Dissipation and friction of a quantum spin system

We investigated the dissipation dynamics of a magnetic tip scanning the surface of a 3D ferromagnetic substrate at a velocity $${{\varvec{v}}}$$ v via a functional integral approach based on the Holstein–Primakoff boson representation of spin operators. The magnetic surface was parameterized by the XXZ model. And the degrees of freedom in the tip were taken as a single spin operator (M). The magnetic surface was coupled to the tip via a local exchange potential $$\frac{w}{2}\delta _{il}$$ w 2 δ il . Moreover, the tip induced a surface potential $$f_{il}$$ f il acting on the magnetic surface spin operators. After tracing over total internal degrees of freedom, we obtained the in–out quantum action. We calculated the imaginary part of the in–out quantum action and the frictional force as functions of speed $$v=|{{\varvec{v}}}|$$ v = | v | . We found that the imaginary part of the in–out quantum action is suppressed as $$v\rightarrow 0$$ v → 0 . The imaginary part of the in–out quantum action is proportional to the probability of ground state decay. Therefore, it implies dissipation of the system. The frictional force linearly depends on the velocity as $$v 0.04$$ v > 0.04 the dependence of the frictional force on v becomes nonlinear. GraphicAbstract