In this work we investigate the possibility of using the reflection algebra as a source of functional equations. More precisely, we obtain functional relations determining the partition function of the six-vertex model with domain-wall boundary conditions and one reflecting end. The model's partition function is expressed as a multiple-contour integral that allows the homogeneous limit to be obtained straightforwardly. Our functional equations are also shown to give rise to a consistent set of partial differential equations for the partition function.Comment: v1: 30 pages. v2: 31 pages, figures added, version accepted for publication in Nucl. Phys.

The spin-\frac{1}{2}12 Heisenberg XXZ chain is a paradigmatic quantum integrable model. Although it can be solved exactly via Bethe ansatz techniques, there are still open issues regarding the spectrum at root of unity values of the anisotropy. We construct Baxter’s Q operator at arbitrary anisotropy from a two-parameter transfer matrix associated to a complex-spin auxiliary space. A decomposition of this transfer matrix provides a simple proof of the transfer matrix fusion and Wronskian relations. At root of unity a truncation allows us to construct the Q operator explicitly in terms of finite-dimensional matrices. From its decomposition we derive truncated fusion and Wronskian relations as well as an interpolation-type formula that has been conjectured previously. We elucidate the Fabricius–McCoy (FM) strings and exponential degeneracies in the spectrum of the six-vertex transfer matrix at root of unity. Using a semicyclic auxiliary representation we give a conjecture for creation and annihilation operators of FM strings for all roots of unity. We connect our findings with the `string-charge duality’ in the thermodynamic limit, leading to a conjecture for the imaginary part of the FM string centres with potential applications to out-of-equilibrium physics.

We study the -analogue of the Haldane–Shastry model, a partially isotropic (xxz-like) long-range spin chain that by construction enjoys quantum-affine (really: quantum-loop) symmetries at finite system size. We derive the pairwise form of the Hamiltonian, found by one of us building on work of D. Uglov, via ‘freezing’ from the affine Hecke algebra. To this end we first obtain explicit expressions for the spin-Macdonald operators of the (trigonometric) spin-Ruijsenaars model. Through freezing these give rise to the higher Hamiltonians of the spin chain, including another Hamiltonian of the opposite ‘chirality’. The sum of the two chiral Hamiltonians has a real spectrum also when $$|\mathsf {q}|=1$$ | q | = 1 , so in particular when is a root of unity. For generic $$\mathsf {q}$$ q the eigenspaces are known to be labelled by ‘motifs’. We clarify the relation between these patterns and the corresponding degeneracies (multiplicities) in the crystal limit $$\textsf {q}\rightarrow \infty $$ q → ∞ . For each motif we obtain an explicit expression for the exact eigenvector, valid for generic , that has (‘pseudo’ or ‘l-’) highest weight in the sense that, in terms of the operators from the monodromy matrix, it is an eigenvector of A and D and annihilated by C. It has a simple component featuring the ‘symmetric square’ of the -Vandermonde polynomial times a Macdonald polynomial—or more precisely its quantum spherical zonal special case. All other components of the eigenvector are obtained from this through the action of the Hecke algebra, followed by ‘evaluation’ of the variables to roots of unity. We prove that our vectors have highest weight upon evaluation. Our description of the exact spectrum is complete. The entire model, including the quantum-loop action, can be reformulated in terms of polynomials. Our main tools are the Y-operators from the affine Hecke algebra. From a more mathematical perspective the key step in our diagonalisation is as follows. We show that on a subspace of suitable polynomials the first M ‘classical’ (i.e. no difference part) Y-operators in N variables reduce, upon evaluation as above, to Y-operators in M variables with parameters at the quantum zonal spherical point.

We perform a numerical study of the F model with domain-wall boundary conditions. Various exact results are known for this particular case of the six-vertex model, including closed expressions for the partition function for any system size as well as its asymptotics and leading finite-size corrections. To complement this picture we use a full lattice multicluster algorithm to study equilibrium properties of this model for systems of moderate size, up to L = 512. We compare the energy to its exactly known large-L asymptotics. We investigate the model's infinite-order phase transition by means of finite-size scaling for an observable derived from the staggered polarization in order to test the method put forward in our recent joint work with Duine and Barkema. In addition we analyze local properties of the model. Our data are perfectly consistent with analytical expressions for the arctic curves. We investigate the structure inside the temperate region of the lattice, confirming the oscillations in vertex densities that were first observed by Syljuåsen and Zvonarev and recently studied by Lyberg et al. We point out "(anti)ferroelectric" oscillations close to the corresponding frozen regions as well as "higher-order" oscillations forming an intricate pattern with saddle-point-like features.

In this paper we extend previous work of Galleas and the author to elliptic sos models. We demonstrate that the dynamical reflection algebra can be exploited to obtain a functional equation characterizing the partition function of an elliptic sos model with domain-wall boundaries and one reflecting end. Special attention is paid to the structure of the functional equation. Through this approach we find a novel multiple-integral formula for that partition function.

A pedagogical introduction to quantum integrability Jules Lamers Contents 1. Introduction 2 2. Bethe's method for the XXZ model 5 2.1 The XXZ spin chain and its symmetries 5 2.2 The coordinate Bethe Ansatz 10 2.3 Results and Bethe-Ansatz equations 13 3. Transfer matrices and the six-vertex model 18 3.1 The six-vertex model 18 3.2 The transfer-matrix method and CBA 22 3.3 Unexpected results 26 4. The quantum inverse-scattering method 30 4.1 Conserved quantities from Lax operators 30 4.2 The Yang-Baxter algebra 33 4.3 The algebraic Bethe Ansatz 41 5. Relation to theoretical high-energy physics 46 5.1 Quantum integrability and 2d QFT 48 5.2 The Bethe/gauge correspondence 51 A. Completeness and the Yang-Yang function 56 B. Computations for the M-particle sector 58 C. Solving the FCR 63 References 65 PoS(Modave2014)001 A pedagogical introduction to quantum integrability Jules Lamers PoS(Modave2014)001 A pedagogical introduction to quantum integrability Jules Lamers Further references. Many important topics in quantum integrability are barely touched in these notes; examples include Baxter's T Q-method, the thermodynamic limit, correlation functions, and quantum groups. Luckily the literature on quantum-integrable models is extensive, ranging from introductory texts to very technical papers. The following references, here ordered alphabetically, have been useful for preparing these notes: • The renowned book by Baxter [1] gives a very detailed account of the CBA and the T Qmethod for several quantum-integrable models in statistical mechanics, including the sixvertex model. The notation is perhaps a bit old fashioned at times. • Faddeev's famous Les Houches lecture notes [3] provide a good basis for the ABA and the XXX model. Some familiarity with quantum integrability may be useful. • Gaudin's book [4] was recently translated into English. Amongst others the XXZ spin chain and the six-vertex model are treated using the CBA, and the thermodynamic limit is studied. • Chapters 1-3 of the book by Gómez, Ruiz-Altaba and Sierra [2] treat the CBA and ABA for the XXZ spin chain and the six-vertex model. The underlying quantum-algebraic structure is pointed out, though perhaps somewhat vaguely at times, and there are nice diagrammatic computations. • Chapters 0-2 of the book by Jimbo and Miwa [5] form a neat concise introduction to statistical physics, the XXZ spin chain and the six-vertex model. Although the ABA is not discussed, the QISM is essentially treated in Sections 2.4-3.3 and 3.7. • Karbach, Hu and Müller [6, 7] have written a nice three-part introduction to the CBA for the XXX model, including a discussion of the low-lying excitations in the physical spectrum for both the ferromagnetic and antiferromagnetic regime.

We present a new and simpler expression for the Hamiltonian of the partially isotropic (xxz-like) version of the Haldane-Shastry model, which was derived by D. Uglov over two decades ago in an apparently little-known preprint. While resembling the pairwise long-range form of the Haldane-Shastry model our formula accounts for the multi-spin interactions obtained by Uglov. Our expression is physically meaningful, makes hermiticity manifest, and is computationally more efficient. We discuss the model's properties, including its limits and (ordinary and quantum-affine) symmetries. In particular we introduce the appropriate notions of translational invariance and momentum. We review the model's exact spectrum found by Uglov for finite spin-chain length, which parallels the isotropic case up to level splitting due to the anisotropy. We also extend the partially isotropic model to higher rank, with SU (n) 'spins', for which the spectrum is determined by sln-motifs.where the products run over k and the harpoons specify the ordering. This expression can be understood as S [i,j]

For systems with infinite-order phase transitions, in which an order parameter smoothly becomes nonzero, a new observable for finite-size scaling analysis is suggested. By construction this new observable has the favourable property of diverging at the critical point. Focussing on the example of the F -model we compare the analysis of this observable with that of another observable, which is also derived from the order parameter but does not diverge, as well as that of the associated susceptibility. We discuss the difficulties that arise in the finite-size scaling analysis of such systems. In particular we show that one may reach incorrect conclusions from large-system size extrapolations of observables that are not known to diverge at the critical point. Our work suggests that one should base finite-size scaling analyses for infinite-order phase transitions only on observables that are guaranteed to diverge.