Integrable Quantum Field Theories Research Articles

The exact spectrum and mirror duality of the (AdS5 × S 5) η superstring

We discuss the spectrum of a string propagating on η-deformed AdS5 × S5 by treating its worldsheet theory as an integrable quantum field theory. The exact S-matrix of this field theory is given by a qdeformation of the AdS5 × S5 worldsheet S-matrix with a real deformation parameter. By considering mirror (double Wick-rotated) versions of these worldsheet theories, we give the thermodynamic Bethe ansatz description of their exact finite-size spectra. Interestingly, this class of models maps onto itself under the mirror transformation. At the string level, this seems to indicate that the light-cone worldsheet theories of strings on particular pairs of backgrounds are related by a double Wick rotation, a feature we call “mirror duality.” We provide a partial verification of these statements at the level of a sigma model by considering reduced actions and their corresponding (mirror) giant magnon solutions.

Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models 2014

Analytical and Numerical Methods for Solving Partial Differential Equations and Integral Equations Arising in Physical Models 2014

Integrable Gross-Neveu models with fermion-fermion and fermion-antifermion pairing

By altering the integrable Gross-Neveu model, the author finds several new integrable quantum field theories in 1+1 dimensions. When the number of fields is large, Cooper pairing is found, which makes these theories exactly solvable toy models for color superconductivity in quantum chromodynamics.

Winding vacuum energies in a deformed [formula omitted] sigma model

We consider the problem of calculating the Casimir energies in the winding sectors of Fateev's SS-model, which is an integrable two-parameter deformation of the O(4) non-linear sigma model in two dimensions. This problem lies beyond the scope of all traditional methods of integrable quantum field theory including the thermodynamic Bethe ansatz and non-linear integral equations. Here we propose a solution based on a remarkable correspondence between classical and quantum integrable systems and express the winding energies in terms of certain solutions of the classical sinh-Gordon equation.

Quench echo and work statistics in integrable quantum field theories.

We propose a boundary thermodynamic Bethe ansatz calculation technique to obtain the Loschmidt echo and the statistics of the work done when a global quantum quench is performed on an integrable quantum field theory. We derive an analytic expression for the lowest edge of the probability density function and find that it exhibits universal features, in the sense that its scaling form depends only on the statistics of excitations. We perform numerical calculations on the sinh-Gordon model, a deformation of the free boson theory, and we obtain that by turning on the interaction the density function develops fermionic properties. The calculations are facilitated by a previously unnoticed property of the thermodynamic Bethe ansatz construction.

From Fuchsian differential equations to integrable QFT

We establish an intriguing correspondence between a special set of classical solutions of the modified sinh-Gordon equation (i.e., Hitchinʼs ‘self-duality’ equations) on a punctured Riemann sphere and a set of stationary states in the finite-volume Hilbert space of the integrable 2D quantum field theory introduced by VA Fateev. An application of this correspondence to the problem of non-perturbative quantization of classically integrable nonlinear sigma models is briefly discussed. A detailed account of the results announced in this communication is contained in separate publications (Bazhanov and Lukyanov 2014 arXiv:1310.4390 [hep-th] and Bazhanov et al 2014 J. High Energy Phys. JHEP09(2014)147).

Boundary state in an integrable quantum field theory out of equilibrium

We study a quantum quench of the mass and the interaction in the Sinh-Gordon model starting from a large initial mass and zero initial coupling. Our focus is on the determination of the expansion of the initial state in terms of post-quench excitations. We argue that the large energy profile of the involved excitations can be relevant for the late time behaviour of the system and common regularization schemes are unreliable. We therefore proceed in determining the initial state by first principles expanding it in a systematic and controllable fashion on the basis of the asymptotic states. Our results show that, for the special limit of pre-quench parameters we consider, it assumes a squeezed state form that has been shown to evolve so as to exhibit the equilibrium behaviour predicted by the Generalized Gibbs Ensemble.

The structure of non-abelian kinks

We consider a class of integrable quantum field theories in 1+1 dimensions whose classical equations have kink solutions with internal collective coordinates that transform under a non-abelian symmetry group. These generalised sine-Gordon theories have been shown to be related to the world sheet theory of the string in the AdS/CFT correspondence. We provide a careful analysis of the boundary conditions at spatial infinity complicated by the fact that they are defined by actions with a WZ term. We go on to describe the local and non-local charges carried by the kinks and end by showing that their structure is perfectly consistent with the exact factorizable S-matrices that have been proposed to describe these theories.

Infinite-time average of local fields in an integrable quantum field theory after a quantum quench.

The infinite-time average of the expectation values of local fields of any interacting quantum theory after a global quench process are key quantities for matching theoretical and experimental results. For quantum integrable field theories, we show that they can be obtained by an ensemble average that employs a particular limit of the form factors of local fields and quantities extracted by the generalized Bethe ansatz.

One-point functions in finite volume/temperature: a case study

We consider finite volume (or equivalently, finite temperature) expectation values of local operators in integrable quantum field theories using a combination of numerical and analytical approaches. It is shown that the truncated conformal space approach, when supplemented with a recently proposed renormalization group, can be sufficiently extended to the low-energy regime that it can be matched with high precision by the low-temperature expansion proposed by Leclair and Mussardo. Besides verifying the consistency of the two descriptions, their combination leads to an evaluation of expectation values which is valid to a very high precision for all volume/temperature scales. As a side result of the investigation, we also discuss some unexpected singularities in the framework recently proposed by Pozsgay and Tak\'acs for the description of matrix elements of local operators in finite volume, and show that while some of these singularities are resolved by the inclusion of the class of exponential finite size corrections known as \mu-terms, these latter corrections themselves lead to the appearance of new singularities. We point out that a fully consistent description of finite volume matrix elements is expected to be free of singularities, and therefore a more complete and systematic understanding of exponential finite size corrections is necessary.

Towards an Operator-Algebraic Construction of Integrable Global Gauge Theories

The recent construction of integrable quantum field theories on two-dimensional Minkowski space by operator-algebraic methods is extended to models with a richer particle spectrum, including finitely many massive particle species transforming under a global gauge group. Starting from a two-particle S-matrix satisfying the usual requirements (unitarity, Yang–Baxter equation, Poincare and gauge invariance, crossing symmetry, . . .), a pair of relatively wedge-local quantum fields is constructed which determines the field net of the model. Although the verification of the modular nuclearity condition as a criterion for the existence of local fields is not carried out in this paper, arguments are presented that suggest it holds in typical examples such as non-linear O(N) σ-models. It is also shown that for all models complying with this condition, the presented construction solves the inverse scattering problem by recovering the S-matrix from the model via Haag–Ruelle scattering theory, and a proof of asymptotic completeness is given.

An operator expansion for integrable quantum field theories

A large class of quantum field theories on 1 + 1 dimensional Minkowski space, namely, certain integrable models, has recently been constructed rigorously by Lechner. However, the construction is very abstract and the concrete form of local observables in these models remains largely unknown. Aiming for more insight into their structure, we establish a series expansion for observables, similar but not identical to the well-known form factor expansion. This expansion will be the basis for a characterization and explicit construction of local observables, to be discussed elsewhere. Here, we establish the expansion independent of the localization aspect, and analyze its behavior under space-time symmetries. We also clarify relations with deformation methods in quantum field theory, specifically, with the warped convolution in the sense of Buchholz and Summers.

New Exact Solutions for a Generalized Double Sinh-Gordon Equation

We study a generalized double sinh-Gordon equation, which has applications in various fields, such as fluid dynamics, integrable quantum field theory, and kink dynamics. We employ the Exp-function method to obtain new exact solutions for this generalized double sinh-Gordon equation. This method is important as it gives us new solutions of the generalized double sinh-Gordon equation.

Spectral expansion for finite temperature two-point functions and clustering

Recently, the spectral expansion of finite temperature two-point functions in integrable quantum field theories was constructed using a finite volume regularization technique and the application of multidimensional residues. In the present work, the original calculation is revisited. By clarifying some details in the residue evaluations, we find and correct some inaccuracies of the previous result. The final result for contributions involving no more than two particles in the intermediate states is presented. The result is verified by proving a symmetry property which follows from the general structure of the spectral expansion, and also by numerical comparison to the discrete finite volume spectral sum. A further consistency check is performed by showing that the expansion satisfies the cluster property up to the order of the evaluation.

Aspects of defects in integrable quantum field theory

Defects are ubiquitous in nature, for example, in the form of dislocations, shocks, bores, or impurities of various kinds, and their descriptions are an important part of any physical theory. But the following question can be asked. What types of defect are allowed, and what are their properties if maintaining integrability within an integrable field theory in two-dimensional space-time is required? We consider a collection of ideas and questions connected with this problem, including examples of integrable defects and the curiously special roles played by energy-momentum conservation and Bäcklund transformations, solitons scattering on defects, and some interesting effects in the framework of the sine-Gordon model, defects in integrable quantum field theory, and the construction of transmission matrices. In conclusion, we remark on algebraic considerations and future research directions.

SPECTRAL ZETA FUNCTIONS OF A 1D SCHRÖDINGER PROBLEM

We study the spectral zeta functions associated to the radial Schrödinger problem with potential V(x) = x2M + αxM-1 + (λ2 - 1/4)/x2. After directly computing some of the zeta functions, we use the quantum Wronskian equation to give sum rules between them, allowing for instances where the explicit form of the zeta functions can be simplified. An immediate application of this work is to derive functional relations and identities involving hypergeometric series, allowing for known identities to be found as instances of more general results. Our work is then extended to a class of related -symmetric eigenvalue problems. Using the fused quantum Wronskian, we give a simple method for indirectly calculating the associated spectral zeta functions. This method is then applied to calculate the nonlocal integrals of motion Gn which appear in an associated integrable quantum field theory.

Integrability of a family of quantum field theories related to sigma models

A method is introduced for constructing lattice discretizations of large classes of integrable quantum field theories. The method proceeds in two steps: The quantum algebraic structure underlying the integrability of the model is determined from the algebra of the interaction terms in the light-cone representation. The representation theory of the relevant quantum algebra is then used to construct the basic ingredients of the quantum inverse scattering method, the lattice Lax matrices and R-matrices. This method is illustrated with four examples: The sinh-Gordon model, the affine sl ( 3 ) Toda model, a model called the fermionic sl ( 2 | 1 ) Toda theory, and the N = 2 supersymmetric sine-Gordon model. These models are all related to sigma models in various ways. The N = 2 supersymmetric sine-Gordon model, in particular, describes the Pohlmeyer reduction of string theory on AdS 2 × S 2 , and is dual to a supersymmetric non-linear sigma model with a sausage-shaped target space.

Review of AdS/CFT Integrability, Chapter III.5: Lüscher Corrections

In integrable quantum field theories, the large volume spectrum is given by the Bethe Ansatz. The leading corrections, due to virtual particles circulating around the cylinder, are encoded in so-called Lüscher corrections. In order to apply these techniques to the AdS/CFT correspondence, one has to generalize these corrections to the case of generic dispersion relations and to multiparticle states. We review these various generalizations and the applications of Lüscher’s corrections to the study of the worldsheet QFT of the superstring in AdS 5 × S 5 and, consequently, to anomalous dimensions of operators in $${\mathcal{N}=4}$$ SYM theory.

Fermionic structure in the sine-Gordon model: Form factors and null-vectors

The form factor bootstrap in integrable quantum field theory allows one to capture local fields in terms of infinite sequences of Laurent polynomials called ‘towers’. For the sine-Gordon model, towers are systematically described by fermions introduced some time ago by Babelon, Bernard and Smirnov. Recently the authors developed a new method for evaluating one-point functions of descendant fields, using yet another fermions which act on the space of local fields. The goal of this paper is to establish that these two fermions are one and the same object. This opens up a way for answering the longstanding question about how to identify precisely towers and local fields.

On sums of tensor and fusion multiplicities

The total multiplicity in the decomposition into irreducibles of the tensor product λ⊗μ of two irreducible representations of a simple Lie algebra is invariant under conjugation of one of them: . This also applies to the fusion multiplicities of affine algebras in conformal WZW theories. In that context, the statement is equivalent to a property of the modular S-matrix, namely Σ(κ) ≔ ∑λSλκ = 0 if κ is a complex representation. Curiously, this vanishing of Σ(κ) also holds when κ is a quaternionic representation. We provide proofs of all these statements. These proofs rely on a case-by-case analysis, maybe overlooking some hidden symmetry principle. We also give various illustrations of these properties in the contexts of boundary conformal field theories, integrable quantum field theories and topological field theories.