The nineteen-vertex model and alternating sign matrices

We reconsider the open XXZ chain of Yang and Fendley. This model possesses a lattice supersymmetry which changes the length of the chain by one site. We perform an algebraic Bethe ansatz analysis of the model and derive the commutation relations of the lattice SUSY operators with the four elements of the open-chain monodromy matrix. Hence we give the action of the SUSY operator on off-shell and on-shell Bethe states. We show that this action generally takes one on-shell Bethe eigenstate to another. The exception is that a zero-energy vacuum state will be a SUSY singlet. The SUSY pairings of Bethe roots we obtain are analogous to those found previously for closed chains by Fendley and Hagendorf by analysing the Bethe equations.

Section: Introductionmentioning

confidence: 99%

Lattice supersymmetry in the open XXZ model: an algebraic Bethe Ansatz analysis

Weston¹,

Yang²

2017

“…(81), (104)- (106) and (107), at two different values of the spectral parameter. We see that for the special value λ = m/2l, within the domain (70), the numerical values for finite systems nicely approach the thermodynamic limit (114). On the other hand, for λ outside the domain (70) the operator Q(λ) is no longer quasi-local and the Mazur bound approaches zero for N → ∞.…”

Section: Mazur Bound For the Spin Drude Weightmentioning

confidence: 76%

Quasi-local conserved charges and spin transport in spin-1 integrable chains

Piroli

Vernier

2016

We consider the integrable one-dimensional spin-1 chain defined by the Zamolodchikov-Fateev (ZF) Hamiltonian. The latter is parametrized, analogously to the XXZ spin-1/2 model, by a continuous anisotropy parameter and at the isotropic point coincides with the well-known spin-1 Babujian-Takhtajan Hamiltonian. Following a procedure recently developed for the XXZ model, we explicitly construct a continuous family of quasi-local conserved operators for the periodic spin-1 ZF chain. Our construction is valid for a dense set of commensurate values of the anisotropy parameter in the gapless regime where the isotropic point is excluded. Using the Mazur inequality, we show that, as for the XXZ model, these quasi-local charges are enough to prove that the high-temperature spin Drude weight is non-vanishing in the thermodynamic limit, thus establishing ballistic spin transport at high temperature.

“…This is discussed in Section 3.5.3, with the normalisation of |φ D defined by (3.41) and (3.78). With this convention, it was shown in [27] that all components of |φ D are polynomials in x with integer coefficients. The following result was also proven in [27].…”

Section: Scalar Productsmentioning

confidence: 99%

“…In this section, we initiate the derivation of the results presented in Section 2. To this end, we employ the same strategy that was used in [27] for the diagonal twist: We generalise the problem and investigate the inhomogeneous transfer matrix of the corresponding nineteen-vertex model.…”

Section: The Nineteen-vertex Modelmentioning

confidence: 99%

Symmetry classes of alternating sign matrices in a nineteen-vertex model

Hagendorf¹,

Morin-Duchesne²

2016

Self Cite

The nineteen-vertex model on a periodic lattice with an anti-diagonal twist is investigated. Its inhomogeneous transfer matrix is shown to have a simple eigenvalue, with the corresponding eigenstate displaying intriguing combinatorial features. Similar results were previously found for the same model with a diagonal twist. The eigenstate for the antidiagonal twist is explicitly constructed using the quantum separation of variables technique. A number of sum rules and special components are computed and expressed in terms of Kuperberg's determinants for partition functions of the inhomogeneous six-vertex model. The computations of some components of the special eigenstate for the diagonal twist are also presented. In the homogeneous limit, the special eigenstates become eigenvectors of the Hamiltonians of the integrable spin-one XXZ chain with twisted boundary conditions. Their sum rules and special components for both twists are expressed in terms of generating functions arising in the weighted enumeration of various symmetry classes of alternating sign matrices (ASMs). These include half-turn symmetric ASMs, quarter-turn symmetric ASMs, vertically symmetric ASMs, vertically and horizontally perverse ASMs and double U-turn ASMs. As side results, new determinant and pfaffian formulas for the weighted enumeration of various symmetry classes of alternating sign matrices are obtained.