Quantum Spin Chains and Riemann Zeta Function with Odd Arguments

TL;DR

This paper explores the connection between the Riemann zeta function with odd arguments and the probability of ferromagnetic string formation in the ground state of the XXX Heisenberg spin 1/2 antiferromagnet, revealing a novel link between number theory and integrable models.

Contribution

It demonstrates that the probability of ferromagnetic string formation can be expressed using the Riemann zeta function with odd arguments in an integrable quantum spin model.

Findings

Probability expressed via Riemann zeta with odd arguments

Connection established between number theory and spin chain models

Analytical results for short string probabilities

Abstract

Riemann zeta function is an important object of number theory. It was also used for description of disordered systems in statistical mechanics. We show that Riemann zeta function is also useful for the description of integrable model. We study XXX Heisenberg spin 1/2 anti-ferromagnet. We evaluate a probability of formation of a ferromagnetic string in the anti-ferromagnetic ground state in thermodynamics limit. We prove that for short strings the probability can be expressed in terms of Riemann zeta function with odd arguments.