In this work, we study the dissipation mechanism and frictional force of a nanometer-sized tip scanning a metal surface via a path integral approach. The metal, with internal degrees of freedom (c, c ^†) and a tip with an internal degree of freedom (d, d ^†) couple with one another by means of an exchanged potential, V. Having integrated out all internal degrees of freedom, we obtain the in-out amplitude. Moreover, we calculate the imaginary part of the in-out amplitude and the frictional force. We find the imaginary part of the in-out amplitude to be positive, and correlated to the sliding velocity in most cases. The frictional force is proportional to the sliding velocity for the case where v < 0.01. However, for cases where v > 0.01, the frictional force demonstrates nonlinear dependence on sliding velocity.

在这项工作中，我们研究了通过路径积分方法扫描金属表面的纳米级尖端的耗散机理和摩擦力。具有内部自由度（c，c ^†）的金属和具有内部自由度（d，d ^†）的尖端通过交换电势V彼此耦合。整合了所有内部自由度后，我们获得了输入-输出幅度。此外，我们计算了进出振幅和摩擦力的虚部。我们发现进出振幅的虚部为正，并且在大多数情况下与滑动速度相关。在v <0.01的情况下，摩擦力与滑动速度成正比。但是，对于v > 0.01的情况，摩擦力表现出对滑动速度的非线性依赖性。