Search

A brief survey of recent results in the study of boundary integrable quantum field theories, indicating some currently open problems. Based on lectures given at the 2000 Eotvos Summer School in Physics on `Nonperturbative QFT methods and their applications'., Comment: 6 pages, Latex, uses sprocl, cite, amssymb

The appearances of complex eigenvalues in the spectra of PT-symmetric quantum-mechanical systems are usually associated with a spontaneous breaking of PT. In this letter we discuss a family of models for which this phenomenon is also linked with an explicit breaking of supersymmetry. Exact level-crossings are located, and connections with N-fold supersymmetry and quasi-exact solvability in certain special cases are pointed out., 11 pages, 12 figures, Latex2e, amssymb, cite and graphicx. v2: a comment and two references added. v3: minor changes, and two references updated

We consider the one-point functions of bulk and boundary fields in the scaling Lee-Yang model for various combinations of bulk and boundary perturbations. The one-point functions of the bulk fields are analysed using the truncated conformal space approach and the form-factor expansion. Good agreement is found between the results of the two methods, though we find that the expression for the general boundary state given by Ghoshal and Zamolodchikov has to be corrected slightly. For the boundary fields we use thermodynamic Bethe ansatz equations to find exact expressions for the strip and semi-infinite cylinder geometries. We also find a novel off-critical identity between the cylinder partition functions of models with differing boundary conditions, and use this to investigate the regions of boundary-induced instability exhibited by the model on a finite strip., Latex, 34 pages, 16 figures

We find the spectrum of boundary bound states for the sine-Gordon model with Dirichlet boundary conditions, closing the bootstrap and providing a complete description of all the poles in the boundary reflection factors. The boundary Coleman-Thun mechanism plays an important role in the analysis. Two basic lemmas are introduced which should hold for any 1+1-dimensional boundary field theory, allowing the general method to be applied to other models., Comment: 26 pages, LaTeX2e, uses amsfonts.sty. V2: typos corrected, and equation numbering changed to agree with published version

We discuss the finite-size properties of a simple integrable quantum field theory in 1+1 dimensions with non-trivial boundary conditions. Novel off-critical identities between cylinder partition functions of models with differing boundary conditions are derived., Comment: 7 pages, 11 figures, JHEP proceedings style. Uses epsfig, amssymb. Talk given at the conference `Nonperturbative Quantum Effects 2000', Paris

We exhibit a relationship between the massless $a_2^{(2)}$ integrable quantum field theory and a certain third-order ordinary differential equation, thereby extending a recent result connecting the massless sine-Gordon model to the Schr\"odinger equation. This forms part of a more general correspondence involving $A_2$-related Bethe ansatz systems and third-order differential equations. A non-linear integral equation for the generalised spectral problem is derived, and some numerical checks are performed. Duality properties are discussed, and a simple variant of the nonlinear equation is suggested as a candidate to describe the finite volume ground state energies of minimal conformal field theories perturbed by the operators $\phi_{12}$, $\phi_{21}$ and $\phi_{15}$. This is checked against previous results obtained using the thermodynamic Bethe ansatz., Comment: 25 pages, Latex 2e, 3 figures, uses graphics, cite. v2: Typos corrected, section 6 expanded and an error fixed,references added v3: Notes and references added, to appear in Nucl. Phys. B

The vacuum expectation values of the so-called Q-operators of certain integrable quantum field theories have recently been identified with spectral determinants of particular Schrodinger operators. In this paper we extend the correspondence to the T-operators, finding that their vacuum expectation values also have an interpretation as spectral determinants. As byproducts we give a simple proof of an earlier conjecture of ours, proved by another route by Suzuki, and generalise a problem in PT symmetric quantum mechanics studied by Bender and Boettcher. We also stress that the mapping between Q-operators and Schrodinger equations means that certain problems in integrable quantum field theory are related to the study of Regge poles in non-relativistic potential scattering., Comment: 33 pages, LaTeX 2e, 7 eps figures, uses amssymb, cite, epsf; v2: typos corrected and one reference added

The spectral determinant $D(E)$ of the quartic oscillator is known to satisfy a functional equation. This is mapped onto the $A_3$-related $Y$-system emerging in the treatment of a certain perturbed conformal field theory, allowing us to give an alternative integral expression for $D(E)$. Generalising this result, we conjecture a relationship between the $x^{2M}$ anharmonic oscillators and the $A_{2M-1}$ TBA systems. Finally, spectral determinants for general $|x|^{\alpha}$ potentials are mapped onto the solutions of nonlinear integral equations associated with the (twisted) XXZ and sine-Gordon models., Comment: 11 pages, Latex 2e, uses amssymb, cite

We point out some problems with the previously-proposed phase diagram of the $Z_6$ spin models. Consideration of the diagram near to the decoupling surface using both exact and approximate arguments suggests a modification which remedies these deficiencies. With the aid of a new parametrisation of the phase space, we study the models numerically, with results which support our conjectures., Comment: 10 pages, Latex 2e, 4 figures, uses epsf, amssymb, cite

The method of analytic continuation is used to find exact integral equations for a selection of finite-volume energy levels for the non-unitary minimal models $M_{2,2N+3}$ perturbed by their $\varphi_{13}$ operators. The N=2 case is studied in particular detail. Along the way, we find a number of general results which should be relevant to the study of excited states in other models., Comment: 64 pages, Latex, 13 figures included (uses epsf). Figures (with slightly higher resolution) also available on request to the Authors

The theory of classically integrable nonlinear wave equations, and the Bethe Ansatz systems describing massive quantum field theories defined on an infinite cylinder, are related by an important mathematical correspondence that still lacks a satisfactory physical interpretation. In this paper we shall describe this link for the case of the classical and quantum versions of the (Tzitz\'eica-)Bullough-Dodd model., Comment: 20 pages

The generally accepted phase diagrams for the discrete $Z_N$ spin models in two dimensions imply the existence of certain renormalisation group flows, both between conformal field theories and into a massive phase. Integral equations are proposed to describe these flows, and some properties of their solutions are discussed. The infrared behaviour in massless and massive directions is analysed in detail, and the techniques used are applied to a number of other models., Comment: 64 pages, latex, 4 eps files, uses epsf.tex. References added and typos corrected; version to appear in Nuclear Physics B

A correspondence between the sextic anharmonic oscillator and a pair of third-order ordinary differential equations is used to investigate the phenomenon of quasi-exact solvability for eigenvalue problems involving differential operators with order greater than two. In particular, links with Bender-Dunne polynomials and resonances between independent solutions are observed for certain second-order cases, and extended to the higher-order problems., 14 pages, Latex

The question of the integrability of real-coupling affine toda field theory on a half-line is addressed. It is found, by examining low-spin conserved charges, that the boundary conditions preserving integrability are strongly constrained. In particular, for the $a_n\ (n>1)$ series of models there can be no free parameters introduced by the boundary condition; indeed the only remaining freedom (apart from choosing the simple condition $\partial_1��=0$), resides in a choice of signs. For a special case of the boundary condition, it is argued that the classical boundary bound state spectrum is closely related to a consistent set of reflection factors in the quantum field theory., 16 pages, TEX (harvmac), DTP-94/7, YITP/U-94-11

The S-matrices for non-simply-laced affine Toda field theories are considered in the context of a generalised bootstrap principle. The S-matrices, and in particular their poles, depend on a parameter whose range lies between the Coxeter numbers of dual pairs of the corresponding non-simply-laced algebras. It is proposed that only odd order poles in the physical strip with positive coefficients throughout this range should participate in the bootstrap. All other singularities have an explanation in principle in terms of a generalised Coleman-Thun mechanism. Besides the S-matrices introduced by Delius, Grisaru and Zanon, the missing case ($f_4^{(1)},e_6^{(2)}$), is also considered and provides many interesting examples of pole generation., Comment: 23 pages including two figures, harvmac

The affine Toda field theories based on the non simply-laced Lie algebras are discussed. By rewriting the S-matrix formulae found by Delius et al, a universal form for the coupling-constant dependence of these models is obtained, and related to various general properties of the classical couplings. This is illustrated via the S-matrix associated with the dual pair of algebras $f_4^{(1)}$ and $e_6^{(2)}$., Comment: 12 pages (harvmac), CERN-TH.6873/93

The bootstrap equations for the ADE series of purely elastic scattering theories have turned out to be intimately connected with the geometry of root systems and the Coxeter element. An informal review of some of this material is given, mentioning also a couple of other contexts — the Pasquier models, and the simply-laced affine Toda field theories — where similar structures are encountered. The relevance of twisted Coxeter elements is indicated, and a construction of these elements inspired by the twisted foldings of the affine Toda models is described.

We propose a class of purely elastic scattering theories generalising the staircase model of Al. B. Zamolodchikov, based on the affine Toda field theories for simply-laced Lie algebras g=A,D,E at suitable complex values of their coupling constants. Considering their Thermodynamic Bethe Ansatz equations, we give analytic arguments in support of a conjectured renormalisation group flow visiting the neighbourhood of each W_g minimal model in turn., 22 pages, Saclay-Bologna preprint SPhT/92-065, DFUB-92-09

The partition functions of Pasquier models on the cylinder, and the associated intertwiners, are considered. It is shown that earlier results due to Saleur and Bauer can be rephrased in a geometrical way, reminiscent of formulae found in certain purely elastic scattering theories. This establishes the positivity of these intertwiners in a general way and elucidates connections between these objects and the finite subgroups of SU(2). It also offers the hope that analogous geometrical structures might lie behind the so-far mysterious results found by Di Francesco and Zuber in their search for generalisations of these models., 18 pages

Starting from a recently-proposed general formula, various properties of the ADE series of purely elastic S-matrices are rederived in a universal way. In particular, the relationship between the pole structure and the bootstrap equations is shown to follow from properties of root systems. The discussion leads to a formula for the signs of the three-point couplings in the simply-laced affine Toda theories, and a simple proof of a result due to Klassen and Melzer of relevance to Thermodynamic Bethe Ansatz calculations., 21 pages

Within the framework of the ODE/IM correspondence, we show that the minimal conformal field theories with c, 12 pages, 3 figures; typos corrected and reference added

Recently, Zamolodchikov has suggested a way of exploring the properties of the Ising model in a background magnetic field, i.e. away from criticaliy(1).

The existence of exact unitary crossing symmetric S -matrices associated with the a, d, e series of Toda field theories is conjectured and the main features illustrated within a discussion of the d 4 case.

The g-function was introduced by Affleck and Ludwig as a measure of the ground state degeneracy of a conformal boundary condition. We consider this function for perturbations of the conformal Yang-Lee model by bulk and boundary fields using conformal perturbation theory, the truncated conformal space approach and the thermodynamic Bethe Ansatz (TBA). We find that the TBA equations derived by LeClair et al describe the massless boundary flows, up to an overall constant, but are incorrect when one considers a simultaneous bulk perturbation; however the TBA equations do correctly give the `non-universal' linear term in the massive case, and the ratio of g-functions for different boundary conditions is also correctly produced. This ratio is related to the Y-system of the Yang-Lee model and by comparing the perturbative expansions of the Y-system and of the g-functions we obtain the exact relation between the UV and IR parameters of the massless perturbed boundary model., LaTeX2e, 36 pages, 19 figures

This article reviews a recently-discovered link between integrable quantum field theories and certain ordinary differential equations in the complex domain. Along the way, aspects of PT-symmetric quantum mechanics are discussed, and some elementary features of the six-vertex model and the Bethe ansatz are explained., 102 pages, 27 figures. v2: added references

We investigate the staircase model, introduced by Aliosha Zamolodchikov through an analytic continuation of the sinh-Gordon S-matrix to describe interpolating flows between minimal models of conformal field theory in two dimensions. Applying the form factor expansion and the c-theorem, we show that the resulting c-function has the same physical content as that found by Zamolodchikov from the thermodynamic Bethe Ansatz. This turns out to be a consequence of a nontrivial underlying mechanism, which leads to an interesting localisation pattern for the spectral integrals giving the multi-particle contributions. We demonstrate several aspects of this form factor relocalisation, which suggests a novel approach to the construction of form factors and spectral sums in integrable renormalisation group flows with non-diagonal scattering., Comment: 37 pages, PDFLatex file with pdf figures

We present evidence for the existence of infinitely-many new families of renormalisation group flows between the nonunitary minimal models of conformal field theory. These are associated with perturbations by the $\phi_{21}$ and $\phi_{15}$ operators, and generalise a family of flows discovered by Martins. In all of the new flows, the finite-volume effective central charge is a non-monotonic function of the system size. The evolution of this effective central charge is studied by means of a nonlinear integral equation, a massless variant of an equation recently found to describe certain massive perturbations of these same models. We also observe that a similar non-monotonicity arises in the more familiar $\phi_{13}$ perturbations, when the flows induced are between nonunitary minimal models., Comment: 31 pages with 8 eps figures. LaTex2e, uses amssymb,cite,graphics.Minor typos corrected