Algebraic Bethe ansatz for the quantum group invariant open XXZ chain at roots of unity

For generic values of q, all the eigenvectors of the transfer matrix of the U_q sl(2)-invariant open spin-1/2 XXZ chain with finite length N can be constructed using the algebraic Bethe ansatz (ABA) formalism of Sklyanin. However, when q is a root of unity (q=exp(i pi/p) with integer p>1), the Bethe equations acquire continuous solutions, and the transfer matrix develops Jordan cells. Hence, there appear eigenvectors of two new types: eigenvectors corresponding to continuous solutions (exact complete p-strings), and generalized eigenvectors. We propose general ABA constructions for these two new types of eigenvectors. We present many explicit examples, and we construct complete sets of (generalized) eigenvectors for various values of p and N.Comment: 50pp, 2 figures, v2: few typos are fixed, Nucl. Phys. B (2016

The symplectic fermion ribbon quasi-Hopf algebra and the SL(2,Z)-action on its centre

We introduce a family of factorisable ribbon quasi-Hopf algebras $Q(N)$ for $N$ a positive integer: as an algebra, $Q(N)$ is the semidirect product of $\mathbb{C}\mathbb{Z}_2$ with the direct sum of a Grassmann and a Clifford algebra in $2N$ generators. We show that $Rep Q(N)$ is ribbon equivalent to the symplectic fermion category $SF(N)$ that was computed by the third author from conformal blocks of the corresponding logarithmic conformal field theory. The latter category in turn is conjecturally ribbon equivalent to representations of $V_{ev}$, the even part of the symplectic fermion vertex operator super algebra. Using the formalism developed in our previous paper we compute the projective $SL(2,\mathbb{Z})$-action on the centre of $Q(N)$ as obtained from Lyubashenko's general theory of mapping class group actions for factorisable finite ribbon categories. This allows us to test a conjectural non-semisimple version of the modular Verlinde formula: we verify that the $SL(2,\mathbb{Z})$-action computed from $Q(N)$ agrees projectively with that on pseudo trace functions of $V_{ev}$.Comment: 75pp; typos fixed, references update

Counting solutions of the Bethe equations of the quantum group invariant open XXZ chain at roots of unity

We consider the sl(2)_q-invariant open spin-1/2 XXZ quantum spin chain of finite length N. For the case that q is a root of unity, we propose a formula for the number of admissible solutions of the Bethe ansatz equations in terms of dimensions of irreducible representations of the Temperley-Lieb algebra; and a formula for the degeneracies of the transfer matrix eigenvalues in terms of dimensions of tilting sl(2)_q-modules. These formulas include corrections that appear if two or more tilting modules are spectrum-degenerate. For the XX case (q=exp(i pi/2)), we give explicit formulas for the number of admissible solutions and degeneracies. We also consider the cases of generic q and the isotropic (q->1) limit. Numerical solutions of the Bethe equations up to N=8 are presented. Our results are consistent with the Bethe ansatz solution being complete.Comment: 34 pages; v2: reference added; v3: two more references added and minor correction

Integrability of rank-two web models

We continue our work on lattice models of webs, which generalise the well-known loop models to allow for various kinds of bifurcations [arXiv:2101.00282, arXiv:2107.10106]. Here we define new web models corresponding to each of the rank-two spiders considered by Kuperberg [arXiv:q-alg/9712003]. These models are based on the $A_2$, $G_2$ and $B_2$ Lie algebras, and their local vertex configurations are intertwiners of the corresponding $q$-deformed quantum algebras. In all three cases we define a corresponding model on the hexagonal lattice, and in the case of $B_2$ also on the square lattice. For specific root-of-unity choices of $q$, we show the equivalence to a number of three- and four-state spin models on the dual lattice. The main result of this paper is to exhibit integrable manifolds in the parameter spaces of each web model. For $q$ on the unit circle, these models are critical and we characterise the corresponding conformal field theories via numerical diagonalisation of the transfer matrix. In the $A_2$ case we find two integrable regimes. The first one contains a dense and a dilute phase, for which we have analytic control via a Coulomb gas construction, while the second one is more elusive and likely conceals non-compact physics. Three particular points correspond to a three-state spin model with plaquette interactions, of which the one in the second regime appears to present a new universality class. In the $G_2$ case we identify four regimes numerically. The $B_2$ case is too unwieldy to be studied numerically in the general case, but it found analytically to contain a simpler sub-model based on generators of the dilute Birman-Murakami-Wenzl algebra.Comment: 69 page

Projective objects and the modified trace in factorisable finite tensor categories

International audienceFor C a factorisable and pivotal finite tensor category over an algebraically closed field of characteristic zero we show: 1) C always contains a simple projective object; 2) if C is in addition ribbon, the internal characters of projective modules span a submodule for the projective SL(2,Z)-action; 3) the action of the Grothendieck ring of C on the span of internal characters of projective objects can be diagonalised; 4) the linearised Grothendieck ring of C is semisimple iff C is semisimple. Results 1-3 remain true in positive characteristic under an extra assumption. Result 1 implies that the tensor ideal of projective objects in C carries a unique-up-to-scalars modified trace function. We express the modified trace of open Hopf links coloured by projectives in terms of S-matrix elements. Furthermore, we give a Verlinde-like formula for the decomposition of tensor products of projective objects which uses only the modular S-transformation restricted to internal characters of projective objects. We compute the modified trace in the example of symplectic fermion categories, and we illustrate how the Verlinde-like formula for projective objects can be applied there

Uqsl2U_q\mathfrak{sl}_2-invariant non-compact boundary conditions for the XXZ spin chain

We introduce new $U_q\mathfrak{sl}_2$-invariant boundary conditions for the open XXZ spin chain. For generic values of $q$ we couple the bulk Hamiltonian to an infinite-dimensional Verma module on one or both boundaries of the spin chain, and for $q=e^{\frac{i\pi}{p}}$ a $2p$-th root of unity $-$ to its $p$-dimensional analogue. Both cases are parametrised by a continuous "spin" $\alpha\in\mathbb{C}$. To motivate our construction, we first specialise to $q=i$, where we obtain a modified XX Hamiltonian with unrolled quantum group symmetry, whose spectrum and scaling limit is computed explicitly using free fermions. In the continuum, this model is identified with the $(\eta,\xi)$ ghost CFT on the upper-half plane with a continuum of conformally invariant boundary conditions on the real axis. The different sectors of the Hamiltonian are identified with irreducible Virasoro representations. Going back to generic $q$ we investigate the algebraic properties of the underlying lattice algebras. We show that if $q^\alpha\notin\pm q^{\mathbb{Z}}$, the new boundary coupling provides a faithful representation of the blob algebra which is Schur-Weyl dual to $U_q\mathfrak{sl}_2$. Then, modifying the boundary conditions on both the left and the right, we obtain a representation of the universal two-boundary Temperley-Lieb algebra. The generators and parameters of these representations are computed explicitly in terms of $q$ and $\alpha$. Finally, we conjecture the general form of the Schur-Weyl duality in this case. This paper is the first in a series where we will study, at all values of the parameters, the spectrum and its continuum limit, the representation content of the relevant lattice algebras and the fusion properties of these new spin chains

Non-semisimple link and manifold invariants for symplectic fermions

We consider the link and three-manifold invariants in [DGGPR], which are defined in terms of certain non-semisimple finite ribbon categories $\mathcal{C}$ together with a choice of tensor ideal and modified trace. If the ideal is all of $\mathcal{C}$, these invariants agree with those defined by Lyubashenko in the 90's. We show that in that case the invariants depend on the objects labelling the link only through their simple composition factors, so that in order to detect non-trivial extensions one needs to pass to proper ideals. We compute examples of link and three-manifold invariants for $\mathcal{C}$ being the category of $N$ pairs of symplectic fermions. Using a quasi-Hopf algebra realisation of $\mathcal{C}$, we find that the Lyubashenko-invariant of a lens space is equal to the order of its first homology group to the power $N$, a relation we conjecture to hold for all rational homology spheres. For $N \ge 2$, $\mathcal{C}$ allows for tensor ideals $\mathcal{I}$ with a modified trace which are different from all of $\mathcal{C}$ and from the projective ideal. Using the theory of pull-back traces and symmetrised cointegrals, we show that the link invariant obtained from $\mathcal{I}$ can distinguish a continuum of indecomposable but reducible objects which all have the same composition series