Asymmetric frictional sliding between incommensurate surfaces

We study the frictional sliding of two ideally incommensurate surfaces with a third incommensurate sheet - a sort of extended lubricant - in between. When the mutual ratios of the three periodicities in this sandwich geometry are chosen to be the golden mean \phi=(1+\sqrt 5)/2, this system is believed to be statically pinned for any choice of system parameters. In the present study we overcome this pinning and force the two "substrates" to slide with a mutual velocity V_ext, analyzing the resulting frictional dynamics. An unexpected feature is an asymmetry of the relative sliding velocity of the intermediate lubricating sheet relative to the two substrates. Strikingly, the velocity asymmetry takes an exactly quantized value which is uniquely determined by the incommensurability ratio, and absolutely insensitive to all other parameters. The reason for quantization of the velocity asymmetry will be addressed. This behavior is compared and contrasted to the corresponding one obtained for a representative cubic irrational, the spiral mean \omega.