---
abstract: 'Some aspects of integrable field theories possessing purely transmitting defects are described. The main example is the sine-Gordon model and several striking features of a classical field theory containing one or more defects are pointed out. Similar features appearing in the associated quantum field theory are also reviewed briefly.'
title: Purely transmitting integrable defects
---

Classical picture
=================

The context of this topic is two-dimensional (i.e. one space – one time) integrable classical (or quantum) field theory, and the basic question concerns how to ‘sew’ together field theories defined in different segments of space.

The setup
---------

The simplest situation has two scalar fields, $u(x,t)$, $x<x_0$ and $v(x,t)$, $x>x_0$, with a Lagrangian density given formally by $$\label{lagrangian} {\cal L}=\theta(x_0-x){\cal L}_u +
\theta(x-x_0){\cal L}_v +\delta (x-x_0) {\cal B}(u,v)\,.$$ The first two terms are the bulk Lagrangian densities for the fields $u$ and $v$ respectively, while the third term provides the sewing conditions; it could in principle depend on $u$, $v$, $u_t$, $v_t$, $u_x$, $v_x$, … but the interesting question is how to choose ${\cal B}$ so that the resulting system remains integrable [@bczlandau].

With free fields, there are many ways to choose ${\cal B}$. For example, $$\label{deltaimpurity} {\cal B}(u,v)=-\sfrac12\,\sigma
uv+\sfrac{1}{2}(u_x+v_x)(u-v)\,,$$ with standard choices for the bulk Lagrangians, leads to the following set of field equations and sewing conditions, $$\begin{array}{rcll}
(\partial^2+m^2)u&=&0\,,\quad&x<x_0\,,\\
(\partial^2+m^2)v&=&0\,,\quad&x>x_0\,,\\
u&=&v\,,\quad&x=x_0\,,\\
v_x-u_x&=&\sigma u\,,\quad& x=x_0\,,
\end{array}$$ implying the fields are continuous with a discontinuity in the derivative. This is an example of a $\delta$-impurity. Typically, the sewing conditions at $x=x_0$ lead to reflection and transmission and (for $\sigma <0$) a bound state. However, if the fields on either side have nonlinear but integrable interactions (e.g. each is a sine-Gordon field), the $\delta$-impurity destroys the integrability (but still interesting [@Goodman02]).

If both $u$ and $v$ are sine-Gordon fields a suitable choice of Lagrangian would be to take $$\label{D}
\begin{array}{rcl}
{\cal B}(u,v)&=&\frac{1}{2}(vu_t-uv_t)+{\cal D}(u,v)\,,\\[9pt]
{\cal D}(u,v)&=&\disty
-2\biggl(\sigma\cos\frac{u+v}2+\frac{1}{\sigma} \cos \frac{u-v}{2}\biggr)\\
\end{array}$$ leading to the set of equations $$\label{sG}
\begin{array}{rcll}
\partial^2 u&=&-\sin u\,,\quad&x<x_0\,,\\
\partial^2 v&=&-\sin v\,,\quad&x>x_0\\[6pt]
u_x&=&\disty
v_t-\sigma \sin \frac{u+v}{2}-\frac{1}{\sigma}\sin\frac{u-v}{2}\,,\quad&x=x_0\\[9pt]
v_x&=&\disty
u_t+\sigma\sin\frac{u+v}{2}-\frac{1}{\sigma}\sin\frac{u-v}{2}\,,\quad&x=x_0\,.
\end{array}$$ (The bulk coupling and mass parameter have been scaled away for convenience.) This set up is not at all the same as the $\delta$-impurity: it is integrable, there is no bound state for any value of the parameter $\sigma$, and typically $u(x_0,t)-v(x_0,t)\ne 0$, implying a discontinuity in the fields. Clearly, the equations (\[sG\]) describe a ‘defect’ (occasionally called a ‘jump-defect’ to distinguish it from other types). Note also that the sewing conditions are strongly reminiscent of a Bäcklund transformation, and would be a Bäcklund transformation if they were not ‘frozen’ at $x=x_0$ (see, for example, [@Backlund]. That this setup is integrable can be verified by constructing Lax pairs using techniques similar to those described in [@bcdr] for boundary situations.

Since the setup (\[sG\]) is local, it is clear there may be many defects, with parameters $\sigma_i$, at different locations $x_i$ along the $x$-axis.

Energy and momentum
-------------------

Time translation invariance is not violated by the defect and therefore there is a conserved energy, which includes a contribution from the defect itself. On the other hand, space translation is violated by a defect and therefore momentum might not be expected to be conserved even allowing for a contribution from the defect. It is worth investigating this in more detail in terms of the quantity ${\cal D}(u,v)$ appearing in (\[D\]).

The momentum carried by the fields on either side of the defect at $x=x_0$ is given by $$\label{momentum}
P=\int_{-\infty}^{x_0}\D x\,u_xu_t+\int_{x_0}^\infty\D x\,v_xv_t$$ and thus, using the defect conditions coming from (\[D\]), $$\label{}
u_x=v_t-\frac{\partial{\cal D}}{\partial u}\,,\quad
v_x=u_t+\frac{\partial{\cal D}}{\partial v}\,,\quad\mathrm{at}\;\,
x=x_0\,,$$ one finds $$\label{Pdot}
\dot P=\left[-v_t \frac{\partial{\cal D}}{\partial u} -
u_t \frac{\partial{\cal D}}{\partial v}-V(u)+W(v)
+\frac{1}{2}\left(\frac{\partial{\cal D}}{\partial u}\right)^2 -
\frac{1}{2}\left(\frac{\partial{\cal D}}{\partial v}\right)^2 \right]_{x_0}.$$ In this expression the fields on either side of the defect have been allowed to have (possibly different) potentials. Clearly, (\[Pdot\]) is not generally a total time-derivative of a functional of the two fields. However, it will be provided, at $x=x_0$, $$\label{Dequations}
\frac{\partial^2{\cal D}}{\partial u^2}=\frac{\partial^2{\cal
D}}{\partial v^2}\,, \qquad \frac{1}{2}\left(\frac{\partial{\cal
D}}{\partial u}\right)^2 - \frac{1}{2}\left(\frac{\partial{\cal
D}}{\partial v}\right)^2 = V(u)-W(v)\,.$$ This set of conditions is satisfied by the sine-Gordon defect function (\[sG\]). However, there are other solutions too, for example Liouville–Liouville, Liouville–massless free, free–free. In fact, in many cases investigated so far, including cases with several scalar fields [@bczlandau; @bcztoda], it turns out that the requirements of integrability coincide with the requirement that there be a modified conserved momentum.

Classical scattering and solitons
---------------------------------

It is not difficult to check that the free-field limit of the sine-Gordon setup, given by $${\cal D}(u,v)\rightarrow \frac{\sigma}{4}(u+v)^2+\frac{1}{4\sigma}(u-v)^2,$$ leads to conditions describing a purely transmitting jump-defect (i.e. no reflection). Given that fact, it is natural to ask what might happen with solitons in the nonlinear sine-Gordon model (for details concerning solitons, see for example [@Scott73].

A soliton travelling in the positive $x$ direction (rapidity $\theta$) is given by expressions $$\E^{\I u/2}=\frac{1+\I E}{1-\I E}\,,\quad  x<x_0\,;\qquad \E^{\I
v/2}=\frac{1+\I zE}{1-\I zE}\,,\quad  x>x_0\,,$$ where $$E=\E^{ax+bt+c}\,,\quad a=\cosh\theta\,,\quad
b=-\sinh\theta\,,\quad \hbox{with $\E^c$ real}.$$ The defect conditions (\[sG\]) are satisfied provided ($\sigma
=\E^{-\eta}$) $$z=\frac{\E^{-\theta}+\sigma}{\E^{-\theta}-\sigma}\equiv \coth
\left(\frac{\eta-\theta}{2}\right),$$ and it is worth noting that $z^2$ would represent the delay experienced by a soliton of rapidity $\theta$ passing another of rapidity $\eta$. As it is, the quantity $z$ may change sign, meaning, in fact, that a soliton can convert to an anti-soliton, or vice-versa, besides being delayed, or even absorbed. In the latter case, the defect gains a unit of topological charge in addition to storing the energy and momentum of the soliton; in the former, the defect gains (or loses) two units of topological charge. Because the defect potential has period $4\pi$, all evenly charged defects have identical energy–momentum, as do all oddly charged defects. A fascinating possibility associated with this type of defect (if it can be realized in practice) would be the capacity to control solitons (see, for example [@cz]). Several defects affect progressing solitons independently; several solitons approaching a defect (inevitably possessing different rapidities) are affected independently, with at most one of the components being absorbed. Notice, too, that the situation is not time-reversal invariant owing to the presence of explicit time derivatives in eqs(\[sG\]). Starting with an odd charged defect, energy–momentum conservation would permit a single soliton to emerge. However, classically, there is nothing to determine the time at which the decay of the defect would occur. In that situation, quantum mechanics should supply a probability for the decay — and indeed it does.

Quantum picture
===============

The transmission matrix
-----------------------

Following the remarks made in the last section one expects two types of transmission matrix, one of them, $^{\rm even}T$, referring to even-labelled defects — and this is expected to be unitary, since these defects cannot decay — and the other, $^{\rm odd}T$, referring to odd-labelled defects. The latter is not expected to be unitary, yet would be expected to be related (via a bootstrap principle) to a complex bound state pole in the former. In fact this is precisely what happens and, remarkably enough, the relevant transmission matrices were described by Konik and LeClair some time ago [@Konik97]. Using roman labels to denote soliton states (taking the value $\pm 1$), and greek labels to label the charge on a defect, and assuming topological charge is conserved in every process, it is expected that both transmission matrices will satisfy ‘triangle’ compatibility relations with the bulk $S$-matrix, for example: $$\label{STT} S_{ab}^{cd}(\theta_1-\theta_2)\,
T_{d\alpha}^{f\beta}(\theta_1)\,T_{c\beta}^{e\gamma}(\theta_2)=
T_{b\alpha}^{d\beta}(\theta_2)\,T_{a\beta}^{c\gamma}(\theta_1)\,
S_{cd}^{ef}(\theta_1-\theta_2)\,.$$ Here, it is supposed the solitons are travelling along the positive $x$-axis ($\theta_1>\theta_2>0$). The bulk $S$-matrix depends on the bulk coupling $\beta$ via the quantity $\gamma=8\pi/\beta^2 -1$, and the conventions used are those adopted in [@bczsg]. The equations (\[STT\]) are well known in many contexts involving the notion of integrability (see [@Jimbo]), but were discussed first with reference to defects by Delfino, Mussardo and Simonetti [@Delf94a]; if the possibility of reflection were to be allowed an alternative framework (such as the one developed by Mintchev, Ragoucy and Sorba [@Mintchev02]), might be more appropriate. Here, the defect is expected to be purely transmitting.

The solution (for general $\beta$, and for even or odd labelled defects — note the labelling is never mixed by (\[STT\])), is given by $$\label{KL}
{\slacs{1.2ex}
T_{a\alpha}^{b\beta}(\theta)=f(q,x)\left(\begin{array}{cc}
\nu^{-1/2}Q^\alpha\delta_\alpha^\beta &
q^{-1/2}\E^{\gamma(\theta-\eta)}\delta_\alpha^{\beta-2}\\[5pt]
q^{-1/2}\E^{\gamma(\theta-\eta)}\delta_\alpha^{\beta+2}&
\nu^{1/2}Q^{-\alpha}\delta_\alpha^\beta\\
\end{array}\right).}$$ In (\[KL\]) a block form has been adopted with the labels $a$, $b$ labelling the four block elements on the right hand side, and where $\nu$ is a free parameter, as is $\eta$ (to be identified with the defect parameter introduced in the previous section), and $$q=\E^{\I\pi\gamma}\,,\quad x=\E^{\gamma\theta}\,,\quad
Q^2=-q=\E^{4\pi^2\I/\beta^2}\,.$$ In addition, $^{\rm even}T$ is a unitary matrix (for real $\theta$), and both types of transmission matrix must be compatible with soliton–anti-soliton annihilation as a virtual process. These two requirements place the following restrictions on the overall factor for the even transmission matrix, $^{\E}f(q,x)$: $$\left\{\begin{array}{l}
{}^{\E}\bar f(q,x)={^{\E}f}(q,qx)\,,\\[5pt]
{}^{\E}f(q,x)\;{^{\E}\!f}(q,qx)\left(1+\E^{2\gamma(\theta-\eta)}\right)=1\,.
\end{array}\right.$$ These do not determine $^{\E}f(q,x)$ uniquely but the ‘minimal’ solution determined by Konik–LeClair has $$\label{KLf} {}^{\E}f(q,x)=
\frac{\E^{\I\pi(1+\gamma)/4}}{1+\I\E^{\gamma(\theta-\eta)}}\,
\frac{r(x)}{\bar r(x)}\,,$$ with ($z=\I\gamma(\theta-\eta)/2\pi$), $$r(x)=\prod_{k=0}^\infty\,
\frac{\Gamma(k\gamma+\sfrac14-z)\,\Gamma((k+1)\gamma+\sfrac34-z)}
{\Gamma((k+\sfrac12)\gamma+\sfrac14-z)\,\Gamma((k+\sfrac12)\gamma+\sfrac34-z)}\,.$$ It is worth noting that the apparent pole in (\[KLf\]) at $1+\I\E^{\gamma(\theta-\eta)}=0$ is actually cancelled by a pole at the same location in $\bar r(x)$. However, there is another pole at $$\theta=\eta -\frac{\I\pi}{2\gamma} \rightarrow \eta\;\;{\rm
as}\;\;\beta\rightarrow 0\,,$$ uncancelled by a zero, and this does actually represent the expected unstable bound state alluded to in the first section.

Several brief remarks are in order. It is clear, on examining (\[KL\]), that the processes in which a classical soliton would inevitably convert to an anti-soliton are clearly dominant even in the quantum theory, yet suppressed if a classical soliton is merely delayed. This much is guaranteed by the factor $\E^{\gamma(\theta-\eta)}$ appearing in the off-diagonal terms. A curious feature is the different way solitons and anti-solitons are treated by the diagonal terms in (\[KL\]). They are treated identically by the bulk $S$-matrix yet one should not be surprised by this since the classical defect conditions (\[sG\]) do not respect all the usual discrete symmetries. Indeed, the dependence of the diagonal entries on the bulk coupling can be demonstrated to follow from the classical picture by using a functional integral type of argument, as explained more fully in [@bczsg]. The sine-Gordon spectrum contains bound states (breathers), and it is interesting to calculate their transmission factors. This much has been done [@bczsg]. However, it would also be interesting to attempt to match these breather transmission factors to perturbative calculations, and this has not yet been done. There are also open questions concerning how to treat defects in motion. From a classical perspective it seems quite natural that defects might move and scatter [@bczsg], however it is less clear how to describe this in the quantum field theory, or indeed to understand what these objects really are.

It is quite remarkable that the simple-looking question asked at the beginning has led to an interesting avenue of enquiry that does not appear to have been explored previously, that links with results, such as (\[KL\]), which had been obtained for seemingly quite different reasons, and that is not yet exhausted (for example, see [@Gomes], for an extension to supersymmetric sine-Gordon).

[**Acknowledgements.** I am very grateful to the organisers for giving me the opportunity to review this material, to Peter Bowcock and to Cristina Zambon for many discussions concerning this topic and for a longstanding collaboration, and to several other members of EUCLID, a Research Training Network funded by the European Commission (contract number HPRN-CT-2002-00325).]{}

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