On the developments of Sklyanin's quantum separation of variables for integrable quantum field theories

We present a microscopic approach in the framework of Sklyanin's quantum separation of variables (SOV) for the exact solution of 1+1-dimensional quantum field theories by integrable lattice regularizations. Sklyanin's SOV is the natural quantum analogue of the classical method of separation of variables and it allows a more symmetric description of classical and quantum integrability w.r.t. traditional Bethe ansatz methods. Moreover, it has the advantage to be applicable to a more general class of models for which its implementation gives a characterization of the spectrum complete by construction. Our aim is to introduce a method in this framework which allows at once to derive the spectrum (eigenvalues and eigenvectors) and the dynamics (time dependent correlation functions) of integrable quantum field theories (IQFTs). This approach is presented for a paradigmatic example of relativistic IQFT, the sine-Gordon model.Comment: 8 pages; invited contribution to the Proceedings of the XVIIth INTERNATIONAL CONGRESS ON MATHEMATICAL PHYSICS, August 2012, Aalborg, Danemark; accepted for publication on the ICMP12 Proceedings by World Scientific. The material here presented is strictly connected to that introduced in arXiv:0910.3173 and arXiv:1204.630

The 8-vertex model with quasi-periodic boundary conditions

We study the inhomogeneous 8-vertex model (or equivalently the XYZ Heisenberg spin-1/2 chain) with all kinds of integrable quasi-periodic boundary conditions: periodic, $\sigma^x$-twisted, $\sigma^y$-twisted or $\sigma^z$-twisted. We show that in all these cases but the periodic one with an even number of sites $\mathsf{N}$, the transfer matrix of the model is related, by the vertex-IRF transformation, to the transfer matrix of the dynamical 6-vertex model with antiperiodic boundary conditions, which we have recently solved by means of Sklyanin's Separation of Variables (SOV) approach. We show moreover that, in all the twisted cases, the vertex-IRF transformation is bijective. This allows us to completely characterize, from our previous results on the antiperiodic dynamical 6-vertex model, the twisted 8-vertex transfer matrix spectrum (proving that it is simple) and eigenstates. We also consider the periodic case for $\mathsf{N}$ odd. In this case we can define two independent vertex-IRF transformations, both not bijective, and by using them we show that the 8-vertex transfer matrix spectrum is doubly degenerate, and that it can, as well as the corresponding eigenstates, also be completely characterized in terms of the spectrum and eigenstates of the dynamical 6-vertex antiperiodic transfer matrix. In all these cases we can adapt to the 8-vertex case the reformulations of the dynamical 6-vertex transfer matrix spectrum and eigenstates that had been obtained by $T$-$Q$ functional equations, where the $Q$-functions are elliptic polynomials with twist-dependent quasi-periods. Such reformulations enables one to characterize the 8-vertex transfer matrix spectrum by the solutions of some Bethe-type equations, and to rewrite the corresponding eigenstates as the multiple action of some operators on a pseudo-vacuum state, in a similar way as in the algebraic Bethe ansatz framework.Comment: 35 page

Antiperiodic XXZ chains with arbitrary spins: Complete eigenstate construction by functional equations in separation of variables

Generic inhomogeneous integrable XXZ chains with arbitrary spins are studied by means of the quantum separation of variables (SOV) method. Within this framework, a complete description of the spectrum (eigenvalues and eigenstates) of the antiperiodic transfer matrix is derived in terms of discrete systems of equations involving the inhomogeneity parameters of the model. We show here that one can reformulate this discrete SOV characterization of the spectrum in terms of functional T-Q equations of Baxter's type, hence proving the completeness of the solutions to the associated systems of Bethe-type equations. More precisely, we consider here two such reformulations. The first one is given in terms of Q-solutions, in the form of trigonometric polynomials of a given degree $N_s$, of a one-parameter family of T-Q functional equations with an extra inhomogeneous term. The second one is given in terms of Q-solutions, again in the form of trigonometric polynomials of degree $N_s$ but with double period, of Baxter's usual (i.e. without extra term) T-Q functional equation. In both cases, we prove the precise equivalence of the discrete SOV characterization of the transfer matrix spectrum with the characterization following from the consideration of the particular class of Q-solutions of the functional T-Q equation: to each transfer matrix eigenvalue corresponds exactly one such Q-solution and vice versa, and this Q-solution can be used to construct the corresponding eigenstate.Comment: 38 page

Paired states on a torus

We analyze the modular properties of the effective CFT description for paired states, proposed in cond-mat/0003453, corresponding to the non-standard filling nu =1/(p+1). We construct its characters for the twisted and the untwisted sector and the diagonal partition function. We show that the degrees of freedom entering our partition function naturally go to complete a Z_2-orbifold construction of the CFT for the Halperin state. Different behaviours for the p even and p odd cases are also studied. Finally it is shown that the tunneling phenomenon selects out a twist invariant CFT which is identified with the Moore-Read model.Comment: 24 pages, 1 figure, Late

A conformal field theory description of magnetic flux fractionalization in Josephson junction ladders

We show how the recently proposed effective theory for a Quantum Hall system at "paired states" filling v=1 (Mod. Phys. Lett. A 15 (2000) 1679; Nucl. Phys. B641 (2002) 547), the twisted model (TM), well adapts to describe the phenomenology of Josephson Junction ladders (JJL) in the presence of defects. In particular it is shown how naturally the phenomenon of flux fractionalization takes place in such a description and its relation with the discrete symmetries present in the TM. Furthermore we focus on closed geometries, which enable us to analyze the topological properties of the ground state of the system in relation to the presence of half flux quanta.Comment: 16 pages, 2 figure, Latex, revised versio

Open spin chains with generic integrable boundaries: Baxter equation and Bethe ansatz completeness from SOV

We solve the longstanding problem to define a functional characterization of the spectrum of the transfer matrix associated to the most general spin-1/2 representations of the 6-vertex reflection algebra for general inhomogeneous chains. The corresponding homogeneous limit reproduces the spectrum of the Hamiltonian of the spin-1/2 open XXZ and XXX quantum chains with the most general integrable boundaries. The spectrum is characterized by a second order finite difference functional equation of Baxter type with an inhomogeneous term which vanishes only for some special but yet interesting non-diagonal boundary conditions. This functional equation is shown to be equivalent to the known separation of variable (SOV) representation hence proving that it defines a complete characterization of the transfer matrix spectrum. The polynomial character of the Q-function allows us then to show that a finite system of equations of generalized Bethe type can be similarly used to describe the complete transfer matrix spectrum.Comment: 28 page

Antiperiodic dynamical 6-vertex model by separation of variables II: Functional equations and form factors

We pursue our study of the antiperiodic dynamical 6-vertex model using Sklyanin's separation of variables approach, allowing in the model new possible global shifts of the dynamical parameter. We show in particular that the spectrum and eigenstates of the antiperiodic transfer matrix are completely characterized by a system of discrete equations. We prove the existence of different reformulations of this characterization in terms of functional equations of Baxter's type. We notably consider the homogeneous functional $T$-$Q$ equation which is the continuous analog of the aforementioned discrete system and show, in the case of a model with an even number of sites, that the complete spectrum and eigenstates of the antiperiodic transfer matrix can equivalently be described in terms of a particular class of its $Q$-solutions, hence leading to a complete system of Bethe equations. Finally, we compute the form factors of local operators for which we obtain determinant representations in finite volume.Comment: 52 page