Integrable Quantum Field Theories Research Articles

Generalizations of the classical Yang-Baxter equation and ${\mathcal O}$O-operators

Tensor solutions (r-matrices) of the classical Yang-Baxter equation (CYBE) in a Lie algebra, obtained as the classical limit of the R-matrix solution of the quantum Yang-Baxter equation, is an important structure appearing in different areas such as integrable systems, symplectic geometry, quantum groups, and quantum field theory. Further study of CYBE led to its interpretation as certain operators, giving rise to the concept of \documentclass[12pt]{minimal}\begin{document}${{\mathcal O}}$\end{document}O-operators. The \documentclass[12pt]{minimal}\begin{document}${\mathcal O}$\end{document}O-operators were in turn interpreted as tensor solutions of CYBE by enlarging the Lie algebra [Bai, C., “A unified algebraic approach to the classical Yang-Baxter equation,” J. Phys. A: Math. Theor. 40, 11073 (2007)]10.1088/1751-8113/40/36/007. The purpose of this paper is to extend this study to a more general class of operators that were recently introduced [Bai, C., Guo, L., and Ni, X., “Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras,” Commun. Math. Phys. 297, 553 (2010)]10.1007/s00220-010-0998-7 in the study of Lax pairs in integrable systems. Relations between \documentclass[12pt]{minimal}\begin{document}${\mathcal O}$\end{document}O-operators, relative differential operators, and Rota-Baxter operators are also discussed.

Higher particle form factors of branch point twist fields in integrable quantum field theories

In this paper we compute higher particle form factors of branch point twist fields. These fields were first described in the context of massive (1+1)-dimensional integrable quantum field theories and their correlation functions are related to the bi-partite entanglement entropy. We find analytic expressions for some form factors and check those expressions for consistency, mainly by evaluating the conformal dimension of the corresponding twist field in the underlying conformal field theory. We find that solutions to the form factor equations are not unique so various techniques need to be used to identify those corresponding to the branch point twist field we are interested in. The models for which we carry out our study are characterized by staircase patterns of various physical quantities as functions of the energy scale. As the latter is varied, the β-function associated with these theories comes close to vanishing at several points between the deep infrared and deep ultraviolet regimes. In other words, renormalization group flows approach the vicinity of various critical points before ultimately reaching the ultraviolet fixed point. This feature provides an optimal way of checking the consistency of higher particle form factor solutions, as the changes on the conformal dimension of the twist field at various energy scales can only be accounted for by considering higher particle form factor contributions to the expansion of certain correlation functions.

Integrable defects in affine Toda field theory and infinite-dimensional representations of quantum groups

Transmission matrices for two types of integrable defect are calculated explicitly, first by solving directly the nonlinear transmission Yang–Baxter equations, and second by solving a linear intertwining relation between a finite-dimensional representation of the relevant Borel subalgebra of the quantum group underpinning the integrable quantum field theory and a particular infinite-dimensional representation expressed in terms of sets of generalised ‘quantum’ annihilation and creation operators. The principal examples analysed are based on the a 2 ( 2 ) and a n ( 1 ) affine Toda models but examples of similar infinite-dimensional representations for quantum Borel algebras for all other affine Toda theories are also provided.

Mean values of local operators in highly excited Bethe states

We consider expectation values of local operators in (continuum) integrable models in asituation when the mean value is calculated in a single Bethe state with a largenumber of particles. We develop a form factor expansion for the thermodynamiclimit of the mean value, which applies whenever the distribution of Bethe roots isgiven by smooth density functions. We present three applications of our generalresult: (i) in the framework of integrable quantum field theory (IQFT) we presenta derivation of the LeClair–Mussardo formula for finite temperature one-pointfunctions. We also extend the results to boundary operators in boundary fieldtheories. (ii) We establish the LeClair–Mussardo formula for the non-relativistic 1DBose gas in the framework of the algebraic Bethe ansatz (ABA). This way weobtain an alternative derivation of the results of Kormos et al for the (temperaturedependent) local correlations using only the concepts of the ABA. (iii) In IQFT weconsider the long-time limit of one-point functions after a certain type of globalquench. It is shown that our general results imply the integral series found byFioretti and Mussardo. We also discuss the generalized eigenstate thermalizationhypothesis in the context of quantum quenches in integrable models. It is shownthat a single mean value always takes the form of a thermodynamic average in ageneralized Gibbs ensemble, although the relation to the conserved charges is ratherindirect.

Form factor expansion for thermal correlators

We consider finite temperature correlation functions in massive integrable quantum fieldtheory. Using a regularization by putting the system in finite volume, we develop a novelapproach (based on multi-dimensional residues) to the form factor expansionfor thermal correlators. The first few terms are obtained explicitly in theorieswith diagonal scattering. We also discuss the validity of the LeClair–Mussardoproposal.

Exact g-function flow between conformal field theories

Exact equations are proposed to describe g-function flows in integrable boundary quantum field theories which interpolate between different conformal field theories in their ultraviolet and infrared limits, extending previous work where purely massive flows were treated. The approach is illustrated with flows between the tricritical and critical Ising models, but the method is not restricted to these cases and should be of use in unravelling general patterns of integrable boundary flows between pairs of conformal field theories.

Finite-temperature dynamical correlations in massive integrable quantum fieldtheories

We consider the finite-temperature frequency and momentum-dependent two-pointfunctions of local operators in integrable quantum field theories. We focus on the casewhere the zero-temperature correlation function is dominated by a delta-function linearising from the coherent propagation of single-particle modes. Our specific examples arethe two-point function of spin fields in the disordered phase of the quantum Ising and theO(3) nonlinear sigma models. We employ a Lehmann representation in terms of the known exactzero-temperature form factors to carry out a low-temperature expansion of two-pointfunctions. We present two different but equivalent methods of regularizing the divergencespresent in the Lehmann expansion: one directly regulates the integral expressions of thesquares of matrix elements in the infinite volume whereas the other operates throughsubtracting divergences in a large, finite volume. Our central results are that thetemperature broadening of the lineshape exhibits a pronounced asymmetry and a shift ofthe maximum upwards in energy (‘temperature-dependent gap’). The field theory resultspresented here describe the scaling limits of the dynamical structure factor in the quantumIsing and integer spin Heisenberg chains. We discuss the relevance of our results for theanalysis of inelastic neutron scattering experiments on gapped spin chain systems such asCsNiCl3 andYBaNiO5.

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One of the simplest examples of a PT-symmetric quantum system is the scaling Yang-Lee model, a quantum field theory with cubic interaction and purely imaginary coupling. We give a historical review of some facts about this model in d <= 2 dimensions, from its original definition in connection with phase transitions in the Ising model and its relevance to polymer physics, to the role it has played in studies of integrable quantum field theory and of PT-symmetric quantum mechanics. We also discuss some more general results on PT-symmetric quantum mechanics and the ODE/IM correspondence, and mention applications to magnetic systems and cold atom physics.

Painlevé III and a singular linear statistics in Hermitian random matrix ensembles, I

In this paper, we study a certain linear statistics of the unitary Laguerre ensembles, motivated in part by an integrable quantum field theory at finite temperature. It transpires that this is equivalent to the characterization of a sequence of polynomials orthogonal with respect to the weight w ( x ) = w ( x , s ) ≔ x α e − x e − s / x , 0 ≤ x < ∞ , α > 0 , s > 0 , namely, the determination of the associated Hankel determinant and recurrence coefficients. Here w ( x , s ) is the Laguerre weight x α e − x perturbed by a multiplicative factor e − s / x , which induces an infinitely strong zero at the origin. For polynomials orthogonal on the unit circle, a particular example where there are explicit formulas, the weight of which has infinitely strong zeros, was investigated by Pollaczek and Szegö many years ago. Such weights are said to be singular or irregular due to the violation of the Szegö condition. In our problem, the linear statistics is a sum of the reciprocal of positive random variables { x j : j = 1 , … , n } ; ∑ j = 1 n 1 / x j . We show that the moment generating function, or the Laplace transform of the probability density function of this linear statistics, can be expressed as the ratio of Hankel determinants and as an integral involving a particular third Painlevé function.

Bipartite Entanglement Entropy in Massive Two-Dimensional Quantum Field Theory

Recently, Cardy, Castro Alvaredo, and the author obtained the first exponential correction to saturation of the bipartite entanglement entropy at large region lengths in massive two-dimensional integrable quantum field theory. It depends only on the particle content of the model, and not on the way particles scatter. Based on general analyticity arguments for form factors, we propose that this result is universal, and holds for any massive two-dimensional model (also out of integrability). We suggest a link of this result with counting pair creations far in the past.

Dynamical correlations of the spin-12Heisenberg XXZ chain in a staggered field

We consider the easy-plane anisotropic spin-$\frac{1}{2}$ Heisenberg chain in combined uniform longitudinal and transverse staggered magnetic fields. The low-energy limit of his model is described by the sine-Gordon quantum field theory. Using methods of integrable quantum field theory we determine the various components of the dynamical structure factor. To do so, we derive explicit expressions for all matrix elements of the low-energy projections of the spin operators involving at most two particles. We discuss applications of our results to experiments on one-dimensional quantum magnets.

Bi-partite Entanglement Entropy in Massive QFT with a Boundary: the Ising Model

In this paper we give an exact infinite-series expression for the bi-partite entanglement entropy of the quantum Ising model both with a boundary magnetic field and in infinite volume. This generalizes and extends previous results involving the present authors for the bi-partite entanglement entropy of integrable quantum field theories, which exploited the generalization of the form factor program to branch-point twist fields. In the boundary case, we isolate in a universal way the part of the entanglement entropy which is related to the boundary entropy introduced by Affleck and Ludwig, and explain how this relation should hold in more general QFT models. We provide several consistency checks for the validity of our form factor results, notably, the identification of the leading ultraviolet behaviour both of the entanglement entropy and of the two-point function of twist fields in the bulk theory, to a great degree of precision by including up to 500 form factor contributions.

Finite-temperature lineshapes in gapped quantum spin chains

We consider the finite-temperature dynamical susceptibility $\ensuremath{\chi}(\ensuremath{\omega},q,T)$ of gapped quantum spin chains such as the integer spin Heisenberg and transverse field Ising models. At zero temperature $\ensuremath{\chi}$ in these models is dominated by a delta-function line arising from the coherent propagation of single-particle modes. Using methods of integrable quantum field theory, we determine the evolution of the lineshape at low temperatures. We show that the lineshape is in general asymmetric in energy. We discuss the application of our results to inelastic neutron-scattering experiments on gapped spin chain systems such as ${\text{Y}}_{2}{\text{BaNiO}}_{5}$ and ${\text{CsNiCl}}_{3}$.

Four-loop perturbative Konishi from strings and finite size effects for multiparticle states

We derive the perturbative four loop anomalous dimension of the Konishi operator in N = 4 SYM theory from the integrable string sigma model by evaluating the finite size effects using Lüscher formulas adapted to multimagnon states at weak coupling. We obtain these multiparticle generalizations of Lüscher formulas by studying certain exactly solvable relativistic integrable quantum field theories. The final result involves a summation of all bound states in the mirror theory and agrees with gauge theory perturbative computations.

On the space of quantum fields in massive two-dimensional theories

For a large class of integrable quantum field theories we show that the S-matrix determines a space of fields which decomposes into subspaces labeled, besides the charge and spin indices, by an integer k. For scalar fields k is non-negative and is naturally identified as an off-critical extension of the conformal level. To each particle we associate an operator acting in the space of fields whose eigenvectors are primary ( k = 0 ) fields of the massive theory. We discuss how the existing results for models as different as Z n , sine-Gordon or Ising with magnetic field fit into this classification.

Finite temperature expectation values of boundary operators

A conjecture is presented for the thermal one-point function of boundary operators in integrable boundary quantum field theories in terms of form factors. It is expected to have applications in studying boundary critical phenomena and boundary flows, which are relevant in the context of condensed matter and string theory. The conjectured formula is verified by a low-temperature expansion developed using finite size techniques, which can also be used to evaluate higher point functions both in the bulk and on the boundary.

Bi-partite entanglement entropy in integrable models with backscattering

In this paper, we generalize the main result of a recent work by J L Cardy and the present authors concerning the bi-partite entanglement entropy between a connected region and its complement. There the expression of the leading-order correction to saturation in the large distance regime was obtained for integrable quantum field theories possessing diagonal scattering matrices. It was observed to depend only on the mass spectrum of the model and not on the specific structure of the diagonal scattering matrix. Here we extend that result to integrable models with backscattering (i.e. with non-diagonal scattering matrices). We use again the replica method, which connects the entanglement entropy to partition functions on Riemann surfaces with two branch points. Our main conclusion is that the mentioned infrared correction takes exactly the same form for theories with and without backscattering. In order to give further support to this result, we provide a detailed analysis in the sine-Gordon model in the coupling regime in which no bound states (breathers) occur. As a consequence, we obtain the leading correction to the sine-Gordon partition function on a Riemann surface in the large distance regime. Observations are made concerning the limit of large number of sheets.

The form factor program: A review and new results, the nested SU(N) off-shell Bethe ansatz and the 1/N expansion

The purpose of the “bootstrap program” for integrable quantum field theories in 1+1 dimensions is to construct a model explicitly in terms of its Wightman functions. We illustrate this program here mainly in terms of the SU(N) Gross-Neveu model. We construct the nested off-shell Bethe ansatz for an SU(N) factoring S-matrix and consider the problem of how to sum over intermediate states in the short-distance limit of the two-point Wightman function for the sinh-Gordon model.

Form factors of boundary exponential operators in the sinh-Gordon model

Using the recently introduced boundary form factor bootstrap equations, the form factors of boundary exponential operators in the sinh-Gordon model are constructed. We also give a general method to evaluate the ultraviolet properties of boundary correlators by extending the bulk cumulant expansion to the boundary case. As an application, the ultraviolet scaling dimension and the normalization of the operators corresponding to the form factor solutions are checked against previously known results for boundary exponential operators. The construction presented in this paper can be applied to determine form factors of relevant primary boundary operators in general integrable boundary quantum field theories.

Form Factors of Branch-Point Twist Fields in Quantum Integrable Models and Entanglement Entropy

In this paper we compute the leading correction to the bipartite entanglement entropy at large sub-system size, in integrable quantum field theories with diagonal scattering matrices. We find a remarkably universal result, depending only on the particle spectrum of the theory and not on the details of the scattering matrix. We employ the "replica trick" whereby the entropy is obtained as the derivative with respect to n of the trace of the n-th power of the reduced density matrix of the sub-system, evaluated at n=1. The main novelty of our work is the introduction of a particular type of twist fields in quantum field theory that are naturally related to branch points in an n-sheeted Riemann surface. Their two-point function directly gives the scaling limit of the trace of the n-th power of the reduced density matrix. Taking advantage of integrability, we use the expansion of this two-point function in terms of form factors of the twist fields, in order to evaluate it at large distances in the two-particle approximation. Although this is a well-known technique, the new geometry of the problem implies a modification of the form factor equations satisfied by standard local fields of integrable quantum field theory. We derive the new form factor equations and provide solutions, which we specialize both to the Ising and sinh-Gordon models.