Microcanonical Truncations of Observables in Quantum Chaotic Systems

We consider the properties of an observable (such as a single spin component that squares to the identity) when expressed as a matrix in the basis of energy eigenstates, and then truncated to a microcanonical slice of energies of varying width. For a quantum chaotic system, we model the unitary or orthogonal matrix that relates the spin basis to the energy basis as a random matrix selected from the appropriate Haar measure. We find that the spectrum of eigenvalues is given by a centered Jacobi distribution that approaches the Wigner semicircle of a random hermitian matrix for small slices. For slices that contain more than half the states, there is a set of eigenvalues of exactly $\pm 1$. The transition to this qualitatively different behavior at half size is similar to that seen in other quantities such as entanglement entropy. Our results serve as a benchmark model for numerical calculations in realistic physical systems.