Scattering theory and correlation functions in statistical models with a line of defect

Abstract

The scattering theory of the integrable statistical models can be generalized to the case of systems with extended lines of defect. This is done by adding the reflection and transmission amplitudes for the interactions with the line of inhomegeneity to the scattering amplitudes in the bulk. The factorization condition for the new amplitudes gives rise to a set of Reflection-Transmission equations. The solutions of these equations in the case of diagonal S-matrix in the bulk are only those with S = ±1. The choice S = −1 corresponds to the Ising model. We compute the exact expressions of the transmission and reflection amplitudes relative to the interaction of the Majorana fermion of the Ising model with the defect. These amplitudes present a weak-strong duality in the coupling constant, the self-dual points being the special values where the defect line acts as a reflecting surface. We also discuss the bosonic case S = 1 which presents instability properties and resonance states. Multi-defect systems which may give rise to a band structure are also considered. The exact expressions of correlation functions is obtained in terms of Form Factors of the bulk theory and matrix elements of the defect operator.

Quantum and classical integrable sine-Gordon model with defect

“…, n + , the set S − has an extra eigenvalue ξ + N − compared with S + and a (1) and (2) there exists some extra defect soliton with a very special behavior. Consider the case (1). If this soliton moves to the right and originally is located on the left half-line x < 0 then it will appear as 2-soliton after defect, i.e.…”

“…Many theoretical studies are dedicated to various models with defects starting from classical and semiclassical to quantum as well as statistical models [1], with several of them devoted exclusively to the sine-Gordon (SG) model with defect (DSG) [2] or inhomogeneity [3], which have enhanced physical importance [4,5]. A specific form of DSG model with a defect at a single point x = 0:…”

Junctions of anyonic Luttinger wires

“…In fact, let U be the unitary matrix diagonalizing U which defines the boundary conditions (32). Since U is symmetric, we can choose U orthogonal, U t = U −1 , and real, U * = U.…”

“…For all these reasons, we investigate in this paper the Tomonaga-Luttinger (TL) model on a junction with an arbitrary number n of arms as depicted in Fig. 1 (a junction with two wires n = 2 can be seen as a defect on the line, a problem that has been largely investigated [29][30][31][32][33][34][35][36][37][38][39][40] in the past). To solve this problem, at the junction we impose conditions that are probably not obvious for an electronic problem, but they show the advantage to be exactly solvable.…”

Purely transmitting integrable defects

“…The bulk S-matrix depends on the bulk coupling β via the quantity γ = 8π/β 2 −1, and the conventions used are those adopted in [9]. The equations (14) are well known in many contexts involving the notion of integrability (see [10]), but were discussed first with reference to defects by Delfino, Mussardo and Simonetti [11]; if the possibility of reflection were to be allowed an alternative framework (such as the one developed by Mintchev, Ragoucy and Sorba [12]), might be more appropriate. Here, the defect is expected to be purely transmitting.…”