Massive Integrable Field Theories Research Articles

Entanglement entropy along a massless renormalisation flow: the tricritical to critical Ising crossover

We study the Rényi entanglement entropies along the massless renormalisation group flow that connects the tricritical and critical Ising field theories. Similarly to the massive integrable field theories, we derive a set of bootstrap equations, from which we can analytically calculate the twist field form factors in a recursive way. Additionally, we also obtain them as a non-trivial ‘roaming limit’ of the sinh-Gordon theory. Then the Rényi entanglement entropies are obtained as expansions in terms of the form factors of these branch point twist fields. We find that the form factor expansion of the entanglement entropy along the flow organises in two different kind of terms. Those that couple particles with the same chirality, and reproduce the entropy of the infrared Ising theory, and those that couple particles with different chirality, which provide the ultraviolet contributions. The massless flow under study possesses a global ℤ2 spin-flip symmetry. We further consider the composite twist fields associated to this group, which enter in the study of the symmetry resolution of the entanglement. We derive analytical expressions for their form factors both from the bootstrap equations and from the roaming limit of the sinh-Gordon theory.

Towards a quadratic Poisson algebra for the subtracted classical monodromy of symmetric space sine-Gordon theories

Symmetric space sine-Gordon theories are two-dimensional massive integrable field theories, generalising the sine-Gordon and complex sine-Gordon theories. To study their integrability properties on the real line, it is necessary to introduce a subtracted monodromy matrix. Moreover, since the theories are not ultralocal, a regularisation is required to compute the Poisson algebra for the subtracted monodromy. In this article, we regularise and compute this Poisson algebra for certain configurations, and show that it can both satisfy the Jacobi identity and imply the existence of an infinite number of conserved quantities in involution.

Entanglement entropy in integrable field theories with line defects. Part I. Topological defect

In this paper and a companion one [1], we study the effect of integrable line defects on entanglement entropy in massive integrable field theories in 1+1 dimensions. The current paper focuses on topological defects that are purely transmissive. Using the form factor bootstrap method, we show that topological defects do not affect the the entanglement entropy in the UV limit and modify slightly the leading exponential correction in the IR. This conclusion holds for both unitary and non-unitary field theories. In contrast, non-topological defects affect the entanglement entropy more significantly both in UV and IR limit and will be studied in the companion paper.

T-duality in massive integrable field theories: the homogeneous and complex sine-Gordon models

The T-duality symmetries of a family of two-dimensional massive integrable field theories defined in terms of asymmetric gauged Wess–Zumino–Novikov–Witten actions modified by a potential are investigated. These theories are examples of massive non-linear sigma models and, in general, T-duality relates two different dual sigma models perturbed by the same potential. When the unperturbed theory is self-dual, the duality transformation relates two perturbations of the same sigma model involving different potentials. Examples of this type are provided by the homogeneous sine-Gordon theories, associated with cosets of the form G / U ( 1 ) r where G is a compact simple Lie group of rank r. They exhibit a duality transformation for each element of the Weyl group of G that relates two different phases of the model. On-shell, T-duality provides a map between the solutions to the equations of motion of the dual models that changes Noether soliton charges into topological ones. This map is carefully studied in the complex sine-Gordon model, where it motivates the construction of Bogomol'nyi-like bounds for the energy that provide a novel characterisation of the already known one-solitons solutions where their classical stability becomes explicit.

Form Factors and Action of on 8-Cycles

Let be a sequence of skew-symmetric polynomials in X 1,⋯,X l satisfying deg X j P n,l ≤n−1, whose coefficients are symmetric Laurent polynomials in z 1 ,⋯,z n . We call p an ∞-cycle if holds for all n,l. These objects arise in integral representations for form factors of massive integrable field theory, i.e., the SU(2)-invariant Thirring model and the sine-Gordon model. The variables α a =−logX a are the integration variables and β j =logz j are the rapidity variables. To each ∞-cycle there corresponds a form factor of the above models. Conjecturally all form-factors are obtained from the ∞-cycles. In this paper, we define an action of on the space of ∞-cycles. There are two sectors of ∞-cycles depending on whether n is even or odd. Using this action, we show that the character of the space of even (resp. odd) ∞-cycles which are polynomials in z 1 ,⋯,z n is equal to the level (−1) irreducible character of with lowest weight −Λ0 (resp. −Λ1). We also suggest a possible tensor product structure of the full space of ∞-cycles.

Massive integrable soliton theories

Massive integrable field theories in 1 + 1 dimensions are defined at the Lagrangian level, whose classical equations of motion are related to the “non-abelian” Toda field equations. They can be thought of as generalizations of the sine-Gordon and complex sine-Gordon theories. The fields of the theories take values in a non-abelian Lie group and it is argued that the coupling constant is quantized, unlike the situation in the sine-Gordon theory, which is a special case since its field takes values in an abelian group. It is further shown that these theories correspond to perturbations of certain coset conformal field theories. The solitons in the theories will, in general, carry non-abelian charges.

A method to determine the operator content of perturbed conformal field theories

A method to determine the full structure of the space of local operators of massive integrable field theories, based on the form factor bootstrap approach, is presented. This method is applied to the integrable perturbations of the Ising conformal point. It is found that the content of local operators can be expressed in terms of fermionic sum representations of the characters χ ̃ ~(q) of the Virasoro irreducible representations of the minimal model M 3,4 . The space of operators factorises into chiral components as Z = Σ χ ̃ (q) χ ̃ ( q) , but with the relation q = q −1 .