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arXiv:1410.6749 · Nuclear Physics B

An integrable nineteen vertex model lying on a hypersurface

Marcio Jose Martins

Published Jan 21, 2015• 6 citations
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Abstract

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.

DOI
10.1016/j.nuclphysb.2015.01.018

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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.
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 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.
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The quantum R matrix for Cartan's exceptional simple Lie algebra G 2 in its seven-dimensional (minimal) representation is presented. The R matrix is determined through the irreducible decomposition of the tensor modules over the quantised universal enveloping algebra U q (G 2 ). This defines a new solvable seven-state 175-vertex model on the planar square lattice whose Boltzmann weights satisfy the Yang-Baxter equation. For the equivalent face model, the local state probability is obtained in terms of a relative of a level-1 string function related to the affine Lie algebra pair B 3 (1) contains/implies G 2 (1) .
A connection between integrability properties and general statistical properties of the spectra of symmetric transfer matrices of the asymmetric eight-vertex model is studied using random matrix theory (eigenvalue spacing distribution and spectral rigidity). For Yang-Baxter integrable cases, including free-fermion solutions, we found a Poissonian behavior, whereas level repulsion close to the Wigner distribution is found for nonintegrable models. For the asymmetric eight-vertex model, however, the level repulsion can also disappear, and the Poisson distribution be recovered on (non Yang-Baxter integrable) algebraic varieties, the so-called disorder varieties. We also present an infinite set of algebraic varieties which are stable under the action of an infinite discrete symmetry group of the parameter space. These varieties are possible loci for free parafermions. Using our numerical criterion, we tested the generic calculability of the model on these algebraic varieties.
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.