This article is based on recent works done in collaboration with M. Mintchev, E. Ragoucy and P. Sorba. It aims at presenting the latest developments in the subject of factorization for integrable field theories with a reflecting and transmitting impurity

Caudrelier, V.

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Caudrelier, V. (2005). Factorization in integrable systems with impurity. Paper presented at the 14th International Colloquium on Quantum Groups, 16 June 2005 - 18 June 2005, Prague, Czech Republic.City Research OnlineOriginal citation: Caudrelier, V. (2005). Factorization in integrable systems with impurity. Paper presented at the 14th International Colloquium on Quantum Groups, 16 June 2005 - 18 June 2005, Prague, Czech Republic.Permanent City Research Online URL: http://openaccess.city.ac.uk/172/ Copyright & reuseCity  University  London has developed City  Research Online  so that  its  users  may access the research outputs of City University London's staff. Copyright © and Moral Rights for this paper are retained by the individual author(s) and/ or other copyright holders. Users may download and/ or print one  copy  of  any  article(s)  in  City  Research  Online  to  facilitate  their  private  study  or  for  non-commercial research. 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In: 14th International Colloquium on Quantum Groups, 2005-06-16 - 2005-06-18, Prague, CZECH REPUBLIC.Permanent City Research Online URL: http://openaccess.city.ac.uk/172/ Copyright & reuseCity  University  London has developed City  Research Online  so that  its  users  may access the research outputs of City University London's staff. Copyright © and Moral Rights for this paper are retained by the individual author(s) and/ or other copyright holders. Users may download and/ or print one  copy  of  any  article(s)  in  City  Research  Online  to  facilitate  their  private  study  or  for  non-commercial research. Users may not engage in further distribution of the material or use it for any profit-making activities or any commercial gain. All material in City Research Online is checked for eligibility for copyright before being made available in the live archive. URLs from City Research Online may be freely distributed and linked to from other web pages. Versions of researchThe version in City Research Online may differ from the final published version. Users are advised to check the Permanent City Research Online URL above for the status of the paper.EnquiriesIf  you have any enquiries about any aspect of City Research Online,  please email  the team at publications@city.ac.uk.arXiv:hep-th/0508157v1  22 Aug 2005LAPTHLaboratoire d’Annecy-le-Vieux de Physique The´oriqueFactorization in integrable systems with impurityV. Caudrelier∗Laboratoire de Physique The´orique LAPTH†9, chemin de Bellevue, BP 110,F-74941 Annecy-le-Vieux Cedex, France.AbstractThis article is based on recent works done in collaboration with M. Mintchev,E´. Ragoucy and P. Sorba. It aims at presenting the latest developments inthe subject of factorization for integrable field theories with a reflecting andtransmitting impurity.PACS numbers: 11.10.Kk, 11.55.Dshep-th/0508157LAPTH-conf-1112/05∗caudreli@lapp.in2p3.fr†UMR 5108 du CNRS, associe´e a` l’Universite´ de Savoie.IntroductionOne particular aspect of quantum integrable field theories is the existence, bydefinition, of countably many independent conserved quantities. In 1 + 1 dimen-sions, nontrivial massive quantum field theories can be studied non-perturbativelythanks to the following results [1]: there is no particle production, the momenta areconserved individually and finally, any N -particle process can be decomposed as asequence of two-particle processes. This last property is the so-called factorizationproperty. It follows that the central ingredient for such theories is the two-bodyscattering matrix which has to satisfy the celebrated (quantum) Yang-Baxter equa-tion [2, 3].It is the purpose of this article to present in a comprehensive way how thesewell-known facts can be generalized to account for the presence of a reflecting andtransmitting defect, called impurity. To this end, we will briefly recast what isknown of factorization for integrable field theories on the whole line and on the half-line. When including an impurity, this has been generalized in two different ways,resulting in an apparent difficulty. We will show how to reconcile the two points ofview.1 Factorization on the whole line1.1 Physical data and Yang-Baxter equationLet us consider a quantum integrable field theory for massive particles withn internal degrees of freedom. One of the essential ingredients is the two-bodyscattering matrix whose coefficients Sβ1β2α1α2(k1, k2), α1, . . . , β2 = 1, . . . , n are functionsof the rapidities (or momenta) k1, k2 ∈ C parametrizing the dispersion relationof the two particles. This matrix encodes the interaction between particles. Forconvenience, in the rest of the article, we will use auxiliary spaces and consider thetwo-body scattering matrix S12(k1, k2) as an element of End(Cn ⊗ Cn)(k1, k2)S12(k1, k2) = Sβ1β2α1α2(k1, k2) Eα1β1 ⊗ Eα2β2 , (1.1)where Eαβ , α, β = 1, . . . , n is the canonical basis of Cn and summation over repeatedindices is implied.The physical unitarity of the total scattering matrix is guaranteed by that of thetwo-body scattering matrixS12(k1, k2)S†12(k1, k2) = I⊗ I , (1.2)where the dagger stands for Hermitian conjugation and I is the n× n unit matrix.In turn, this is implied by two conditions that are usually imposed for convenienceUnitarity S12(k1, k2)S21(k2, k1) = I⊗ I , (1.3)Hermitian analyticity S†12(k1, k2) = S21(k2, k1) . (1.4)1In this context, the factorization property of the underlying theory is representedby the Yang-Baxter equation that we require for the two-body scattering matrixS12(k1, k2)S13(k1, k3)S23(k2, k3) = S23(k2, k3)S13(k1, k3)S12(k1, k2) . (1.5)This has to be understood as an identity in End(Cn ⊗ Cn ⊗ Cn)(k1, k2, k3).1.2 Algebraic setupIt has been realized in [4, 5] that the previous two-body scattering matrix couldbe taken as the central piece of an algebraic setup analogous to the Heisenbergalgebra but aiming at describing the asymptotic states of an interacting theorycharacterized by S12(k1, k2). The consistency of this approach is ensured by theabove conditions imposed on S12(k1, k2). This is known as the Zamolodchikov-Faddeev (ZF) algebra [4, 5]. This approach has been rigorously investigated in [6]where the Fock space representation of the ZF algebra was explicitly constructed.The ZF algebra has proved extremely fruitful in the Quantum Inverse Scatteringmethod (see e.g. [5]).In this context, the quantum nonlinear Schro¨dinger equation is the paradigm ofquantum integrable field theories (see e.g. [7] and references therein). Indeed, usingthe ZF algebra for this system, it is possible to reconstruct the canonical quantumfield satisfying the nonlinear evolution equation. One also easily gets the N -particlescattering matrix elements and the time-dependent correlation functions (at zerotemperature). Finally, this algebraic approach enables one to identify the symmetryalgebra of the system [8, 9]. This powerful approach makes it very tempting togeneralize the algebraic setup when investigating systems with boundary or impurity.2 Generalizing factorizationConsider a quantum integrable system with a (fixed) boundary. We are interestedin the factorization property of the theory knowing that, in addition to the two-bodyscattering matrix S12(k1, k2), one has to take into account the reflection matrixR(k).The latter represents the scattering properties of a particle characterized by k withthe boundary. Generalizing the ideas of [4] to a system with boundary, I. Cherednikdiscovered in [10] that the reflection matrix should satisfy the reflection equation.The factorization property means here that the N -particle scattering matrix is builtout of S12(k1, k2) and R(k) only. In analogy with the properties listed in the firstsection for S12(k1, k2) one requires the following properties for the physical dataR(k)Unitarity R(k)R(−k) = I , (2.1)Hermitian analyticity R†(k) = R(−k) , (2.2)Reflection equation S12(k1, k2)R1(k1)S21(k2,−k1)R2(k2) (2.3)= R2(k2)S12(k1,−k2)R1(k1)S21(−k2,−k1) .22.1 Algebraic point of viewAs explained above, it is very promising to design an algebraic approach forsystems with boundary. Two approaches have been developed so far. The first oneis based on a boundary operator [11,12] which is added to the usual ZF generators.The second and more recent approach relies on the so-called boundary algebra [13]which also includes an additional generator accounting for the boundary but deeplymodifies the defining relations of the original ZF algebra. Each approach has its owninterests. For example, the first one has been convenient for the bootstrap programwith boundary (see e.g. [11, 14]) while the second is most useful in the quantuminverse scattering method with boundary (see e.g. [15]). Let us stress however thatthere is no rigorous justification of the first approach (as admitted by the authorsof [11] themselves) as opposed to the second one [13].Nevertheless, there is no difference between the two approaches at the level ofthe equations imposed on the physical data. Both consistently reproduce the setsof relations (1.3-1.5) and (2.1-2.3).Both approaches were naturally generalized to the case of a defect (or impurity).In this case, the possibility for transmission is encoded in an additional matrix:the transmission matrix T (k). At the algebraic level, the differences are even moreimportant than in the boundary case. Indeed, in the first approach, the boundaryoperator is replaced by a single defect operator added to the usual ZF algebra, giv-ing rise to a defect algebra [16] while in the second approach, the boundary algebrais turned into a reflection-transmission algebra where two additional generators ac-count for the presence of the impurity [17]. The status of the two approaches isparallel to the boundary case. The first one remains quite formal and has been usedfor various computations of statistical physics [16]. Let us stress however that astrong restriction discovered in [18] holds in this context, limiting the study of inte-grable systems with reflection and transmission essentially to noninteracting ones.The second approach stands on mathematical foundations involving the explicit con-struction of Fock representations [17]. It proved fundamental in the quantum inversescattering method applied to the nonlinear Schro¨dinger equation with impurity [19],which constitutes the first known example of this kind, and in the investigation ofthe corresponding symmetry algebra [20].Nevertheless, the situation is completely different from the boundary case assoon as one is interested in the factorization property for the physical data since thetwo approaches seem to give different physical equations. This is explained in therest of this article.2.2 Going back to physical dataWe are concerned with the analogs of the Yang-Baxter and reflection equationswhen transmission is allowed. Let us emphasize that these reflection-transmissionquantum Yang-Baxter equations (RTQYBE) are the crucial elements for physics3in the sense that they encode the factorization property of any N -particle processand allow for physical computations. The previous algebraic setups are convenienttheoretical tools for which the only constraints are self-consistency and consistentreproduction of the RTQYBE.Let us focus now on the equations obtained in the first approach. To be accurate,introduce the Lorentz (Galilean) invariant two-body scattering matrix S12(k1 − k2).Introduce also two reflection matrices R+(k), R−(k) and two transmission matricesT+(k), T−(k), as is done in [18]. They account for possible different bahaviours onthe left and on the right of the impurity. Invoking the consistency and the associa-tivity of the corresponding defect algebra, one recovers the fundamental relations(1.3-1.5) and deduces the following typical relationsS12(k1 − k2)R+1 (−k1)S21(k1 + k2)R+2 (−k2)= R+2 (−k2)S12(k1 + k2)R+1 (−k1)S21(k1 − k2) , (2.4)T+1 (k1)S21(k2 − k1)R+2 (−k2) = R+2 (−k2)T+1 (k1)S21(−k2 − k1) , (2.5)S12(k1 − k2)T+1 (k1)T+2 (k2) = T+2 (k2)T+1 (k1)S12(k1 − k2) , (2.6)R+(−k)R+(k) + T−(k)T+(k) = I , R−(k)T+(k) + T+(−k)R+(k) = 0 . (2.7)We refer the reader to the original works for the remaining relations. What isimportant here is to note that the relations involving S12 and T+ are cubic, in thatthey involve three terms on each side.Now, let us move on to the equations obtained from the second approach. Thephysical data is given by SAB(k1, k2), R(k), T (k) (where we labelled the auxiliaryspaces by letters for later convenience). One may wonder about the possibility ofdifferent left and right reflection and transmission but we will see in the examplebelow that this is intrinsically encoded in this approach. The sought relations appearas Fock representations of the RT algebra [17] and readSAB(k1, k2)RA(k1)SBA(k2,−k1)RB(k2)= RB(k2)SAB(k1,−k2)RB(k1)SBA(−k2,−k1) , (2.8)SAB(k1, k2)RA(k1)SBA(k2,−k1)T B(k2)= T B(k2)SAB(k1, k2)RA(k1)SBA(k2,−k1) , (2.9)SAB(k1, k2)T A(k1)SBA(k2, k1)T B(k2)= T B(k2)SAB(k1, k2)T A(k1)SBA(k2, k1) , (2.10)R(k)R(−k) + T (k)T (k) = I , R(k)T (−k) + T (k)R(k) = 0 . (2.11)Here the equations involving transmission are quartic. It seems that the twoapproaches are incompatible as long as one is interested in the physical equationsfor reflection and transmission. But how to decide which one is correct?4One can notice that both approaches agree on the reflection equation ((2.4) and(2.8)) and on the unitarity relations ((2.7) and (2.11)). Forgetting the algebraicsetup, one can think of starting from these equations to get some insight. This hasbeen first presented in [17] and detailed in [21]. It is argued that solving (2.11) forT (k) in terms of R(k) and using the reflection equation, one gets that T (k) mustsatisfy the quartic relations (2.9) and (2.10) and not the cubic ones.2.3 Reconciling points of viewThe above conclusions were at the heart of an apparent controversy. In [21], thesituation has been fully clarified. It appears that there is no contradiction since thefirst approach is merely a particular case of the second one. Indeed, making thefollowing particular choice for the scattering data of the second approach∗SAB(k1, k2) =S12(k1 − k2) 0 0 00 I⊗ I 0 00 0 I⊗ I 00 0 0 S21(k2 − k1) . (2.12)R(k) =(R+(−k) 00 R−(−k)), T (k) =(0 T−(k)T+(k) 0), (2.13)and plugging in (2.8-2.11), one recovers the equations for the data of the first ap-proach.ConclusionsThe reflection-transmission quantum Yang-Baxter equations have the form (2.8-2.11). They are naturally obtained in the Reflection-Transmission (RT) algebrasapproach [17] and reproduce as a particular case the original equations obtainedin [16, 18]. However, they do not suffer from the same limitation and allow forquantum interacting integrable systems with reflection and transmission. This wasfirst illustrated in [19] using the well-known nonlinear Schro¨dinger equation. Itappears that the way out of free models is not due to the more general dependence inthe momenta allowed for in the RT approach. This is discussed and illustrated in [21]where explicit solutions with a Lorentz invariant two-body scattering matrix areestablished. The key point lies in the quartic relations involving transmission. In thisrespect, we remark that the physical data used to solve the nonlinear Schro¨dingerequation with impurity satisfies the quartic but not the cubic relations.This shows that the RT algebra approach offers new interesting possibilities inthe study of integrable systems with impurity.∗Here, if the labels 1 and 2 correspond to the auxiliary space Cn then the labels A and Bcorrespond to the auxiliary space C2 ⊗ Cn.5The author is grateful to the organizers of the XIV th International Colloquium onIntegrable Systems held in Prague in June 2005.References[1] S. Parke: Nucl. Phys. B174 (1980) 166.[2] C.N. Yang: Phys. Rev. Lett. 19 (1967) 1312 ; Phys. Rev. 168 (1968) 1920.[3] R.J. Baxter: Ann. Phys. 70 (1972) 193.[4] A.B. Zamolodchikov, Al.B. Zamolodchikov: Annals of Phys. 120 (1979) 253.[5] L.D. Faddeev: Sov. Sci. Rev. C1 (1980) 107.[6] A. Liguori, M. Mintchev: Comm. Math. Phys. 169 (1995) 635.[7] B. Davies: J. Phys. A14 (1981) 2631.[8] S. Murakami, M. Wadati: J. Phys. 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Factorization in integrable systems with impurity

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