Principal Chiral Field Research Articles

Integrability and renormalizability for the fully anisotropic SU(2) principal chiral field and its deformations

For the class of 1 + 1 dimensional field theories referred to as the non-linear sigma models, there is known to be a deep connection between classical integrability and one-loop renormalizability. In this work, the phenomenon is reviewed on the example of the so-called fully anisotropic SU(2) Principal Chiral Field (PCF). Along the way, we discover a new classically integrable four parameter family of sigma models, which is obtained from the fully anisotropic SU(2) PCF by means of the Poisson-Lie deformation. The theory turns out to be one-loop renormalizable and the system of ODEs describing the flow of the four couplings is derived. Also provided are explicit analytical expressions for the full set of functionally independent first integrals (renormalization group invariants).

Resurgence and 1/N Expansion in Integrable Field Theories

In theories with renormalons the perturbative series is factorially divergent even after restricting to a given order in 1/N, making the 1/N expansion a natural testing ground for the theory of resurgence. We study in detail the interplay between resurgent properties and the 1/N expansion in various integrable field theories with renormalons. We focus on the free energy in the presence of a chemical potential coupled to a conserved charge, which can be computed exactly with the thermodynamic Bethe ansatz (TBA). In some examples, like the first 1/N correction to the free energy in the non-linear sigma model, the terms in the 1/N expansion can be fully decoded in terms of a resurgent trans-series in the coupling constant. In the principal chiral field we find a new, explicit solution for the large N free energy which can be written as the median resummation of a trans-series with infinitely many, analytically computable IR renormalon corrections. However, in other examples, like the Gross-Neveu model, each term in the 1/N expansion includes non-perturbative corrections which can not be predicted by a resurgent analysis of the corresponding perturbative series. We also study the properties of the series in 1/N. In the Gross-Neveu model, where this is convergent, we analytically continue the series beyond its radius of convergence and show how the continuation matches with known dualities with sine-Gordon theories.

Renormalons in integrable field theories

In integrable field theories in two dimensions, the Bethe ansatz can be used to compute exactly the ground state energy in the presence of an external field coupled to a conserved charge. We generalize previous results by Volin and we extract analytic results for the perturbative expansion of this observable, up to very high order, in various asymptotically free theories: the non-linear sigma model and its supersymmetric extension, the Gross-Neveu model, and the principal chiral field. We study the large order behavior of these perturbative series and we give strong evidence that, as expected, it is controlled by renormalons. Our analysis is sensitive to the next-to-leading correction to the asymptotics, which involves the first two coefficients of the beta function. We also show that, in the supersymmetric non-linear sigma model, there is no contribution from the first IR renormalon, in agreement with general arguments.

Assembling integrable \u03c3-models as affine Gaudin models

We explain how to obtain new classical integrable field theories by assembling two affine Gaudin models into a single one. We show that the resulting affine Gaudin model depends on a parameter γ in such a way that the limit γ → 0 corresponds to the decoupling limit. Simple conditions ensuring Lorentz invariance are also presented. A first application of this method for σ-models leads to the action announced in [1] and which couples an arbitrary number N of principal chiral model fields on the same Lie group, each with a Wess-Zumino term. The affine Gaudin model descriptions of various integrable σ-models that can be used as elementary building blocks in the assembling construction are then given. This is in particular used in a second application of the method which consists in assembling N − 1 copies of the principal chiral model each with a Wess-Zumino term and one homogeneous Yang-Baxter deformation of the principal chiral model.

Integrable Coupled σ Models.

A systematic procedure for constructing classical integrable field theories with arbitrarily many free parameters is outlined. It is based on the recent interpretation of integrable field theories as realizations of affine Gaudin models. In this language, one can associate integrable field theories with affine Gaudin models having arbitrarily many sites. We present the result of applying this general procedure to couple together an arbitrary number of principal chiral model fields on the same Lie group, each with a Wess-Zumino term.

On the Yang–Baxter Poisson algebra in non-ultralocal integrable systems

A common approach to the quantization of integrable models starts with the formal substitution of the Yang–Baxter Poisson algebra with its quantum version. However it is difficult to discern the presence of such an algebra for the so-called non-ultralocal models. The latter includes the class of non-linear sigma models which are most interesting from the point of view of applications. In this work, we investigate the emergence of the Yang–Baxter Poisson algebra in a non-ultralocal system which is related to integrable deformations of the Principal Chiral Field.

Free field representation of the ZF algebra of the SU(N) × SU(N) PCF model**Dedicated to the memory of Petr Petrovich Kulish

A free field representation of the Zamolodchikov-Faddeev algebra of the SU(N) × SU(N) principal chiral field model is constructed, and used to derive an integral representation for form factors of a multi-parameter family of exponential fields.

Formal Diagonalisation of Lax-Darboux Schemes

We discuss the concept of Lax-Darboux scheme and illustrate it on well known examples associated with the Nonlinear Schrodinger (NLS) equation. We explore the Darboux links of the NLS hierarchy with the hierarchy of Heisenberg model, principal chiral field model as well as with differential-difference integrable systems (including the Toda lattice and differential-difference Heisenberg chain) and integrable partial difference systems. We show that there exists a transformation which formally diagonalises all elements of the Lax-Darboux scheme simultaneously. It provides us with generating functions of local conservation laws for all integrable systems obtained. We discuss the relations between conservation laws for systems belonging to the Lax-Darboux scheme.

Finite size spectrum of SU(N) principal chiral field from discrete Hirota dynamics

Using recently proposed method of discrete Hirota dynamics for integrable (1+1)D quantum field theories on a finite space circle of length L we derive and test numerically a finite system of nonlinear integral equations for the exact spectrum of energies of SU(N)×SU(N) principal chiral field model as functions of mL, where m is the mass scale. We propose a determinant solution of the underlying Y-system, or Hirota equation, in terms of Wronskian determinants of N×N matrices parameterized by N−1 functions of the spectral parameter θ with the known analytic properties at finite L. Although the method works in principle for any state, the explicit equations are written for states in the U(1) sector only. For N>2, we encounter and clarify a few subtleties in these equations related to the presence of bound states, absent in the previously considered N=2 case. As a demonstration of efficiency of our method, we solve these equations numerically for a few low-lying states at N=3 in a wide range of mL.

ODE/IM correspondence for the Fateev model

The Fateev model is somewhat special among two-dimensional quantum field theories. For different values of the parameters,it can be reduced to a variety of integrable systems. An incomplete list of the reductions includes O(3) and O(4) non-linear sigma models and their continuous deformations (2D and 3D sausages, anisotropic principal chiral field), the Bukhvostov-Lipatov model, the N=2 supersymmetric sine-Gordon model, as well as the integrable perturbed SU_2(n)\otimes SU_2(p-2)/SU_2(n+p-2) coset CFT. The model possesses a mysterious symmetry structure of the exceptional quantum superalgebras U_q{\hat{\big(D(2|1;\alpha)}}\big). In this work, we propose the ODE/IM correspondence between the Fateev model and a certain generalization of the classical problem of constant mean curvature embedding of a thrice-punctured sphere in AdS_3.

A solution to the non-Abelian duality problem

Dualities uniquely excel at resolving non-perturbative aspects of complex phase diagrams of interacting, Landau or topologically ordered, systems. However, traditional duality transformations fail for systems like the Heisenberg model and non-Abelian gauge theories. The bond-algebraic theory of quantum and classical dualities provides a solution to this long-standing conundrum, the so-called non-Abelian duality problem, by embedding traditional dualities into a more general transformation scheme that always preserves locality in any number of dimensions. Remarkably, it turns out to be unimportant whether a modelʼs group of symmetries is Abelian or non-Abelian. The capability of the bond-algebraic approach to handle finite and infinite systems with arbitrary boundary conditions has recently led to the discovery of holographic symmetries, relating topological order, edge states, and generalized order parameters. We discuss the interplay between these distinguished boundary symmetries and our solution to the non-Abelian duality problem. To illustrate our technique we present, among others, novel dualities for the SU(2) principal chiral field and both U(1) and SU(2) generalizations of the planar quantum compass model of orbital ordering.

YANGIANS IN INTEGRABLE FIELD THEORIES, SPIN CHAINS AND GAUGE-STRING DUALITIES

In the following paper, which is based on the author's PhD thesis submitted to Imperial College London, we explore the applicability of Yangian symmetry to various integrable models, in particular, in relation with S-matrices. One of the main themes in this work is that, after a careful study of the mathematics of the symmetry algebras one finds that in an integrable model, one can directly reconstruct S-matrices just from the algebra. It has been known for a long time that S-matrices in integrable models are fixed by symmetry. However, Lie algebra symmetry, the Yang–Baxter equation, crossing and unitarity, which constrain the S-matrix in integrable models, are often taken to be separate, independent properties of the S-matrix. Here, we construct scattering matrices purely from the Yangian, showing that the Yangian is the right algebraic object to unify all required symmetries of many integrable models. In particular, we reconstruct the S-matrix of the principal chiral field, and, up to a CDD factor, of other integrable field theories with 𝔰𝔲(n) symmetry. Furthermore, we study the AdS/CFT correspondence, which is also believed to be integrable in the planar limit. We reconstruct the S-matrices at weak and at strong coupling from the Yangian or its classical limit. We give a pedagogical introduction into the subject, presenting a unified perspective of Yangians and their applications in physics. This paper should hence be accessible to mathematicians who would like to explore the application of algebraic objects to physics as well as to physicists interested in a deeper understanding of the mathematical origin of physical quantities.

Leading infrared logarithms for the σ-model with fields on an arbitrary Riemann manifold

We derive non-linear recursion equation for the leading infrared logarithms (LL) in four dimensional sigma-model with fields on an arbitrary Riemann manifold. The derived equation allows one to compute leading infrared logarithms to essentially unlimited loop order in terms of geometric characteristics of the Riemann manifold. We reduce the solution of the SU(oo) principal chiral field in arbitrary number of dimensions in the LL approximation to the solution of very simple recursive equation. This result paves a way to the solution of the model in arbitrary number of dimensions at N-->oo

Quasideterminant multisoliton solutions of a supersymmetric chiral field model in two dimensions

The Darboux transformation of a supersymmetric principal chiral field model in two dimensions is studied. We obtain exact superfield multisoliton solutions of the model by means of iterated Darboux transformation and express them in terms of quasideterminants.

Decompositions of the loop algebra over so(4) and integrable models of the chiral equation type

Decompositions of the loop algebra over so(4) are considered and the exactly integrable nonlinear hyperbolic systems of the principal chiral field equation type are analyzed. A new example of such a system is found and the Lax representation for this example is constructed.

Asymptotic Behavior of Soliton Solutions with a DoubleSpectral Parameter for Principal Chiral Field**The projectsupported by Chinese National Research Project ``Nonlinear Science''

The soliton solutions with a double spectral parameter for theprincipal chiral field are derived by Darboux transformation. Theasymptotic behavior of the solutions as time tends to infinity isobtained and the speeds of the peaks in the asymptotic solutionsare not constants.

New types of spatial structures in multisublattice antiferromagnets

A broad class of three-dimensional space structures in multisublattice antiferromagnets was found in the isotropic approximation (the principal chiral field model on the SU(2) group). According to the Andreev-Marchenko theory, this approximation is applicable to spin glasses and provides qualitative understanding of structures in real multisublattice antiferromagnets. Special substitutions were used to reduce the equations of the model to new equations with simple geometric interpretation. A differential geometry method was applied to obtain various structure types (some of which were determined by arbitrary functions), including localized and nonlocalized textures, structures with the degree of mapping equal to one, antiferromagnetic “targets” and three-dimensional sources, and two-and three-dimensional vortex and spiral structures. Possibilities for experimentally checking the presence of localized, vortex, and spiral structures in antiferromagnets were demonstrated.

Liouville integrability of the finite dimensional Hamiltonian systems derived from principal chiral field

For finite dimensional Hamiltonian systems derived from 1+1 dimensional integrable systems, if they have Lax representations, then the Lax operator creates a set of conserved integrals. When these conserved integrals are in involution, it is believed quite popularly that there will be enough functionally independent ones among them to guarantee the Liouville integrability of the Hamiltonian systems, at least for those derived from physical problems. In this article, we give a counterexample based on the U(2) principal chiral field. It is proved that the finite dimensional Hamiltonian systems derived from the U(2) principal chiral field are Liouville integrable. Moreover, their Lax operator gives a set of involutive conserved integrals, but they are not enough to guarantee the integrability of the Hamiltonian systems.

ON SOLITON SOLUTIONS OF NONLINEAR SIGMA MODELS OF SYMMETRIC SPACES

The inverse scattering transform technique of Belinskii–Zakharov for the integration of nonlinear σ model equations is reviewed. N-soliton solutions of the principal chiral field equations are given. The explicit two-complex pole soliton solutions of vacuum and electro-vacuum Ernst equations are constructed.

Anti-self-dual Yang–Mills equations on noncommutative space–time

By replacing the ordinary product with the so-called ☆-product, one can construct an analog of the anti-self-dual Yang–Mills (ASDYM) equations on the noncommutative R 4 . Many properties of the ordinary ASDYM equations turn out to be inherited by the ☆-product ASDYM equation. In particular, the twistorial interpretation of the ordinary ASDYM equations can be extended to the noncommutative R 4 , from which one can also derive the fundamental structures for integrability such as a zero-curvature representation, an associated linear system, the Riemann–Hilbert problem, etc. These properties are further preserved under dimensional reduction to the principal chiral field model and Hitchin’s Higgs pair equations. However, some structures relying on finite dimensional linear algebra break down in the ☆-product analogs.