Quantum Fields on Star Graphs

We construct canonical quantum fields which propagate on a star graph modeling a quantum wire. The construction uses a deformation of the algebra of canonical commutation relations, encoding the interaction in the vertex of the graph. We discuss in this framework the Casimir effect and derive the correction to the Stefan-Boltzmann law induced by the vertex interaction. We also generalize the algebraic setting for covering systems with integrable bulk interactions and solve the quantum non-linear Schroedinger model on a star graph.Comment: LaTex 23+1 pages, 4 figure

Spontaneous symmetry breaking in the non-linear Schrodinger hierarchy with defect

We introduce and solve the one-dimensional quantum non-linear Schrodinger (NLS) equation for an N-component field defined on the real line with a defect sitting at the origin. The quantum solution is constructed using the quantum inverse scattering method based on the concept of Reflection-Transmission (RT) algebras recently introduced. The symmetry of the model is generated by the reflection and transmission defect generators defining a defect subalgebra. We classify all the corresponding reflection and transmission matrices. This provides the possible boundary conditions obeyed by the canonical field and we compute these boundary conditions explicitly. Finally, we exhibit a phenomenon of spontaneous symmetry breaking induced by the defect and identify the unbroken generators as well as the exact remaining symmetry.Comment: discussion on symmetry breaking has been improved and examples adde

The quantum non-linear Schrodinger model with point-like defect

We establish a family of point-like impurities which preserve the quantum integrability of the non-linear Schrodinger model in 1+1 space-time dimensions. We briefly describe the construction of the exact second quantized solution of this model in terms of an appropriate reflection-transmission algebra. The basic physical properties of the solution, including the space-time symmetry of the bulk scattering matrix, are also discussed.Comment: Comments on the integrability and the impurity free limit adde

Interplay between Zamolodchikov-Faddeev and Reflection-Transmission algebras

We show that a suitable coset algebra, constructed in terms of an extension of the Zamolodchikov-Faddeev algebra, is homomorphic to the Reflection-Transmission algebra, as it appears in the study of integrable systems with impurity.Comment: 8 pages; a misprint in eq. (2.14) and (2.15) has been correcte

Quantum field theory on quantum graphs and application to their conductance

We construct a bosonic quantum field on a general quantum graph. Consistency of the construction leads to the calculation of the total scattering matrix of the graph. This matrix is equivalent to the one already proposed using generalized star product approach. We give several examples and show how they generalize some of the scattering matrices computed in the mathematical or condensed matter physics litterature. Then, we apply the construction for the calculation of the conductance of graphs, within a small distance approximation. The consistency of the approximation is proved by direct comparison with the exact calculation for the `tadpole' graph.Comment: 32 pages; misprints in tree graph corrected; proofs of consistency and unitarity adde

Vertex operators for quantum groups and application to integrable systems

Starting with any R-matrix with spectral parameter, obeying the Yang-Baxter equation and a unitarity condition, we construct the corresponding infinite dimensional quantum group U_{R} in term of a deformed oscillators algebra A_R. The realization we present is an infinite series, very similar to a vertex operator. Then, considering the integrable hierarchy naturally associated to A_{R}, we show that U_{R} provides its integrals of motion. The construction can be applied to any infinite dimensional quantum group, e.g. Yangians or elliptic quantum groups. Taking as an example the R-matrix of Y(N), the Yangian based on gl(N), we recover by this construction the nonlinear Schrodinger equation and its Y(N) symmetry.Comment: 19 pages, no figure, Latex2e Error in theorem 3.3 and lemma 3.1 correcte

R-matrix presentation for (super)-Yangians Y(g)

We give a unified RTT presentation of (super)-Yangians Y(g) for so(n), sp(2n) and osp(m|2n).Comment: 9 page

Non-diagonal solutions of the reflection equation for the trigonometric An−1(1)A^{(1)}_{n-1} vertex model

We obtain a class of non-diagonal solutions of the reflection equation for the trigonometric $A^{(1)}_{n-1}$ vertex model. The solutions can be expressed in terms of intertwinner matrix and its inverse, which intertwine two trigonometric R-matrices. In addition to a {\it discrete} (positive integer) parameter $l$, $1\leq l\leq n$, the solution contains $n+2$ {\it continuous} boundary parameters.Comment: Latex file, 14 pages; V2, minor typos corrected and a reference adde

Analytical Bethe ansatz in gl(N) spin chains

We present a global treatment of the analytical Bethe ansatz for gl(N) spin chains admitting on each site an arbitrary representation. The method applies for closed and open spin chains, and also to the case of soliton non-preserving boundaries.Comment: Talk given at Integrable Systems, Prague (Czech Republic), 16--18 June 2005 Integrable Models and Applications, EUCLID meeting, Santiago (Spain), 12--16 Sept. 200

W-superalgebras as truncation of super-Yangians

We show that some finite W-superalgebras based on gl(M|N) are truncation of the super-Yangian Y(gl(M|N)). In the same way, we prove that finite W-superalgebras based on osp(M|2n) are truncation of the twisted super-Yangians Y(gl(M|2n))^{+}. Using this homomorphism, we present these W-superalgebras in an R-matrix formalism, and we classify their finite-dimensional irreducible representations.Comment: Latex, 32 page