Noncompact Heisenberg spin magnets from high-energy QCD: III. Quasiclassical approach

TL;DR

This paper applies a quasiclassical approach to analyze the spectral problem of the noncompact SL(2,C) Heisenberg spin magnet, revealing a hidden symmetry and matching the spectrum with exact solutions.

Contribution

It introduces a novel quasiclassical quantization method involving both alpha- and beta-periods on a Riemann surface for the noncompact spin magnet.

Findings

Quasiclassical spectrum agrees with exact results.

Quantization involves full modular group of the spectral curve.

Reveals hidden symmetry in the energy spectrum.

Abstract

The exact solution of the noncompact SL(2,C) Heisenberg spin magnet reveals a hidden symmetry of the energy spectrum. To understand its origin, we solve the spectral problem for the model within quasiclassical approach. In this approach, the integrals of motion satisfy the Bohr-Sommerfeld quantization conditions imposed on the orbits of classical motion. In the representation of the separated coordinates, the latter wrap around a Riemann surface defined by the spectral curve of the model. A novel feature of the obtained quantization conditions is that they involve both the alpha- and beta-periods of the action differential on the Riemann surface, thus allowing us to find their solutions by exploring the full modular group of the spectral curve. We demonstrate that the quasiclassical energy spectrum is in a good agreement with the exact results.