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Canadian Journal of Physics

A modified Bethe Ansatz for the two-dimensional dimer problem

Jeffrey Robert Schmidt

Published Jul 1, 2007• 5 citations
Publisher

Abstract

The two-dimensional dimer problem is studied using the tools of modern solvable models. The transfer matrix is constructed from local monodromy matrices, and is diagonalized by a modified Bethe Ansatz that does not begin with a Bethe ground state, but rather with a pair of states transformed one into the other by the transfer matrix. The resulting Bethe eigenvectors have a structure very different from those constructed in other models from a ground state, and have some properties in common with the free-fermion eigenvectors of the squared transfer matrix encountered in the literature. PACS Nos.: 05.50+q, 02.10.Ox, 02.30.Ik, 05.20.–y

DOI
10.1139/p07-075

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The two-dimensional dimer problem is studied using the tools of modern solvable models. The transfer matrix is constructed from local monodromy matrices, and is diagonalized by a modified Bethe Ansatz that does not begin with a Bethe ground state, but rather with a pair of states transformed one into the other by the transfer matrix. The resulting Bethe eigenvectors have a structure very different from those constructed in other models from a ground state, and have some properties in common with the free-fermion eigenvectors of the squared transfer matrix encountered in the literature. PACS Nos.: 05.50+q, 02.10.Ox, 02.30.Ik, 05.20.–y
Introduction The algebraic Bethe Ansatz is presented in this chapter. This is an important generalization of the coordinate Bethe Ansatz presented in Part I, and is one of the essential achievements of QISM. The algebraic Bethe Ansatz is based on the idea of constructing eigenfunctions of the Hamiltonian via creation and annihilation operators acting on a pseudovacuum. The matrix elements of the monodromy matrix play the role of these operators. The transfer matrix (the sum of the diagonal elements of the monodromy matrix) commutes with the Hamiltonian; thus constructing eigenfunctions of τ(µ) determines the eigenfunctions of the Hamiltonian. The basis of the algebraic Bethe Ansatz is stated in section 1. The commutation relations between matrix elements of the monodromy matrix are specified by the R-matrix. The explicit form of the commutation relations allows the construction of eigenfunctions of the transfer matrix (the trace of the monodromy matrix). (Recall that the Hamiltonian may also be obtained from the transfer matrix via the trace identities.) Further developments of the algebraic Bethe Ansatz necessary for the computation of correlation functions are given in section 2. The general scheme is illustrated with some examples in section 3. The NS model, the sine-Gordon model and spin models are considered in detail. The Pauli principle for interacting one-dimensional bosons plays an important role in constructing the ground state of the system and is discussed in section 4. The eigenvalues of the shift operator acting on the monodromy matrix are calculated in section 5.
A crash course on the Yang–Baxter equation and its applications in the statistical mechanics of lattice vertex models is presented. Using the simple example of a one-dimensional lattice gas, the basic terminology and standard mathematical procedures of statistical mechanics are illustrated. The coproduct notation is introduced via a discussion of the two-dimensional dimer model, and the transfer matrix formulation is elaborated further. The algebraic Bethe Ansatz (ABA) is introduced in the context of the ice model, leading to the Yang–Baxter equations. The motivation for introducing the ABA is developed, emphasizing the similarity to the quantum oscillator problem, in features such as the generation of “excited states”, and the determination of the eigenvalue spectrum. Finally, these methods from the theory of lattice models are applied to the case of a quantum many-body problem, the Heisenberg chain. From the condition that the monodromy matrices satisfy the Yang–Baxter equation, a complete set of mutually commuting operators is derived, and the Heisenberg chain problem is completely solved.PACS Nos.: 02.20.Uw, 24.10.Cn, 87.10.Hk, 05.20.–y
A general method is proposed for constructing the Bethe ansatz equations of integrable models without U(1) symmetry. As an example, the exact spectrum of the XXZ spin ring with a Möbius-like topological boundary condition is derived by constructing a modified T-Q relation based on the functional connection between the eigenvalues of the transfer matrix and the quantum determinant of the monodromy matrix. With the exact solution, the elementary excitations of the topological XX spin ring are discussed in detail. It is found that the excitation spectrum indeed shows a nontrivial topological nature.
The algebraic Bethe Ansatz is carried out for models (dimer models on a hexagonal lattice) whose matrix of local Boltzmann weights fails to satisfy the Yang–Baxter equation. The operator algebra is constructed instead by a generalizable alternative process using only the coproduct decomposition of the operators. The new technique makes it possible to solve many lattice N-body models whose local Boltzmann weights fail to meet the algebraic requirements of the Yang–Baxter equation.PACS Nos.: 02.20.Uw, 24.10.Cn, 87.10.Hk, 05.20.–y
We study integrable models solvable by the nested algebraic Bethe ansatz and possessing the GL(3)-invariant R-matrix.We consider a composite model where the total monodromy matrix of the model is presented as a product of two partial monodromy matrices.Assuming that the last ones can be expanded into series with respect to the inverse spectral parameter we calculate matrix elements of the local operators in the basis of the transfer matrix eigenstates.We obtain determinant representations for these matrix elements.Thus, we solve the inverse scattering problem in a weak sense.
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.