RIGGED STRING CONFIGURATIONS, BETHE ANSATZ QUBITS, AND CONSERVATION OF PARITY

摘要

Bethe Ansatz solutions for the Heisenberg Hamiltonian of a one - dimensional magnetic ring of N nodes, each with the spin 1/2, within the XXX model,have been presented as some composite systems, in a spirit of quantum information theory. The constituents are single - node spin states, which organize into strings of various length, and "seas of holes". The former are responsible for dynamics, whereas the latter determine the range of riggings for strings. Another aim was to demonstrate a unification of Bethe Ansatz eigenstates by means of Galois symmetries of finite field extensions. The key observation is that the original eigenproblem is expressible in integers, and thus, for a finite fixed N, the splitting field of the characteristic polynom of the Heisenberg Hamiltonian is also finite. The Galois group of the latter field permutes, by definition, roots of this polynom, which implies permutation of eigenstates. General considerations are demonstrated on the example of heptagon(N= 7),which admits an implementation of a collection of arithmetic qubits, and also demonstrates a special case of degeneration of the spectrum of the Hamiltonian, resulting from conservation of parity, within the realm of rigged string configurations.