Overcoming the wall in the semiclassical baker's map

Abstract

A major barrier in semiclassical calculations is the sheer number of terms that contribute as time increases; for classically chaotic dynamics, the proliferation is exponential. We have been able to overcome this ``exponential wall'' for the baker's map, using an analogy with spin-chain partition functions. The semiclassical sum is contracted so that only of order N T^(3/2) operations are needed for an N-state system evolved for T time steps. This is typically less than the computational load for quantum propagation, T N^2. This method enables us to obtain semiclassical results up to the Heisenberg time, which in our example would have required 10^90 terms if we were to evaluate the sum exactly. This calculation in turn provides new insight as to the accuracy of the semiclassical approximation at long times. The semiclassical result is often correct; its breakdown is nonuniform.