\ensuremath{\rightarrow}K$ (i.e., maps without any topological defects), with an partition function that is independent of space-time triangulation, then the corresponding nonlinear $\ensuremath{\sigma}$-model will be called topological nonlinear $\ensuremath{\sigma}$-model which is exactly soluble. These exactly soluble models suggest that phase transitions induced by fluctuations with no topological defects usually produce a topologically ordered state and are topological phase transitions. In contrast, phase transitions induced by fluctuations with all topological defects give rise to trivial product states and are not topological phase transitions. Under the classification conjecture of Lan-Kong-Wen [Phys. Rev. X 8, 021074 (2018)], it is shown that, if $K$ is a space with only nontrivial first homotopy group $G$, which is finite, then these topological nonlinear $\ensuremath{\sigma}$-models can already realize all $3+1\text{D}$ bosonic topological orders without emergent fermions, which are described by Dijkgraaf-Witten theory with gauge group ${\ensuremath{\pi}}_{1}(K)=G$. Under the similar conjecture, we show that the $3+1\text{D}$ bosonic topological orders with emergent fermions can be realized by topological nonlinear $\ensuremath{\sigma}$-models with ${\ensuremath{\pi}}_{1}(K)=$ finite groups, ${\ensuremath{\pi}}_{2}(K)={Z}_{2}$, and ${\ensuremath{\pi}}_{ng2}(K)=0$. A subset of these topological nonlinear $\ensuremath{\sigma}$-models corresponds to 2-gauge theories, which realize and may classify bosonic topological orders with emergent fermions that have no emergent Majorana zero modes at triple string intersections.

The first analytic topologically non-trivial solutions in the (3+1)-dimensional gauged non-linear sigma model representing multi-solitons at finite volume with manifest ordered structures generating their own electromagnetic field are presented. The complete set of seven coupled non-linear field equations of the gauged non-linear sigma model together with the corresponding Maxwell equations are reduced in a self-consistent way to just one linear Schrodinger-like equation in two dimensions. The corresponding two dimensional periodic potential can be computed explicitly in terms of the solitons profile. The present construction keeps alive the topological charge of the gauged solitons. Both the energy density and the topological charge density are periodic and the positions of their peaks show a crystalline order. These solitons describe configurations in which (most of) the topological charge and total energy are concentrated within three-dimensional tube-shaped regions. The electric and magnetic fields vanish in the center of the tubes and take their maximum values on their surface while the electromagnetic current is contained within these tube-shaped regions. Electromagnetic perturbations of these families of gauged solitons are shortly discussed., Comment: 18 pages, 22 figures, accepted for publication on EUROPEAN PHYSICAL JOURNAL C

We review various connections between condensed matter systems with the Nambu-Jona Lasinio model and nonlinear sigma models. The field theoretical description of interacting systems offers a systematic framework to describe the dynamical generation of condensates. Resent findings of a duality between the Nambu-Jona Lasinio model and the nonlinear sigma model enables us to investigate various properties underlying both theories. In this review we mainly focus on inhomogeneous condensations in static situations. The various methods developed in the Nambu-Jona Lasinio model reveal the inhomogeneous phase structures and also yield new inhomogeneous solutions in the nonlinear sigma model owing to the duality. The recent progress on interacting systems in finite systems is also reviewed., Comment: 24pages, 10 figures, Invited review paper commissioned by Symmetry. Comments warmly welcome

The harmonic map is introduced and several physical applications are presented. The classical nonlinear σ model can be looked at as the embedding of a two-dimensional surface in a threedimensional sphere, which is itself embedded in a four-dimensional space. A system of nonlinear evolution equations are obtained by working out the zero curvature condition for the Gauss equations relevant to this geometric formulation.

In [1], here below referred as I, we have written the set of non-linear integral equations(NLIEs) governing the finite size effects of the vacuum as well as the thermodynamics for the integrable deformation of O(3) non-linear sigma model (NLSM), getting it from a manipulation, inspired by those introduced years ago by Suzuki [3, 4], of the larger set of Thermodynamic Bethe Ansatz (TBA) equationsof the model, known since the original paper by Fateev, Onofri and Zamolodchikov [2]. However, one can realize that (I3.24)5, a crucial relation in our derivation of the sausage model NLIE, is not well-defined because neither Q nor Q¯ are analytic on the real axis. Hence Q˜ and Q˜¯ cannot be interpreted as Fourier transforms along the real line. In this addendum we examine this problem carefully and show that the derivation of the sausage model NLIE remains valid in spite of this potential difficulty.

We study the role of zero-point quantum fluctuations in a range of magnetic states which on the classical level are close to spin-aligned ferromagnets. These include Skyrmion textures characterized by non-zero topological charge, and topologically-trivial spirals arising from the competition of the Heisenberg and Dzyaloshinskii-Moriya interactions. For the former, we extend our previous results on quantum exactness of classical Bogomolny-Prasad-Sommerfield (BPS) ground-state degeneracies to the general case of K\"ahler manifolds, with a specific example of Grassmann manifolds $\mathrm{Gr(M,N)}$. These are relevant to quantum Hall ferromagnets with $\mathrm{N}$ internal states and integer filling factor $\mathrm{M}$. A promising candidate for their experimental implementation is monolayer graphene with $\mathrm{N=4}$ corresponding to spin and valley degrees of freedom at the charge neutrality point with $\mathrm{M=2}$ filled Landau levels. We find that the vanishing of the zero-point fluctuations in taking the continuum limit occurs differently in the case of BPS states compared to the case of more general smooth textures, with the latter exhibiting more pronounced lattice effects. This motivates us to consider the vanishing of zero-point fluctuations in such near-ferromagnets more generally. We present a family of lattice spin models for which the vanishing of zero-point fluctuations is evident, and show that some spirals can be thought of as having nonzero but weak zero-point fluctuations on account of their closeness to this family. Between them, these instances provide concrete illustrations of how the Casimir energy, dependent on the full UV-structure of the theory, evolves as the continuum limit is taken., Comment: 12 pages, 8 figures

In this paper the domain wall solutions of a Ginzburg-Landau non-linear $\mathbb{S}^2$-sigma hybrid model are exactly calculated. There exist two types of basic domain walls and two families of composite domain walls. The domain wall solutions have been identified by using a Bogomolny arrangement in a system of sphero-conical coordinates on the sphere $\mathbb{S}^2$. The stability of all the domain walls is also investigated., 23 pages, 5 figures

In this work, taking advantage of the Generalized Hedgehog Ansatz, we construct new self-gravitating solitons in a cavity with mirror-like boundary conditions for the SU(2) Non-linear Sigma Model and Skyrme model. For spherically symmetric spacetimes, we are able to reduce the system to three independent equations that are numerically integrated. There are two branches of well-behaved solutions. The first branch is defined for arbitrary values of the Skyrme coupling and therefore also leads to a gravitating soliton in the Non-linear Sigma Model, while the second branch exists only for non-vanishing Skyrme coupling. The solutions are quasi-static and in the first branch are characterized by two integration constants that correspond to the frequency of the phase of the Skyrme field and the value of the Skyrme profile at the origin, while in the second branch the latter is the unique parameter characterizing the solutions. These parameters determine the size of the cavity, the redshift at the boundary of the cavity, the energy of the scalar field and the charge associated to a U(1) global symmetry. We also show that within this ansatz, assuming analyticity of the matter fields, there are no spherically symmetric black hole solutions., V3: 15 pages and 12 figures. We thank an anonymous referee for pointing out an error in our numerical evidence for the finiteness of the mass and the charge. In this version we have confined the system in a cavity which allows to construct new solutions including also the Skyrme term

We construct exact, regular and topologically non-trivial\ configurations of the coupled Einstein-nonlinear sigma model in (3+1) dimensions. The ansatz for the nonlinear $SU(2)$ field is regular everywhere and circumvents Derrick's theorem because it depends explicitly on time, but in such a way that its energy-momentum tensor is compatible with a stationary metric. Moreover, the $SU(2)$ configuration cannot be continuously deformed to the trivial Pion vacuum as it possesses a non-trivial winding number. We reduce the full coupled 4D Einstein nonlinear sigma model system to a single second order ordinary differential equation. When the cosmological constant vanishes, such master equation can be further reduced to an Abel equation. Two interesting regular solutions correspond to a stationary traversable wormhole (whose only \textquotedblleft exotic matter" is a negative cosmological constant) and a (3+1)-dimensional cylinder whose (2+1)-dimensional section is a Lorentzian squashed sphere. The Klein-Gordon equation in these two families of spacetimes can be solved in terms of special functions. The angular equation gives rise to the Jacobi polynomials while the radial equation belongs to the Poschl-Teller family. The solvability of the Poschl-Teller problem implies non-trivial quantization conditions on the parameters of the theory., Comment: 14 pages, to appear in Physical Review D

We investigate the consistency conditions for matter fields coupled to the four-dimensional ( N = 1 supersymmetric) C P ( 1 ) nonlinear sigma model (the coset space SU ( 2 ) G / U ( 1 ) H ). We find that consistency requires that the U ( 1 ) H charge of the matter be quantized, in units of half of the U ( 1 ) H charge of the Nambu–Goldstone (NG) boson, if the matter has a nonsingular kinetic term and the dynamics respect the full group SU ( 2 ) G . We can then take the linearly realized group U ( 1 ) H to comprise the weak hypercharge group U ( 1 ) Y of the Standard Model. Thus we have charge quantization without a Grand Unified Theory (GUT), completely avoiding problems like proton decay, doublet–triplet splitting, and magnetic monopoles. We briefly investigate the phenomenological implications of this model-building framework. The NG boson is fractionally charged and completely stable. It can be naturally light, avoiding constraints while being a component of dark matter or having applications in nuclear physics. We also comment on the extension to other NLSMs on coset spaces, which will be explored more fully in a followup paper.

Homogeneous Ricci solitons on four-dimensional Lie groups with a left-invariant Riemannian metric are studied. The absence of nontrivial homogeneous invariant Ricci solitons is proved. The algebraic soliton equations are solved in terms of the structure constants of the metric Lie algebra.

It is well known that sigma models in symmetric spaces accept equivalent descriptions in terms of integrable systems such as the sine-Gordon equation through Pohlmeyer reduction. In this paper, we study the mapping between known solutions of the Euclidean O(3) non-linear sigma model, such as instantons, merons and elliptic solutions that interpolate between the latter and solutions of the Pohlmeyer reduced theory, namely the sinh-Gordon equation. It turns out that instantons do not have a counterpart, merons correspond to the ground state, while the class of elliptic solutions is characterized by a two to one correspondence between solutions in the two descriptions., 27 pages, 3 figures

International audience; The recently proposed renormalization group improved optimized perturbation theory is employed to evaluate the pressure of the two dimensional non linear sigma model at finite temperatures. We explicitly show how this powerful resummation method can turn the lowest (one loop) perturbative contribution to the pressure, which is not RG invariant, into a non perturbative quantity exhibiting scale invariance.

We present lattice results for simulations of the O(3) non-linear sigma model at finite chemical potential. The complex action problem is overcome by a dual variable representation of the model. We discuss two aspects of the theory at finite density: 1) The relation of the finite density data to scattering phases and wave-functions. 2) The phase-structure of the theory as a function of chemical potential and the possibility of a Kosterlitz-Thouless phase transition.

We show that there is a topological (Berry phase) term in the non-linear sigma model description of the SO(5) spin chain. It distinguishes the linear and projective representations of the SO(5) symmetry group, in exact analogy to the well-known theta-term of the SO(3) spin chain. The presence of the topological term is due to the fact that pi(2)(SO(5)/SO(3) x SO(2)) = Z. We discuss the implication of our results on the spectra of the SO(5) spin chain, and connect it with a recent solvable SO(5) spin model which exhibits valence bond solid ground state and edge degeneracy. (c) 2011 Elsevier B.V. All rights reserved.

The quantal symmetry property of the C P1 nonlinear σ model with Maxwell non-Abelian Chern-Simons terms in (2+1) dimension is studied. In the Coulomb gauge, the system is quantized by using the Faddeev–Senjanovic (FS) path-integral formalism. Based on the quantaum Noether theorem, the quantal conserved angular momentum is derived and the fractional spin at the quantum level in this system is presented.

Nonlinear integral equations are proposed for the description of the full finite size spectrum of the 2-dimensional O ( 3 ) nonlinear σ-model in a periodic box. Numerical results for the energy eigenvalues are compared to the rotator spectrum and perturbation theory for small volumes and with the recently proposed generalized Luscher formulas at large volumes.

In this paper we investigate the non-linear sigma model arising in the reduction of D = 5 supergravity to D = 3, and present the application of this sigma model to supersymmetric black ring solutions in five-dimensional minimal supergravity. With the ansatz of stationary solutions with $R \times U(1)\times U(1)$ isometry, we obtain a two-dimensional Lagrangian corresponding to geodesic motion of a string-like object on the coset $G_{2(+2)}/SO(4)$, and study the algebra of conserved charges and supersymmetry constraints of supersymmetric black rings. We also obtain the semi-classical wave function of supersymmetric black rings., 15 pages

The purpose of this review is to provide a comprehensive pedagogical introduction into Keldysh technique for interacting out-of-equilibrium fermionic and bosonic systems. The emphasis is placed on a functional integral representation of underlying microscopic models. A large part of the review is devoted to derivation and applications of the non-linear sigma-model for disordered metals and superconductors. We discuss such topics as transport properties, mesoscopic effects, counting statistics, interaction corrections, kinetic equation, etc. The sections devoted to disordered superconductors include Usadel equation, fluctuation corrections, time-dependent Ginzburg-Landau theory, proximity and Josephson effects, etc. (This review is a substantial extension of arXiv:cond-mat/0412296.), Review: 103 pages, 19 figures

In the framework of Faddeev-Senjanovic (FS) path-integral quantization, CP 1 nonlinear σ model coupled to Non-Abelian Chern-Simons (CS) fields is quantized. Generalized canonical Ward identities (WI) are deduced from the invariance of the canonical effective action under gauge transformations, which are obtained from the generators of gauge transformations, including all first-class constraints, in Dirac’s sense. The generalized canonical WI has brief form and is equivalent to canonical WI under gauge transformations in Dirac’s sense.