Abstract
The spectral form factor (SFF) is an important diagnostic of energy level
repulsion in random matrix theory (RMT) and quantum chaos. The short-time
behavior of the SFF as it approaches the RMT result acts as a diagnostic of the
ergodicity of the system as it approaches the thermal state. In this work we
observe that for systems without time-reversal symmetry, there is a second
break from the RMT result at late time around the Heisenberg time. Long after
thermalization has taken hold, and after the SFF has agreed with the RMT result
to high precision for a time of order the Heisenberg time, the SFF of a large
system will briefly deviate from the RMT behavior in a way exactly determined
by its early time thermalization properties. The conceptual reason for this
second deviation is the Riemann-Siegel lookalike formula, a resummed expression
for the spectral determinant relating late time behavior to early time spectral
statistics. We use the lookalike formula to derive a precise expression for the
late time SFF for semi-classical quantum chaotic systems, and then confirm our
results numerically for more general systems.