XXZ chain with a boundary

The XXZ spin chain with a boundary magnetic field h is considered, using the vertex operator approach to diagonalize the hamiltonian. We find explicit bosonic formulas for the two vacuum vectors with zero particle content. There are three distinct regions when h ⩾ 0, in which the structure of the vacuum states is different. Excited states are given by the action of vertex operators on the vacuum states. We derive the boundary S-matrix and present an integral formula for the correlation functions. The boundary magnetization exhibits boundary hysteresis. We also discuss the rational limit, the XXX model.     

Field-theoretical three-body relativistic equations for the multichannel πN↔γN↔ππN↔γπN reactions

A new kind of the relativistic three-body equations for the coupled πN and γN scattering reactions with the ππN and γπN three particle final states are suggested. These equations are derived in the framework of the standard field-theoretical S-matrix approach in the time-ordered three-dimensional form. Therefore, corresponding relativistic covariant equations are three-dimensional from the beginning and the considered formulation is free of the ambiguities which appear due to a three dimensional reduction of the four dimensional Bethe-Salpeter equations. The solutions of the considered equations satisfy the unitarity condition and they are exactly gauge invariant even after the truncation of the multiparticle (n>3) intermediate states. Moreover, the form of these three-body equations does not depend on the choice of the model Lagrangian and it is the same for the formulations with and without quark degrees of freedom. The effective potential of the suggested equations is defined by the vertex functions with two on-mass shell particles. It is emphasized that these INPUT vertex functions can be constructed from experimental data. Special attention is given to the construction of the intermediate on shell and off shell Δ resonance states. These intermediate Δ states are obtained after separation of the Δ resonance pole contributions in the intermediate πN Green function. The resulting amplitudes for the Δ⇔Nπ; Δ⇔Nγ; Δ^′⇔Δγ transition have the same structure as the vertex functions for transitions between the on-mass shell particle states with spin 1/2 and 3/2. Therefore it is possible to introduce the real value for the magnetic momenta for the Δ^′⇔Δγ transition amplitudes in the same way as it is done for the N^′⇔Nγ vertex function.     

Difference equations in spin chains with a boundary

Correlation functions and form factors in vertex models or spin chains are known to satisfy certain difference equations called the quantum Knizhnik-Zamolodchikov equations. We find similar difference equations for the case of semi-infinite spin chain systems with integrable boundary conditions. We derive these equations using the properties of the vertex operators and the boundary vacuum state, or alternatively through corner transfer matrix arguments for the eight-vertex model with a boundary. The spontaneous boundary magnetization is found by solving such difference equations. The boundary S-matrix is also proposed and compared, in the sine-Gordon limit, with Ghoshal-Zamolodchikov's result. The axioms satisfied by the form factors in the boundary theory are formulated.     

Operator approach to boundary Liouville theory

We propose new methods for calculation of the discrete spectrum, the reflection amplitude and the correlation functions of boundary Liouville theory on a strip with Lorentzian signature. They are based on the structure of the vertex operator V=e^-φ in terms of the asymptotic operators. The methods first are tested for the particle dynamics in the Morse potential, where similar structures appear. Application of our methods to boundary Liouville theory reproduces the known results obtained earlier in the bootstrap approach, but there can arise a certain extension when the boundary parameters are near to critical values. Namely, in this case we have found up to four different equidistant series of discrete spectra, and the reflection amplitude is modified, respectively.     

The twisted quantum affine algebra Uq(A2(2)) and correlation functions of the Izergin-Korepin model

We derive the exchange relations of the vertex operators of U_q(A₂⁽²⁾) and show that these vertex operators give the bosonization of the Izergin-Korepin model. We give an integral expression of the correlation functions of the Izergin-Korepin model and derive the difference equations which they satisfy.     

On Conformal Field Theory of SLE(κ,ρ)

SLE(κ ρ), a generalization of chordal Schramm-Löwner evolution (SLE), is discussed from the point of view of statistical mechanics and conformal field theory (CFT). Certain ratios of CFT correlation functions are shown to be martingales. The interpretation is that SLE(κ ρ) describes an interface in a statistical mechanics model whose boundary conditions are created in the Coulomb gas formalism by vertex operators with charges α _j = $\alpha_j = \frac{\rho_j}{2 \sqrt{\kappa}}SLE(κ ρ), a generalization of chordal Schramm-Löwner evolution (SLE), is discussed from the point of view of statistical mechanics and conformal field theory (CFT). Certain ratios of CFT correlation functions are shown to be martingales. The interpretation is that SLE(κ ρ) describes an interface in a statistical mechanics model whose boundary conditions are created in the Coulomb gas formalism by vertex operators with charges α _j = $\alpha_j = \frac{\rho_j}{2 \sqrt{\kappa}}SLE(κ ρ), a generalization of chordal Schramm-L?wner evolution (SLE), is discussed from the point of view of statistical mechanics and conformal field theory (CFT). Certain ratios of CFT correlation functions are shown to be martingales. The interpretation is that SLE(κ ρ) describes an interface in a statistical mechanics model whose boundary conditions are created in the Coulomb gas formalism by vertex operators with charges α _j = . The total charge vanishes and therefore the partition function has a simple product form. We also suggest a generalization of SLE(κ ρ)     

Conformal algebra and multipoint correlation functions in 2D statistical models

Based on the conformal algebra approach, a general technique is given for the calculation of multipoint correlation functions in 2D statistical models at the critical point. Particular conformal operator algebras are found for operators of the 2D q-component Potts model (1 < q < 4), and the O(N) model (0 < N < 2) at the critical point. A number of four-point correlation functions are calculated for these models.     

N=2 boundary supersymmetry in integrable models and perturbed boundary conformal field theory

Boundary integrable models with N=2 supersymmetry are considered. For the simplest boundary N=2 superconformal minimal model with a Chebyshev bulk perturbation we show explicitly how fermionic boundary degrees of freedom arise naturally in the boundary perturbation in order to maintain integrability and N=2 supersymmetry. A new boundary reflection matrix is obtained for this model and N=2 boundary superalgebra is studied. A factorized scattering theory is proposed for a N=2 supersymmetric extension of the boundary sine-Gordon model with either (i) fermionic or (ii) bosonic and fermionic boundary degrees of freedom. Exact results are obtained for some quantum impurity problems: the boundary scaling Lee–Yang model, a massive deformation of the anisotropic Kondo model at the filling values g=2/(2n+3) and the boundary Ashkin–Teller model.     

Conformal field theories near a boundary in general dimensions

The implications of restricted conformal invariance under conformal transformations preserving a plane boundary are discussed for general dimensions d. Calculations of the universal function of a conformal invariant ξ which appears in the two-point function of scalar operators in conformally invariant theories with a plane boundary are undertaken to first order in the ge = 4 − d expansion for the operator φ² in φ⁴ theory. The form for the associated functions of ξ for the two-point functions for the basic field φ^α and the auxiliary field λ in the N → ∞ limit of the O(N) nonlinear sigma model for any d in the range 2 < d < 4 are also rederived. These results are obtained by integrating the two-point functions over planes parallel to the boundary, defining a restricted two-point function which may be obtained more simply. Assuming conformal invariance this transformation can be inverted to recover the full two-point function. Consistency of the results is checked by considering the limit d → 4 and also by analysis of the operator product expansions for φ^αφ^β and λλ. Using this method the form of the two-point function for the energy-momentum tensor in the conformal O(N) model with a plane boundary is also found. General results for the sum of the contributions of all derivative operators appearing in the operator product expansion, and also in a corresponding boundary operator expansion, to the two-point functions are also derived making essential use of conformal invariance.     

Vacuum instability in scalar field theories

The class of scalar field theories with interaction gφ^2N?1, are studied using the semi-classical approximation. The imaginary part of the vertex functions generated by tunnelling out of the metastable ground state is calculated to first order. Using this result, the leading asymptotic behaviour of the renormalisation group β function for φ³ field theory is obtained in six dimensions. The validity of this result is discussed in view of the extra singularities which appear when the theory is just renormalisable. The structure of the perturbation expansion for n component φ³ theory is also discussed, and cases in which these theories yield perturbation expansions which are Borel summable, are pointed out.     

Exact solution for the spin-s XXZ quantum chain with non-diagonal twists

We study integrable vertex models and quantum spin chains with toroidal boundary conditions. An interesting class of such boundaries is associated with non-diagonal twist matrices. For such models there are no trivial reference states upon which a Bethe ansatz calculation can be constructed, in contrast to the well-known case of periodic boundary conditions. In this paper we show how the transfer matrix eigenvalue expression for the spin-s XXZ chain twisted by the charge-conjugation matrix can in fact be obtained. The technique used is the generalization to spin-s of the functional relation method based on “pair propagation through a vertex”. The Bethe ansatz-type equations obtained reduce, in the case of lattice size N = 1, to those recently found for the Hofstadter problem of Bloch electrons on a square lattice in a magnetic field.     

N = ∞ phase transition in a class of exactly soluble model lattice gauge theories

An exactly soluble class of model U(N) lattice gauge theories is considered. The ground state is discussed as a separable N-fermion problem solved by mathieu functions. Some exact correlation functions are presented. The N = ∞ limit exhibits a third order phase transition demarcating the strong and weak phases at (g²N)^?1 ≈ 0.55.     

Boundary Correlation Functions of the gl(1|1) Supersymmetric Vertex Model

The gl(1|1) supersymmetric vertex model with domain wall boundary conditions (DWBC) on an N×N square lattice is considered. We derive the reduction formulae for the one-point boundary correlation functions of the model. The determinant representation for the boundary correlation functions is also obtained.     

A study of meson-baryon couplings in the constituent quark model

Within the framework of the constituent quark model we discuss the effects of different types of meson-baryon-baryon vertex operators on the form factors and the coupling strengths of the lowest-lying positive and negative parity non-strange baryons. We compare the quark pair creation model (³P₀-model) with the SU(6)-model in which mesons are treated as elementary fields that directly couple to the quarks. The latter model is employed both in the so-called static limit and in a modification motivated by Galilei invariance. It is demonstrated that the inclusion of non-static effects simulates some features of the³P₀ vertex. Especially the reaction πN→ππN is found to be very sensitive to the different assumptions on the dynamics of theq¯q pair creation process. More indirect hints for the internal structure of the mesons might be obtained from the predicted asymmetry for the two form factors ofN→Δ+π andΔ→N+π, which occurs in the³P₀-model, only.     

Exact one-point function of N=1 super-Liouville theory with boundary

In this paper, exact one-point functions of N=1 super-Liouville field theory in two-dimensional space–time with appropriate boundary conditions are presented. Exact results are derived both for the theory defined on a pseudosphere with discrete (NS) boundary conditions and for the theory with explicit boundary actions which preserves super conformal symmetries. We provide various consistency checks. We also show that these one-point functions can be related to a generalized Cardy conditions along with corresponding modular S-matrices. Using this result, we conjecture the dependence of the boundary two-point functions of the (NS) boundary operators on the boundary parameter.     

Ultraviolet asymptotics of the Green and vertex functions in supergauge invariant field theory

Ultraviolet asymptotics of the Green and vertex functions in supergauge invariant field theory for the SU(N) group is established. Renormalization of the gauge field coupling constant is calculated and the condition for asymtotic freedom is found.     

Vertex QCD sum rules,quark model and pion wave function

It is shown that the predictions of the QCD sum rules (SR) and the quark model for the low energy pion wave function can be reconciled within the so-called vertex SR in exclusive kinematics. In contrast to the standard procedure, non-singular terms asx→0 in correlation functions are summed up by means of the conformal symmetry arguments. The soft contribution into the pion form factor is argued to dominate at least up toQ ²～4–6 GeV²     

Wave function(al)s in the large-N limit

The energy spectrum and correlation functions in various quantum theories have been determined within the large-N limit. Here we study the wave functions. Explicit results are presented for a quantum mechanical rotor, invariant under adjoint transformations of U(N), as well as for O(N) invariant vector models in quantum mechanics and quantum field theory.