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arXiv:0802.3802 · Brazilian Journal of Physics

The Matrix Product Ansatz for integrable U(1)N models in Lunin-Maldacena backgrounds

Matheus J. Lazo

Published Jun 1, 2008• 2 citations
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Abstract

We obtain through a Matrix Product Ansatz (MPA) the exact solution of the most general N-state spin chain with U(1)N symmetry and nearest neighbour interaction. In the case N = 6 this model contain as a special case the integrable SO(6) spin chain related to the one loop mixing matrix for anomalous dimensions in N = 4 SYM, dual to type IIB string theory in the generalised Lunin-Maldacena backgrounds. This MPA is construct by a map between scalar fields and abstract operators that satisfy an appropriate associative algebra. We analyses the Yang-Baxter equation in the N = 3 sector and the consistence of the algebraic relations among the matrices defining the MPA and find a new class of exactly integrable model unknown up to now.

DOI
10.1590/s0103-97332008000200005

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We obtain through a Matrix Product Ansatz (MPA) the exact solution of the most general N-state spin chain with U(1)N symmetry and nearest neighbour interaction. In the case N = 6 this model contain as a special case the integrable SO(6) spin chain related to the one loop mixing matrix for anomalous dimensions in N = 4 SYM, dual to type IIB string theory in the generalised Lunin-Maldacena backgrounds. This MPA is construct by a map between scalar fields and abstract operators that satisfy an appropriate associative algebra. We analyses the Yang-Baxter equation in the N = 3 sector and the consistence of the algebraic relations among the matrices defining the MPA and find a new class of exactly integrable model unknown up to now.
We obtain through a Matrix Product Ansatz (MPA) the exact solution of the most general N-state spin chain with U(1)N symmetry and nearest neighbour interaction. In the case N = 6 this model contain as a special case the integrable SO(6) spin chain related to the one loop mixing matrix for anomalous dimensions in N =4 SYM, dual to type IIB string theory in the generalised Lunin-Maldacena backgrounds. This MPA is construct by a map between scalar fields and abstract operators that satisfy an appropriate associative algebra. We analyses the Yang-Baxter equation in the N = 3 sector and the consistence of the algebraic relations among the matrices defining the MPA and find a new class of exactly integrable model unknown up to now.
We obtain through a Matrix Product Ansatz the exact solution of the most general inhomogeneous spin chain with nearest neighbor interaction and with U(1)² and U(1)³ symmetries. These models are related to the one loop mixing matrix of the Leigh-Strassler deformed N = 4 SYM theory, dual to type IIB string theory in the generalized Lunin-Maldacena backgrounds, in the sectors of two and three kinds of fields, respectively. The solutions presented here generalizes the results obtained by the author in a previous work for homogeneous spins chains with U(1)N symmetries in the sectors of N = 2 and N = 3.
Most of the exact solutions of quantum one-dimensional Hamiltonians are obtained thanks to the success of the Bethe ansatz on its several formulations. According to this ansatz, the amplitudes of the eigenfunctions of the Hamiltonian are given by a sum of permutations of appropriate plane waves. In this paper, alternatively, we present a matrix product ansatz that asserts that those amplitudes are given in terms of a matrix product. The eigenvalue equation for the Hamiltonian defines the algebraic properties of the matrices defining the amplitudes. The consistency of the commutativity relations among the elements of the algebra implies the exact integrability of the model. The matrix product ansatz we propose allows an unified and simple formulation of several exact integrable Hamiltonians. In order to introduce and illustrate this ansatz we present the exact solutions of several quantum chains with one and two global conservation laws and periodic boundaries such as the XXZ chain, spin-1 Fateev–Zamolodchikov model, Izergin–Korepin model, Sutherland model, t – J model, Hubbard model, etc. Formulation of the matrix product ansatz for quantum chains with open ends is also possible. As an illustration we present the exact solution of an extended XXZ chain with z -magnetic fields at the surface and arbitrary hard-core exclusion among the spins.
Proposed is a method to construct a stationary state of one-dimensional integrable systems in the form of products of matrices. This is a generalization of the so-called “matrix product ansatz (MPA)”. The key idea is that the matrices are chosen to constitute the Zamolodchikov-Faddeev algebra (ZF-algebra) for the R-matrix and the K-matrix, which are the solutions of the Yang-Baxter equation (YBE) and the reflection equation (RE). It is shown that a matrix product state gives a simultaneous stationary state of commuting operators which are expressed in terms of the R-matrices and the K-matrices. As an example, a solution for the isotropic Heisenberg spin chain with non-diagonal boundary fields is given. The connection to the conventional MPA is clarified. Applications to other models and the relationship to the algebraic Bethe ansatz (ABA) are also discussed.
Abstract We classify all fundamental integrable spin chains with two-dimensional (2D) local Hilbert space which have regular R -matrices of difference form. This means that the R -matrix underlying the integrable structures is of the form and reduces to the permutation operator at some particular point. We find a total of 14 independent solutions, eight of which correspond to well-known eight or lower vertex models. The remaining six models appear to be new and some have peculiar properties such as not being diagonalizable or being nilpotent. Furthermore, for even R -matrices, we find a bijection between solutions of the Yang–Baxter equation (YBE) and the graded YBE which extends our results to the graded 2D case.
We consider integrable Matrix Product States (MPS) in integrable spin chains and show that they correspond to “operator valued” solutions of the so-called twisted Boundary Yang-Baxter (or reflection) equation. We argue that the integrability condition is equivalent to a new linear intertwiner relation, which we call the “square root relation”, because it involves half of the steps of the reflection equation. It is then shown that the square root relation leads to the full Boundary Yang-Baxter equations. We provide explicit solutions in a number of cases characterized by special symmetries. These correspond to the “symmetric pairs” (SU(N),SO(N)) and (SO(N),SO(D) \otimes ⊗ SO(N-D)), where in each pair the first and second elements are the symmetry groups of the spin chain and the integrable state, respectively. These solutions can be considered as explicit representations of the corresponding twisted Yangians, that are new in a number of cases. Examples include certain concrete MPS relevant for the computation of one-point functions in defect AdS/CFT.