Quantum chaos: Approach to classicality In 
https://arxiv.org/abs/physics/0105072
the authors state that 
The entire field of quantum chaos is founded upon one
property of the classical limit, however, which is that
convergence to classicality as  $\hbar \to 0$ is logarithmically 
slow if classical trajectories diverge exponentially.
I would like a reference which discusses this statement in more detail.
 A: The best reference can be found in Joseph Emerson's thesis on chaos and the quantum-classical correspondence. In particular, note that the latter part of the property quoted in the paper you have cited is a condition in classical chaos, as stated on page 7 of Emerson's thesis:

Classical chaos is characterized by “extreme sensitivity to initial conditions.” This property is normally identified with an exponential divergence, on average, in the separation between initially nearby classical trajectories.

I believe the logarithmic convergence you are talking about is introduced on page iii, and is discussed throughout the text:

In classically chaotic regimes, the widths of initially narrow quantum states grow, on average, exponentially with time, until saturation at the system size. This initial spreading rate is well approximated by the classical Lyapunov exponent when the accessible classical phase space is predominantly chaotic. Because of the exponential growth rate of the quantum variance, the Ehrenfest regime is delimited by a break-time that grows logarithmically with increasing quantum numbers.

For a general discussion of quantum ergodicity (which might also assist you in understanding the quantum-classical correspondence in quantum chaos) is this introduction to the subject by Anatoli Polkovnikov. It is very readable, and one of the most comprehensive reviews on the subjects I know of. 
