| Abstract: |
| We establish a KAM theorem for constructing analytic quasi-periodic equilibrium
configurations in the classical anisotropic Heisenberg XYZ spin chain with
quasi-periodic coupling and a small external magnetic field. The equilibrium
equations reduce to a second-order nonlinear matrix difference equation with
quasi-periodic coefficients. By introducing auxiliary parameters and employing a
factorization method, we reformulate the problem into two first-order
equations. Under Diophantine conditions on the frequency and appropriate
non-degeneracy assumptions, we prove the existence of analytic quasi-periodic
solutions via a Nash-Moser iteration scheme. The proof relies on solving twisted
cohomology equations and a reducibility lemma. As an application, we obtain the
existence of quasi-periodic spin patterns for the Heisenberg XYZ model with
small quasi-periodic perturbations. |
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