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Acta Physica Polonica A

Heisenberg and Bethe Field Extensions Applied to Magnetic Rings

Grzegorz Banaszak, Dorota Blinkiewicz, Piotr Krasoń, Jan Milewski

Published Mar 1, 2018• 1 citation
Publisher

Abstract

We consider striking connections between the theory of homogenous isotropic Heisenberg ring (XXX-model) and algebraic number theory.We explain the nature of these connections especially applications of Galois theory for computation of the spectrum of the Heisenberg operators and Bethe parameters.The solutions of the Heisenberg eigenproblem and Bethe Ansatz generate interesting families of algebraic number fields.Galois theory yields additional symmetries which not only simplify the analysis of the model but may lead to new applications and horizons.

DOI
10.12693/aphyspola.133.441

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We analyse the number eld-theoretic properties of solutions of the eigenproblem of the Heisenberg Hamiltonian for the magnetic hexagon with the single-node spin 1/2 and isotropic exchange interactions.It follows that eigenenergies and eigenstates are expressible within an extension of the prime eld Q of rationals of degree 2 3 and 2 4 , respectively.In quantum information setting, each real extension of rank 2 represents an arithmetic qubit.We demonstrate in detail some actions of the Galois group on the eigenproblem.
Abstract In this paper and upcoming ones, we initiate a systematic study of Bethe ansatz equations for integrable models by modern computational algebraic geometry. We show that algebraic geometry provides a natural mathematical language and powerful tools for understanding the structure of solution space of Bethe ansatz equations. In particular, we find novel efficient methods to count the number of solutions of Bethe ansatz equations based on Gröbner basis and quotient ring. We also develop analytical approach based on companion matrix to perform the sum of on-shell quantities over all physical solutions without solving Bethe ansatz equations explicitly. To demonstrate the power of our method, we revisit the completeness problem of Bethe ansatz of Heisenberg spin chain, and calculate the sum rules of OPE coefficients in planar N=4 \mathcal{N}=4 N=4 N = 4 super-Yang-Mills theory.
In this paper and upcoming ones, we initiate a systematic study of Bethe ansatz equations for integrable models by modern computational algebraic geometry. We show that algebraic geometry provides a natural mathematical language and powerful tools for understanding the structure of solution space of Bethe ansatz equations. In particular, we find novel efficient methods to count the number of solutions of Bethe ansatz equations based on Gröbner basis and quotient ring. We also develop analytical approach based on companion matrix to perform the sum of on-shell quantities over all physical solutions without solving Bethe ansatz equations explicitly. To demonstrate the power of our method, we revisit the completeness problem of Bethe ansatz of Heisenberg spin chain, and calculate the sum rules of OPE coefficients in planar N=4 super-Yang-Mills theory.