Quantum group symmetry of integrable systems with or without boundary

We present a construction of integrable hierarchies without or with boundary, starting from a single R-matrix, or equivalently from a ZF algebra. We give explicit expressions for the Hamiltonians and the integrals of motion of the hierarchy in term of the ZF algebra. In the case without boundary, the integrals of motion form a quantum group, while in the case with boundary they form a Hopf coideal subalgebra of the quantum group.Comment: 14 page

Remarks on the Letter of the Patriarch Theophylact to Tsar Peter in the Context of Certain Byzantine and Slavic Anti-heretic Texts

Translated by Marek MajerThe Letter of patriarch Theophylact to tsar Peter is the oldest, but seemingly not the most informative Greek source for the history of Bogomilism. It is in essence a standard document, a typical product of the patriarch’s chancery; it is not conceived as an in-depth investigation into the theological minutiae pertaining to the cosmogony, dogmas and social doctrines of the heretics and the orthodox Church, but rather as a practical tutorial on how to thwart any given neo-Manichaean dualist heresy. It brings to light the fact that Bogomilism, the ‘new’ heresy was treated as an ‘old’ one – as a ‘reactivation’ of earlier gnostic-dualist and neo-Manichaean movements. The letter also features a peculiar innovative feature, though not one directly related to the Bogomil heresy itself: the degree of commitment to preaching the dogmas of the heresy is used for differentiating the situation of the followers. The analysis of the Letter of patriarch Theophylact to tsar Peter raises the more general issue concerning the detailed study of Byzantine and Slavic liturgical texts as a source of information on neo-Manichaean doctrines

Quantum Fields on Star Graphs with Bound States at the Vertex

We investigate the propagation of a massless scalar field on a star graph, modeling the junction of $n$ quantum wires. The vertex of the graph is represented by a point-like impurity (defect), characterized by a one-body scattering matrix. The general case of off-critical scattering matrix with bound and/or antibound states is considered. We demonstrate that the contribution of these states to the scalar field is fixed by causality (local commutativity), which is the key point of our investigation. Two different regimes of the theory emerge at this stage. If bound sates are absent, the energy is conserved and the theory admits unitary time evolution. The behavior changes if bound states are present, because each such state generates a kind of damped harmonic oscillator in the spectrum of the field. These oscillators lead to the breakdown of time translation invariance. We study in both regimes the electromagnetic conductance of the Luttinger liquid on the quantum wire junction. We derive an explicit expression for the conductance in terms of the scattering matrix and show that antibound and bound states have a different impact, giving raise to oscillations with exponentially damped and growing amplitudes respectively.Comment: LaTex 1+29 pages, 2 figures: Expanded version with new title and abstract; clarifying comments, fig.2 and references added; final version to appear in J. Math. Phy

Bosonization and Vertex Algebras with Defects

The method of bosonization is extended to the case when a dissipationless point-like defect is present in space-time. Introducing the chiral components of a massless scalar field, interacting with the defect in two dimensions, we construct the associated vertex operators. The main features of the corresponding vertex algebra are established. As an application of this framework we solve the massless Thirring model with defect. We also construct the vertex representation of the sl(2) Kac-Moody algebra, describing the complex interplay between the left and right sectors due to the interaction with the defect. The Sugawara form of the energy-momentum tensor is also explored.Comment: 23 pages, 1 figur

Luttinger Liquid in Non-equilibrium Steady State

We propose and investigate an exactly solvable model of non-equilibrium Luttinger liquid on a star graph, modeling a multi-terminal quantum wire junction. The boundary condition at the junction is fixed by an orthogonal matrix S, which describes the splitting of the electric current among the leads. The system is driven away from equilibrium by connecting the leads to heat baths at different temperatures and chemical potentials. The associated non-equilibrium steady state depends on S and is explicitly constructed. In this context we develop a non-equilibrium bosonization procedure and compute some basic correlation functions. Luttinger liquids with general anyon statistics are considered. The relative momentum distribution away from equilibrium turns out to be the convolution of equilibrium anyon distributions at different temperatures. Both the charge and heat transport are studied. The exact current-current correlation function is derived and the zero-frequency noise power is determined.Comment: LaTex, 1+28 pages, 7 figure

Quantum Field Theories and Critical Phenomena on Defects

We construct and investigate quantum fields induced on a d-dimensional dissipationless defect by bulk fields propagating in a (d+1)-dimensional space. All interactions are localized on the defect. We derive a unitary non-canonical quantum field theory on the defect, which is analyzed both in the continuum and on the lattice. The universal critical behavior of the underlying system is determined. It turns out that the O(N)-symmetric phi^4 theory, induced on the defect by massless bulk fields, belongs to the universality class of particular d-dimensional spin models with long-range interactions. On the other hand, in the presence of bulk mass the critical behavior crossovers to the one of d-dimensional spin models with short-range interactions.Comment: 20 pages, references adde

Correlation functions of one-dimensional anyonic fluids

A universal description of correlation functions of one-dimensional anyonic gapless systems in the low-momentum regime is presented. We point out a number of interesting features, including universal oscillating terms with frequency proportional to the statistical parameter and beating effects close to the fermion points. The results are applied to the one-dimensional anyonic Lieb-Liniger model and checked against the exact results in the impenetrable limit.Comment: 4 pages, 1 figur

Algebraic approach to multiple defects on the line and application to Casimir force

An algebraic framework for quantization in presence of arbitrary number of point-like defects on the line is developed. We consider a scalar field which interacts with the defects and freely propagates away of them. As an application we compute the Casimir force both at zero and finite temperature. We derive also the charge density in the Gibbs state of a complex scalar field with defects. The example of two delta-defects is treated in detail.Comment: 24 pages, 10 figure

Non-linear quantum noise effects in scale invariant junctions

We study non-equilibrium steady state transport in scale invariant quantum junctions with focus on the particle and heat fluctuations captured by the two-point current correlation functions. We show that the non-linear behavior of the particle current affects both the particle and heat noise. The existence of domains of enhancement and reduction of the noise power with respect to the linear regime are observed. The impact of the statistics is explored. We demonstrate that in the scale invariant case the bosonic particle noise exceeds the fermionic one in the common domain of heat bath parameters. Multi-lead configurations are also investigated and the effect of probe terminals on the noise is discussed.Comment: LaTex, 1+20 pages, 10 figure