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Derivation of Matrix Product Ansatz for the Heisenberg Chain from Algebraic Bethe Ansatz

Hosho Katsura, Isao Maruyama

Published Jan 1, 2010• 37 citations
OpenAlex

Abstract

We derive a matrix product representation of the Bethe ansatz state for the XXX and XXZ spin-1/2 Heisenberg chains using the algebraic Bethe ansatz. In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices which act on ${\bar {\mathscr H}}$, the tensor product of auxiliary spaces. By changing the basis in ${\bar {\mathscr H}}$, we derive explicit finite-dimensional representations for the matrices. These matrices are the same as those appearing in the recently proposed matrix product ansatz by Alcaraz and Lazo [Alcaraz F C and Lazo M J 2006 {\it J. Phys. A: Math. Gen.} \textbf{39} 11335.] apart from normalization factors. We also discuss the close relation between the matrix product representation of the Bethe eigenstates and the six-vertex model with domain wall boundary conditions [Korepin V E 1982 {\it Commun. Math. Phys.}, \textbf{86} 391.] and show that the change of basis corresponds to a mapping from the six-vertex model to the five-vertex model.

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Scalar Products of the Asymmetric XXZ Chain

Yasuhiro Fujii, Miki Wadati

The asymmetric XXZ chain is studied in the framework of the algebraic Bethe ansatz. The model is one of the Heisenberg XXZ spin chains with Dzyaloshinsky–Moriya interactions and is mapped onto the asymmetric six-vertex model. We calculate the scalar products that constitute various correlation functions of the model, by the Korepin's method. They are expressed in terms of determinants of the dual quantum fields.

Journal of the Physical Society of Japan•Jul 15, 1999•3 citations
We derive a matrix product representation of the Bethe ansatz state for the XXX and XXZ spin-1/2 Heisenberg chains using the algebraic Bethe ansatz. In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices which act on Hˉ{\bar {\mathscr H}}Hˉ, the tensor product of auxiliary spaces. By changing the basis in Hˉ{\bar {\mathscr H}}Hˉ, we derive explicit finite-dimensional representations for the matrices. These matrices are the same as those appearing in the recently proposed matrix product ansatz by Alcaraz and Lazo [Alcaraz F C and Lazo M J 2006 {\it J. Phys. A: Math. Gen.} \textbf{39} 11335.] apart from normalization factors. We also discuss the close relation between the matrix product representation of the Bethe eigenstates and the six-vertex model with domain wall boundary conditions [Korepin V E 1982 {\it Commun. Math. Phys.}, \textbf{86} 391.] and show that the change of basis corresponds to a mapping from the six-vertex model to the five-vertex model.
We derive a matrix product representation of the Bethe ansatz state for the XXX and XXZ spin-1 2 Heisenberg chains using the algebraic Bethe ansatz.In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices which act on H, the tensor product of auxiliary spaces.By changing the basis in H, we derive explicit finite-dimensional representations for the matrices.These matrices are the same as those appearing in the recently proposed matrix product ansatz by Alcaraz and Lazo (2006 J. Phys.A: Math.Gen. 39 11335) apart from normalization factors.We also discuss the close relation between the matrix product representation of the Bethe eigenstates and the six-vertex model with domain wall boundary conditions (Korepin 1982 Commun.Math.Phys.86 391) and show that the change of basis corresponds to a mapping from the six-vertex model to the five-vertex model.
Using the algebraic Bethe Ansatz, we derive a matrix product representation of the exact Bethe-Ansatz states of the six-vertex Heisenberg chain (either XXX or XXZ and spin-1/2) with open boundary conditions. In this representation, the components of the Bethe eigenstates are expressed as traces of products of matrices that act on a tensor product of auxiliary spaces. As compared to the matrix product states of the same Heisenberg chain but with periodic boundary conditions, the dimension of the exact auxiliary matrices is enlarged as if the conserved number of spin-flips considered would have been doubled. This result is generic for any non-nested integrable model, as is clear from our derivation, and we further show this by providing an additional example of the same matrix product state construction for a well-known model of a gas of interacting bosons. Counterintuitively, the matrices do not depend on the spatial coordinate despite the open boundaries, and thus they suggest generic ways of exploiting (emergent) translational invariance both for finite size and in the thermodynamic limit.
The Bethe Ansatz is a method for constructing exact eigenstates of quantum-integrable spin chains. Recently, deterministic quantum algorithms, referred to as “algebraic Bethe circuits”, have been developed to prepare Bethe states for the spin- 1/2 1 / 2 XXZ model. These circuits represent a unitary formulation of the standard algebraic Bethe Ansatz, expressed using matrix-product states that act on both the spin chain and an auxiliary space. In this work, we systematize these previous results, and show that algebraic Bethe circuits can be derived by a change of basis in the auxiliary space. The new basis, identical to the “F-basis” known from the theory of quantum-integrable models, generates the linear superposition of plane waves that is characteristic of the coordinate Bethe Ansatz. We explain this connection, highlighting that certain properties of the F-basis (namely, the exchange symmetry of the spins) are crucial for the construction of algebraic Bethe circuits. We demonstrate our approach by presenting new quantum circuits for the inhomogeneous spin- 1/2 1 / 2 XXZ model.
Recently it was shown that the eigenfunctions for the the asymmetric exclusion problem and several of its generalizations as well as a huge family of quantum chains, like the anisotropic Heisenberg model, Fateev- Zamolodchikov model, Izergin-Korepin model, Sutherland model, t-J model, Hubbard model, etc, can be expressed by a matrix product ansatz. Differently from the coordinate Bethe ansatz, where the eigenvalues and eigenvectors are plane wave combinations, in this ansatz the components of the eigenfunctions are obtained through the algebraic properties of properly defined matrices. In this work, we introduce a formulation of a matrix product ansatz for the six-vertex model with periodic boundary condition, which is the paradigmatic example of integrability in two dimensions. Remarkably, our studies of the six-vertex model are in agreement with the conjecture that all models exactly solved by the Bethe ansatz can also be solved by an appropriated matrix product ansatz.
The spectra of recently constructed auxiliary matrices for the six-vertex model respectively the spin s = 1/2 Heisenberg chain at roots of unity are investigated. Two conjectures are formulated both of which are proved for N = 3 and are verified numerically for several examples with N > 3. The first conjecture identifies an Abelian subset of auxiliary matrices whose eigenvalues are polynomials in the spectral variable. The zeros of these polynomials are shown to fall into two sets. One consists of the solutions to the Bethe ansatz equations which determine the eigenvalues of the six-vertex transfer matrix. The other set of zeros contains the complete strings which encode the information on the degeneracies of the model due to the loop symmetry present at roots of 1. The second conjecture then states a polynomial identity which relates the complete string centres to the Bethe roots allowing one to determine the dimension of the degenerate eigenspaces. Its proof for N = 3 involves the derivation of a new functional equation for the auxiliary matrices and the six-vertex transfer matrix. Moreover, it is demonstrated in several explicit examples that the complete strings coincide with the classical analogue of the Drinfeld polynomial. The latter is used to classify the finite-dimensional irreducible representations of the loop algebra . This suggests that the constructed auxiliary matrices not only enable one to solve the six-vertex model but also completely characterize the decomposition of its eigenspaces into irreducible representations of the underlying loop symmetry.