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arXiv:1010.1274 · Journal of Physics A Mathematical and Theoretical

The Yang–Baxter equation for \mathcal {PT} invariant 19-vertex models

Rodrigo A. Pimenta, M J Martins

Published Feb 4, 2011• 16 citations
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Abstract

We study the solutions of the Yang–Baxter equation associated with 19-vertex models invariant by the parity-time symmetry from the perspective of algebraic geometry. We determine the form of the algebraic curves constraining the respective Boltzmann weights and find that they possess a universal structure. This allows us to classify the integrable manifolds into four different families reproducing three known models, besides uncovering a novel 19-vertex model in a unified way. The introduction of the spectral parameter on the weights is made via the parameterization of the fundamental algebraic curve which is a conic. The diagonalization of the transfer matrix of the new vertex model and its thermodynamic limit properties are discussed. We point out a connection between the form of the main curve and the nature of the excitations of the corresponding spin-1 chains.

DOI
10.1088/1751-8113/44/8/085205

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We investigate the Yang-Baxter algebra for U(1)\mathrm{U}(1)U(1) invariant three-state vertex models whose Boltzmann weights configurations break explicitly the parity-time reversal symmetry. We uncover two families of regular Lax operators with nineteen non-null weights which ultimately sit on algebraic plane curves with genus five. We argue that these curves admit degree two morphisms onto elliptic curves and thus they are bielliptic. The associated R\mathrm{R}R-matrices are non-additive in the spectral parameters and it has been checked that they satisfy the Yang-Baxter equation. The respective integrable quantum spin-1 Hamiltonians are exhibited.
We have found a family of solvable nineteen vertex model with statistical configurations invariant by the time reversal symmetry within a systematic study of the respective Yang–Baxter relation. The Boltzmann weights sit on a degree seven algebraic threefold which is shown birationally equivalent to the three-dimensional projective space. This permits to write parameterized expressions for both the transition operator and the R-matrix depending on three independent affine spectral parameters. The Hamiltonian limit tells us that the azimuthal magnetic field term is connected with the asymmetry among two types of spectral variables. The absence of magnetic field defines a physical submanifold whose geometrical properties are remarkably shown to be governed by a quartic K3 surface. This expands considerably the class of irrational manifolds that could emerge in the theory of quantum integrable models.
A new type of Yang-Baxter equation (YBE) for RRR-operators depending on three spectral parameters is constructed from the star-triangle relation for the chiral Potts model. As the ZNZ_NZN​ symmetric generalization to the Ising model, its Boltzmann weights are known to depend on two variables describing a curve with genus larger than one for N>2N>2N>2, except for the self-dual point corresponding to the Fateev-Zamolodchikov chain. Combined with the fact that quantum Hamiltonians of edge-type models such as the Ising model contain both nearest-neighbor interaction and onsite potential terms, this leads naturally to an additional spectral parameter in the associated RRR-operator. The construction extends the edge-vertex correspondence of solvable lattice models, and provides a bridge between the Bazhanov-Stroganov four-parameter RRR-matrix---realized as an intertwiner of cyclic representations of Uq(sl2)U_q(\mathfrak{sl}_2)Uq​(sl2​) at a root of unity---and Shastry's two-parameter RRR-operator obtained from the decorated YBE.
It has been recently discovered in the context of the six vertex or XXZ model in the fundamental representation that new symmetries arise when the anisotropy parameter (q+q−1)/2(q+q^{-1})/2(q+q−1)/2 is evaluated at roots of unity qN=1q^{N}=1qN=1. These new symmetries have been linked to an U(A1(1))U(A^{(1)}_1)U(A1(1)​) invariance of the transfer matrix and the corresponding spin-chain Hamiltonian.In this paper these results are generalized for odd primitive roots of unity to all vertex models associated with trigonometric solutions of the Yang-Baxter equation by invoking representation independent methods which only take the algebraic structure of the underlying quantum groups Uq(hatg)U_q(\\hat g)Uq​(hatg) into account. Here hatg\\hat ghatg is an arbitrary Kac-Moody algebra. Employing the notion of the boost operator it is then found that the Hamiltonian and the transfer matrix of the integrable model are invariant under the action of U(hatg)U(\\hat{g})U(hatg). For the simplest case hatg=A1(1)\\hat g=A_1^{(1)}hatg=A1(1)​ the discussion is also extended to even primitive roots of unity.
In this paper we study, in the manner of Baxter [R. J. Baxter Exactly Solved Models in Statistical Mechanics (Academic, New York, 1982)], the possible integrable manifolds of the asymmetric eight-vertex model. As expected, they occur when the Boltzmann weights are either symmetric or satisfy the free-fermion condition; but our analysis clarifies the reason why both manifolds need to share a universal invariant. We also show that the free-fermion condition implies three distinct classes of integrable models.
In this paper we investigate the mathematical properties of the integrability of the symmetric six-vertex model towards the view of algebraic geometry. We show that the algebraic variety originated from Baxter’s commuting transfer method is birationally isomorphic to a ubiquitous threefold known as Segre cubic primal. This relation makes it possible to present the most generic solution for the Yang–Baxter triple associated to this lattice model. The respective -matrix and Lax operators are parameterized by three independent affine spectral variables.