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Abstract: We consider the partition function of the boundary coset sigma model on an annulus, based on the lattice regularization introduced in the companion paper. Using results for the action of and on the corresponding spin chain, as well as mini-superspace and small calculations, we conjecture the full spectrum and set of degeneracies on the entire critical line. Potential relationship with the Gross–Neveu model is also discussed. [Copyright &y& Elsevier]

Abstract: We define and study a lattice model which we argue is in the universality class of the supercoset sigma model for a large range of values of the coupling constant . In this first paper, we analyze in details the symmetries of this lattice model, in particular the decomposition of the space of the quantum spin chain as a bimodule over and its commutant, the Brauer algebra . It turns out that is a nonsemisimple module for both and . The results are used in the companion paper to elucidate the structure of the (boundary) conformal field theory. [Copyright &y& Elsevier]

Abstract: We study a model of densely packed self-avoiding loops on the annulus, related to the Temperley–Lieb algebra with an extra idempotent boundary generator. Four different weights are given to the loops, depending on their homotopy class and whether they touch the outer rim of the annulus. When the weight of a contractible bulk loop , this model is conformally invariant for any real weight of the remaining three parameters. We classify the conformal boundary conditions and give exact expressions for the corresponding boundary scaling dimensions. The amplitudes with which the sectors with any prescribed number and types of non-contractible loops appear in the full partition function Z are computed rigorously. Based on this, we write a number of identities involving Z which hold true for any finite size. When the weight of a contractible boundary loop y takes certain discrete values, with r integer, other identities involving the standard characters of the Virasoro algebra are established. The connection with Dirichlet and Neumann boundary conditions in the model is discussed in detail, and new scaling dimensions are derived. When q is a root of unity and , exact connections with the type RSOS model are made. These involve precise relations between the spectra of the loop and RSOS model transfer matrices, valid in finite size. Finally, the results where are related to the theory of Temperley–Lieb cabling. [Copyright &y& Elsevier]

Abstract: We set up a strategy for studying large families of logarithmic conformal field theories by using the enlarged symmetries and non-semisimple associative algebras appearing in their lattice regularizations (as discussed in a companion paper [N. Read, H. Saleur, Enlarged symmetry algebras of spin chains, loop models, and S-matrices, cond-mat/0701259]). Here we work out in detail two examples of theories derived as the continuum limit of XXZ spin-1/2 chains, which are related to spin chains with supersymmetry algebras and , respectively, with open (or free) boundary conditions in all cases. These theories can also be viewed as vertex models, or as loop models. Their continuum limits are boundary conformal field theories (CFTs) with central charge and respectively, and in the loop interpretation they describe dense polymers and the boundaries of critical percolation clusters, respectively. We also discuss the case of dilute (critical) polymers as another boundary CFT with . Within the supersymmetric formulations, these boundary CFTs describe the fixed points of certain nonlinear sigma models that have a supercoset space as the target manifold, and of Landau–Ginzburg field theories. The submodule structures of indecomposable representations of the Virasoro algebra appearing in the boundary CFT, representing local fields, are derived from the lattice. A central result is the derivation of the fusion rules for these fields. [Copyright &y& Elsevier]

Abstract: Motivated by a careful analysis of the Laplacian on the supergroup we formulate a proposal for the state space of the WZNW model. We then use properties of characters to compute the partition function of the theory. In the special case of level the latter is found to agree with the properly regularized partition function for the continuum limit of the integrable super-spin chain. Some general conclusions applicable to other WZNW models (in particular the case ) are also drawn. [Copyright &y& Elsevier]

Abstract: We solve in this paper the problem of the antiferromagnetic transition for the Q-state Potts model (defined geometrically for Q generic using the loop/cluster expansion) on the square lattice. This solution is based on the detailed analysis of the Bethe ansatz equations (which involve staggered source terms of the type “real” and “anti-string”) and on extensive numerical diagonalization of transfer matrices. It involves subtle distinctions between the loop/cluster version of the model, and the associated RSOS and (twisted) vertex models. The essential result is that the twisted vertex model on the transition line has a continuum limit described by two bosons, one which is compact and twisted, and the other which is not, with a total central charge , for . The non-compact boson contributes a continuum component to the spectrum of critical exponents. For Q generic, these properties are shared by the Potts model. For Q a Beraha number, i.e., with n integer, and in particular Q integer, the continuum limit is given by a “truncation” of the two boson theory, and coincides essentially with the critical point of parafermions . Moreover, the vertex model, and, for Q generic, the Potts model, exhibit a first-order critical point on the transition line—that is, the antiferromagnetic critical point is not only a point where correlations decay algebraically, but is also the locus of level crossings where the derivatives of the free energy are discontinuous. In that sense, the thermal exponent of the Potts model is generically equal to . Things are however profoundly different for Q a Beraha number. In this case, the antiferromagnetic transition is second order, with the thermal exponent determined by the dimension of the parafermion, . As one enters the adjacent “Berker–Kadanoff” phase, the model flows, for t odd, to a minimal model of CFT with central charge , while for t even it becomes massive. This provides a physical realization of a flow conjectured long ago by Fateev and Zamolodchikov in the context of integrable perturbations. Finally, though the bulk of the paper concentrates on the square-lattice model, we present arguments and numerical evidence that the antiferromagnetic transition occurs as well on other two-dimensional lattices. [Copyright &y& Elsevier]

Abstract: We present a complete solution of the WZW model on the supergroup . Our analysis begins with a careful study of its minisuperspace limit (“harmonic analysis on the supergroup”). Its spectrum is shown to contain indecomposable representations. This is interpreted as a geometric signal for the appearance of logarithms in the correlators of the full field theory. We then discuss the representation theory of the current algebra and propose an Ansatz for the state space of the WZW model. The latter is established through an explicit computation of the correlation function. We show in particular, that the 4-point functions of the theory factorize on the proposed set of states and that the model possesses an interesting spectral flow symmetry. The note concludes with some remarks on generalizations to other supergroups. [Copyright &y& Elsevier]

Abstract: We discuss the relationship between the phase diagram of the state Potts model, the arboreal gas model, and the supersphere sigma model . We identify the Potts antiferromagnetic critical point with the critical point of the arboreal gas (at negative tree fugacity), and with a critical point of the sigma model. We show that the corresponding conformal theory on the square lattice has a non-linearly realized symmetry, and involves non-compact degrees of freedom, with a continuous spectrum of critical exponents. The role of global topological properties in the sigma model transition is discussed in terms of a generalized arboreal gas model. [Copyright &y& Elsevier]

As a step to understand general patterns of integrability in 1+1 quantum field theories with supergroup symmetry, we study in details the case of OSP(1/2). Our results include the solutions of natural generalizations of models with ordinary group symmetry: the UOSP(1/2)k WZW model with a current–current perturbation, the UOSP(1/2) principal chiral model, and the UOSP(1/2)⊗UOSP(1/2)/UOSP(1/2) coset models perturbed by the adjoint. Graded parafermions are also discussed. A pattern peculiar to supergroups is the emergence of another class of models, whose simplest representative is the OSP(1/2)/OSP(0/2) sigma model, where the (non unitary) orthosymplectic symmetry is realized non-linearly (and can be spontaneously broken). For most models, we provide an integrable lattice realization. We show in particular that integrable osp(1/2) spin chains with integer spin flow to UOSP(1/2) WZW models in the continuum limit, hence providing what is to our knowledge the first physical realization of a super WZW model. [Copyright &y& Elsevier]

Abstract: In the context of Conformal Field Theory (CFT), many results can be obtained from the representation theory of the Virasoro algebra. While the interest in Logarithmic CFTs has been growing recently, the Virasoro representations corresponding to these quantum field theories remain dauntingly complicated, thus hindering our understanding of various critical phenomena. We extend in this paper the construction of Read and Saleur (2007) [1,2], and uncover a deep relationship between the Virasoro algebra and a finite-dimensional algebra characterizing the properties of two-dimensional statistical models, the so-called blob algebra (a proper extension of the Temperley–Lieb algebra). This allows us to explore vast classes of Virasoro representations (projective, tilting, generalized staggered modules, etc.), and to conjecture a classification of all possible indecomposable Virasoro modules (with, in particular, Jordan cells of arbitrary rank) that may appear in a consistent physical Logarithmic CFT where Virasoro is the maximal local chiral algebra. As by-products, we solve and analyze algebraically quantum-group symmetric XXZ spin chains and supersymmetric spin chains with extra spins at the boundary, together with the “mirror” spin chain introduced by Martin and Woodcock (2003) [3]. [Copyright &y& Elsevier]

Abstract: In this paper we continue the investigation of partition functions of critical systems on a rectangle initiated in [R. Bondesan, et al., Nucl. Phys. B 862 (2012) 553–575]. Here we develop a general formalism of rectangle boundary states using conformal field theory, adapted to describe geometries supporting different boundary conditions. We discuss the computation of rectangular amplitudes and their modular properties, presenting explicit results for the case of free theories. In a second part of the paper we focus on applications to loop models, discussing in details lattice discretizations using both numerical and analytical calculations. These results allow to interpret geometrically conformal blocks, and as an application we derive new probability formulas for self-avoiding walks. [Copyright &y& Elsevier]

Abstract: Work of the last few years has shown that the key algebraic features of Logarithmic Conformal Field Theories (LCFTs) are already present in some finite lattice systems (such as the XXZ spin-1/2 chain) before the continuum limit is taken. This has provided a very convenient way to analyze the structure of indecomposable Virasoro modules and to obtain fusion rules for a variety of models such as (boundary) percolation etc. LCFTs allow for additional quantum numbers describing the fine structure of the indecomposable modules, and generalizing the ‘b-number’ introduced initially by Gurarie for the case. The determination of these indecomposability parameters (or logarithmic couplings) has given rise to a lot of algebraic work, but their physical meaning has remained somewhat elusive. In a recent paper, a way to measure b for boundary percolation and polymers was proposed. We generalize this work here by devising a general strategy to compute matrix elements of Virasoro generators from the numerical analysis of lattice models and their continuum limit. The method is applied to XXZ spin-1/2 and spin-1 chains with open (free) boundary conditions. They are related to and -invariant superspin chains and to non-linear sigma models with supercoset target spaces. These models can also be formulated in terms of dense and dilute loop gas. We check the method in many cases where the results were already known analytically. Furthermore, we also confront our findings with a construction generalizing Gurarieʼs, where logarithms emerge naturally in operator product expansions to compensate for apparently divergent terms. This argument actually allows us to compute indecomposability parameters in any logarithmic theory. A central result of our study is the construction of a Kac table for the indecomposability parameters of the logarithmic minimal models and . [Copyright &y& Elsevier]

Abstract: The sigma models on projective superspaces with topological angle flow to non-unitary, logarithmic conformal field theories in the low-energy limit. In this paper, we determine the exact spectrum of these theories for all open boundary conditions preserving the full global symmetry of the model, generalizing recent work on the particular case [C. Candu et al., JHEP 1002 (2010) 015]. In the sigma model setting, these boundary conditions are associated with complex line bundles, and are labelled by an integer, related with the exact value of θ. Our approach relies on a spin chain regularization, where the boundary conditions now correspond to the introduction of additional edge states. The exact values of the exponents then follow from a lengthy algebraic analysis, a reformulation of the spin chain in terms of crossing and non-crossing loops (represented as a certain subalgebra of the Brauer algebra), and earlier results on the so-called one- and two-boundary Temperley–Lieb algebras (also known as blob algebras). A remarkable result is that the exponents, in general, turn out to be irrational. The case has direct applications to the spin quantum Hall effect, which will be discussed in a sequel. [Copyright &y& Elsevier]

Abstract: A good understanding of conformal field theory (CFT) at is vital to the physics of disordered systems, as well as geometrical problems such as polymers and percolation. Steady progress has shown that these CFTs should be logarithmic, with indecomposable operator product expansions, and indecomposable representations of the Virasoro algebra. In one of the earliest papers on the subject, V. Gurarie introduced a single parameter b to quantify this indecomposability in terms of the logarithmic partner t of the stress–energy tensor T. He and A. Ludwig conjectured further that for polymers and for percolation. While a lot of physics may be hidden behind this parameter — which has also given rise to a lot of discussions — it had remained very elusive up to now, due to the lack of available methods to measure it experimentally or numerically, in contrast say with the central charge. We show in this paper how to overcome the many difficulties in trying to measure b. This requires control of a lattice scalar product, lattice Jordan cells, together with a precise construction of the state . The final result is that for polymers. For percolation, we find that within an XXZ or supersymmetric representation. In the geometrical representation, we do not find a Jordan cell for at level two (finite-size Hamiltonian and transfer matrices are fully diagonalizable), so there is no b in this case. [Copyright &y& Elsevier]

Abstract: We study the conformal boundary conditions of the dilute model in two dimensions. A pair of mutually dual solutions to the boundary Yang–Baxter equations are found. They describe anisotropic special transitions, and can be interpreted in terms of symmetry breaking interactions in the model. We identify the corresponding boundary condition changing operators, Virasoro characters, and conformally invariant partition functions. We compute the entropies of the conformal boundary states, and organize the flows between the various boundary critical points in a consistent phase diagram. The operators responsible for the various flows are identified. Finally, we discuss the relation to open boundary conditions in the model, and present new crossing probabilities for Ising domain walls. [Copyright &y& Elsevier]

Abstract: We study the two-boundary extension of a loop model—corresponding to the dense phase of the model, or to the state Potts model—in the critical regime . This model is defined on an annulus of aspect ratio τ. Loops touching the left, right, or both rims of the annulus are distinguished by arbitrary (real) weights which moreover depend on whether they wrap the periodic direction. Any value of these weights corresponds to a conformally invariant boundary condition. We obtain the exact seven-parameter partition function in the continuum limit, as a function of τ, by a combination of algebraic and field theoretical arguments. As a specific application we derive some new crossing formulae for percolation clusters. [Copyright &y& Elsevier]

Abstract: The antiferromagnetic critical point of the Potts model on the square lattice was identified by Baxter [R.J. Baxter, Proc. R. Soc. London A 383 (1982) 43] as a staggered integrable six-vertex model. In this work, we investigate the integrable structure of this model. It enables us to derive some new properties, such as the Hamiltonian limit of the model, an equivalent vertex model, and the structure resulting from the symmetry. Using this material, we discuss the low-energy spectrum, and relate it to geometrical excitations. We also compute the critical exponents by solving the Bethe equations for a large lattice width N. The results confirm that the low-energy spectrum is a collection of continua with typical exponent gaps of order . [Copyright &y& Elsevier]

Abstract: By a combination of analytical and numerical techniques, we analyze the continuum limit of the integrable superspin chain. We discover profoundly new features, including a continuous spectrum of conformal weights, whose numerical evidence is infinite degeneracies of the scaled gaps in the thermodynamic limit. This indicates that the corresponding conformal field theory has a non compact target space (even though our lattice model involves only finite-dimensional representations). We argue that our results are compatible with this theory being the level , “ WZW model” (whose precise definition requires some care). In doing so, we establish several new results for this model. With regard to potential applications to the spin quantum Hall effect, we conclude that the continuum limit of the integrable superspin chain is not the same as (and is in fact very different from) the continuum limit of the corresponding chain with two-superspin interactions only, which is known to be a model for the spin quantum Hall effect. The study of possible RG flows between the two theories is left for further study. [Copyright &y& Elsevier]