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Banach Center Publications

Bethe Ansatz, Galois symmetries, and finite quantum systems

Tadeusz Lulek

Published Jan 1, 2016• 1 citation
Publisher

Abstract

We demonstrate some applications of arithmetic structures in Bethe Ansatz — the famous substitution which yields an exact solution of a quantum $N$-body problem for the linear magnetic ring of $N$ spins $1/2$ within the XXX model. We point out the purely

DOI
10.4064/bc109-0-7

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We demonstrate some applications of arithmetic structures in Bethe Ansatz — the famous substitution which yields an exact solution of a quantum NNN-body problem for the linear magnetic ring of NNN spins 1/21/21/2 within the XXX model. We point out the purely
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.
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.
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.