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arXiv:1201.5627 · Physical Review B

Algebraic Bethe ansatz and tensor networks

V. Murg, V. E. Korepin, Frank Verstraete

Published Jul 20, 2012• 45 citations
arXivHTMLPDF

Abstract

The algebraic Bethe ansatz is a prosperous and well-established method for solving one-dimensional quantum models exactly. The solution of the complex eigenvalue problem is thereby reduced to the solution of a set of algebraic equations. Whereas the spectrum is usually obtained directly, the eigenstates are available only in terms of complex mathematical expressions. This makes it very hard, in general, to extract properties from the states, for example, correlation functions. In our work, we apply the tools of tensor-network states to describe the eigenstates approximately as matrix product states. From the matrix product state expression, we then obtain observables like the structure factor, dimer-dimer correlation functions, chiral correlation functions, and one-particle Green function directly.

DOI
10.1103/physrevb.86.045125

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The algebraic Bethe ansatz is a prosperous and well-established method for solving one-dimensional quantum models exactly. The solution of the complex eigenvalue problem is thereby reduced to the solution of a set of algebraic equations. Whereas the spectrum is usually obtained directly, the eigenstates are available only in terms of complex mathematical expressions. This makes it very hard, in general, to extract properties from the states, for example, correlation functions. In our work, we apply the tools of tensor-network states to describe the eigenstates approximately as matrix product states. From the matrix product state expression, we then obtain observables like the structure factor, dimer-dimer correlation functions, chiral correlation functions, and one-particle Green function directly.
We consider several exactly solvable models of strongly correlated electrons in one dimension, such as the Heisenberg XXX model, the supersymmetric t-J model and the Hubbard model. These models can be solved by using the method of graded algebraic Bethe ansatz. We use it to design graded tensor networks which can be contracted approximately to obtain a Matrix Product State. This overcomes a major shortcoming of current density matrix renormalization group (DMRG) methods which work well on the ground states, but have difficulty working with the excited states of such models. In addition, observables such as correlation functions are important as they are experimentally measurable, but have been analytically described in the double scaling limit only. Moreover, these analytical results are mostly expressed in the form of determinants, which are numerically inefficient to compute. With the tensor network description of the spin models, we can efficiently compute any expectation value of the eigenstates on finite length lattices for direct comparison with laboratory results. As a proof of principle, we calculate correlation functions of ground states and excited states of such models on finite lattices of lengths in an intermediate regime which are of experimental interest. | 108 pages
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.
The Bethe ansatz in its several formulations is a common tool for the exact solution of one-dimensional quantum Hamiltonians. This ansatz asserts that several eigenfunctions of the Hamiltonians are given in terms of a sum of permutations of plane waves. We present results that induce us to expect that, alternatively, the eigenfunctions of all the exact integrable quantum chains can also be expressed by a matrix product ansatz. In this ansatz several components of the eigenfunctions are obtained through the algebraic properties of properly defined matrices. This ansatz allows an unified formulation of several exact integrable Hamiltonians. We show how to formulate this ansatz for a large family of quantum chains such as the anisotropic Heisenberg model, Fateev–Zamolodchikov model, Izergin–Korepin model, Sutherland model, t – J model, Hubbard model, etc.
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.
Abstract We give a pedagogical introduction to the Bethe ansatz techniques in integrable QFTs and spin chains. We first discuss and motivate the general framework of asymptotic Bethe ansatz for the spectrum of integrable QFTs in large volume, based on the exact S -matrix. Then we illustrate this method in several concrete theories. The first case we study is the SU (2) chiral Gross–Neveu model. We derive the Bethe equations via algebraic Bethe ansatz, solving in the process the Heisenberg XXX spin chain. We discuss this famous spin chain model in some detail, covering in particular the coordinate Bethe ansatz, some properties of Bethe states, and the classical scaling limit leading to finite-gap equations. Then we proceed to the more involved SU (3) chiral Gross–Neveu model and derive the Bethe equations using nested algebraic Bethe ansatz to solve the arising SU (3) spin chain. Finally we show how a method similar to the Bethe ansatz works in a completely different setting, namely for the 1D oscillator in quantum mechanics.