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Journal of Physics A Mathematical and Theoretical

q -bosons, Toda lattice, Pieri rules and Baxter q -operator*

Antoine Duval, Vincent Pasquier

Published Mar 8, 2016• 5 citations
Publisher

Abstract

Abstract We use the Pieri rules to recover the q -boson model and show it is equivalent to a discretized version of the relativistic Toda chain. We identify its semi infinite transfer matrix and the corresponding Baxter Q -matrix with half vertex operators related by an ω -duality transformation. We observe that the scalar product of two higher spin XXZ wave functions can be expressed with a Gaudin determinant.

DOI
10.1088/1751-8113/49/15/154006

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Abstract We use the Pieri rules to recover the q -boson model and show it is equivalent to a discretized version of the relativistic Toda chain. We identify its semi infinite transfer matrix and the corresponding Baxter Q -matrix with half vertex operators related by an ω -duality transformation. We observe that the scalar product of two higher spin XXZ wave functions can be expressed with a Gaudin determinant.
We construct the Baxter operator [Formula: see text] for the [Formula: see text]-Toda chain and the Toda2 chain (the Toda chain in the second Hamiltonian structure). Our construction builds on the relation between the Baxter operator and Bäcklund transformations that were unravelled in [13]. We construct a number of quantum intertwiners ensuring the commutativity of [Formula: see text] with the transfer matrix of the models and of the [Formula: see text]’s between each other. Most importantly, [Formula: see text] is modular invariant in the sense of Faddeev. We derive the Baxter equation for the eigenvalues [Formula: see text] of [Formula: see text] and show that these are entire functions of [Formula: see text]. This last property will ultimately lead to the quantization of the spectrum for the considered Toda chains, in a subsequent publication [1]. This work is dedicated to the memory of L. D. Faddeev
Matrix-product operators (MPOs) appear throughout the study of integrable lattice models, notably as the transfer matrices. They can also be used as transformations to construct dualities between such models, both invertible (including unitary) and non-invertible (including discrete gauging). We analyse how the local Yang--Baxter integrable structures are modified under such dualities. We see that the Rˇ\check{R}Rˇ-matrix, that appears in the baxterization approach to integrability, transforms in a simple manner. We further argue for a broad class of MPO dualities that the usual Yang--Baxter RRR-matrix should be extended. This extended RRR-matrix satisfies modified algebraic relations, including a modified Yang--Baxter algebra previously identified in the unitary case, that altogether give a local integrable structure underlying the commuting transfer matrices of the dual model. We illustrate these results with two case studies, analysing an invertible unitary MPO and a non-invertible MPO both applied to the canonical XXZ spin chain. The former is the cluster entangler, arising in the study of symmetry-protected topological phases, while the latter is the Kramers--Wannier duality. We show several results for MPOs with exact MPO inverses that are of independent interest.
We implement the quantum inverse scattering method for the 4-vertex model. In comparison to previous works of the author which examined the 6-vertex, and 20-vertex, models, the 4-vertex model exhibits different characteristics, ranging from L-operators expressed in terms of projectors and Pauli matrices to algebraic and combinatorial properties, including Poisson structure and boxed plane partitions. With far fewer computations with an L-operator provided for the 4-vertex model by Bogoliubov in 2007, in comparison to those for L-operators of the 6, and 20, vertex models, from lower order expansions of the transfer matrix we derive a system of relations from the structure of operators that can be leveraged for studying characteristics of the higher-spin XXX chain in the weak finite volume limit. In comparison to quantum inverse scattering methods for the 6, and 20, vertex models which can be used to further study integrability, and exact solvability, an adaptation of such an approach for the 4-vertex model can be used to approximate, asymptotically in the weak finite volume limit, sixteen brackets which generate the Poisson structure. From explicit relations for operators of the 4-vertex transfer matrix, we conclude by discussing corresponding aspects of the Yang-Baxter algebra, which is closely related to the operators obtained from products of L-operators for approximating the transfer, and quantum monodromy, matrices. The structure of computations from L-operators of the 4-vertex model directly transfers to L-operators of the higher-spin XXX chain, revealing a similar structure of another Yang-Baxter algebra of interest.
In this paper we review the theory of the Yang–Baxter equation related to the 6-vertex model and its higher spin generalizations. We employ a 3D approach to the problem. Starting with the 3D R-matrix, we consider a two-layer projection of the corresponding 3D lattice model. As a result, we obtain a new expression for the higher spin R-matrix associated with the affine quantum algebra Uq(sl(2)ˆ). In the simplest case of the spin s=1/2 this R-matrix naturally reduces to the R-matrix of the 6-vertex model. Taking a special limit in our construction we also obtain new formulas for the Q-operators acting in the representation space of arbitrary (half-)integer spin. Remarkably, this construction can be naturally extended to any complex values of spin s. We also give all functional equations satisfied by the transfer-matrices and Q-operators.