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arXiv:0911.1881

Norm of Bethe Wave Function as a Determinant

V. E. Korepin

Published Nov 10, 2009• 2 citations
arXivHTMLPDF

Abstract

This is a historical note. Bethe Ansatz solvable models are considered, for example XXZ Heisenberg anti-ferromagnet and Bose gas with delta interaction. Periodic boundary conditions lead to Bethe equation. The square of the norm of Bethe wave function is equal to a determinant of linearized system of Bethe equations (determinant of matrix of second derivatives of Yang action). The proof was first published in Communications in Mathematical Physics, vol 86, page 391 in l982. Also domain wall boundary conditions for 6 vertex model were discovered in the same paper [see Appendix D]. These play an important role for algebraic combinatorics: alternating sign matrices, domino tiling and plane partition. Many publications are devoted to six vertex model with domain wall boundary conditions.

DOI
10.48550/arxiv.0911.1881

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This is a historical note. Bethe Ansatz solvable models are considered, for example XXZ Heisenberg anti-ferromagnet and Bose gas with delta interaction. Periodic boundary conditions lead to Bethe equation. The square of the norm of Bethe wave function is equal to a determinant of linearized system of Bethe equations (determinant of matrix of second derivatives of Yang action). The proof was first published in Communications in Mathematical Physics, vol 86, page 391 in l982. Also domain wall boundary conditions for 6 vertex model were discovered in the same paper [see Appendix D]. These play an important role for algebraic combinatorics: alternating sign matrices, domino tiling and plane partition. Many publications are devoted to six vertex model with domain wall boundary conditions.
Bethe Ansatz solvable models are considered, like XXZ Heisenberg\nanti-ferromagnet and Bose gas with delta interaction. Periodic boundary conditions lead to Bethe\nequation. The square of the norm of Bethe wave function is equal to a determinant of linearized\nsystem of Bethe equations (determinant of matrix of second derivatives of Yang action). The proof\nwas first published in Communications in Mathematical Physics, vol 86, page 391 in l982. Also\ndomain wall boundary conditions for 6 vertex model were discovered in the same paper [see\nAppendix D]. These play an important role for algebraic combinatorics: alternating sign matrices,\ndomino tiling and plane partition.
The algebraic Bethe ansatz is one of the main approaches to solving integrable models in statistical mechanics. This thesis is a detailed introduction to aspects of this technique. In chapter 1, following Takhtajan1, we fully explain all preliminaries that are required by this technique in the context of the 8-vertex model, including the solution of the Yang-Baxter equations in terms of Boltzmann weights that are parameterized by elliptic function. In chapter 2, also following Takhtajan, we restrict our attention to the special case of the 6-vertex model, with Boltzmann weights parameterized by hyperbolic trigonometric functions, formulate the model in detail on a finite lattice with periodic boundary conditions, and show how the algebraic Bethe ansatz can be used to characterize the eigenvectors and eigenvalues, in the homogeneous case (all vertical rapidities are equal, and all horizontal rapidities are also equal). In chapter 3, following Bogoliubov et al2, we consider the 6-vertex model in inhomogeneous case (all rapidities are now independenti variables) but with domain wall boundary conditions. Using the algebraic Bethe ansatz, and Izergin’s result for the partition function, we calculate boundary 1-point functions of the six-vertex model on a finite lattice and represent them in determinant form. Takhatajan, L. J. Introduction to Agebraic Bethe Ansatz, Lecture Notes in Physics, Exactly solvable problems in condensed matter and relativistic field theory 2 Boundary correlation functions of the six-vertex model J. Phys. A: Math. Gen. 35, 5525-5541 (2002)
The algebraic Bethe Ansatz is a powerful method for the calculation of quantum correlation functions. In Part III we shall start this calculation. We shall arrive at an extremely important conclusion: quantum correlation functions can be represented as determinants of certain matrices. The dimension of these matrices is equal to the number of particles in the ground state. In order to better understand the nature of this matrix, one should look once more through section 6 of Chapter VI and section 10 of Chapter VII, where the determinant representation of the partition function for the six-vertex model with a special type of boundary conditions (domain wall) was obtained. Starting from determinant formulæe (VII.10.1) and (VII.10.2) and using also the formula for the determinant of the sum of two matrices (in the Appendix to Chapter IX), one can reproduce all the determinant formulæe in Part III. In the thermodynamic limit, the correlation functions will be represented as determinants of a Predholm integral operator. In Part IV, we shall explain how to use this determinant representation to write down the differential equation for a quantum correlation function. In Part IV we shall also discover that the differential equation for a quantum correlation function is closely related to the initial classical nonlinear differential equation, which was quantized (in our example this is the nonlinear Schrödinger equation). The quantum correlation functions play the role of the τ functions of a classical differential equation.
In this note, we consider the six-vertex model with domain wall boundary conditions, defined on an M×M lattice, in the inhomogeneous case where the partition function depends on 2M inhomogeneities λj and μk. For a particular choice of the set of λj we find a new determinant representation for the partition function, which allows evaluation of the bulk free energy in the thermodynamic limit. This provides a new connection between two types of determinant formulas. We also show in a special case that spin correlations on the horizontal line going through the center coincide with the ones for periodic boundary conditions.
In this paper, we review a few known facts on the coordinate Bethe ansatz. We present a detailed construction of the Bethe ansatz vector $\psi $ and energy $\Lambda $, which satisfy $V\psi =\Lambda \psi $, where VVV is the transfer matrix of the six-vertex model on a finite square lattice with periodic boundary conditions for weights a=b=1a=b=1a=b=1 and c>0c>0c>0. We also show that the same vector $\psi $ satisfies $H\psi =E\psi $, where HHH is the Hamiltonian of the XXZ model (which is the model for which the Bethe ansatz was first developed), with a value EEE computed explicitly. Variants of this approach have become central techniques for the study of exactly solvable statistical mechanics models in both the physics and mathematics communities. Our aim in this paper is to provide a pedagogically-minded exposition of this construction, aimed at a mathematical audience. It also provides the opportunity to introduce the notation and framework which will be used in a subsequent paper by the authors [5] that amounts to proving that the random-cluster model on Z2\mathbb{Z}^{2}Z2 with cluster weight q>4q>4q>4 exhibits a first-order phase transition.
It is shown that the transfer matrix of the inhomogeneous nineteen-vertex model with certain diagonal twisted boundary conditions possesses a simple eigenvalue. This is achieved through the identification of a simple and completely explicit solution of its Bethe equations. The corresponding eigenvector is computed by means of the algebraic Bethe ansatz and both a simple component and its square norm are expressed in terms of the Izergin–Korepin determinant. In the homogeneous limit, the vector coincides with a supersymmetry singlet of the twisted spin-one XXZ chain. It is shown that in a natural polynomial normalisation scheme its square norm and the simple component coincide with generating functions for weighted enumeration of alternating sign matrices.