Modular Hamiltonian and Typicality of Degenerate States

Abstract

Free fermions exhibit Eigenstate Typicality (ET) meaning that the reduced density matrix, $\rho$, of a small spatial subsystem of a {\sl single}, typical, excited energy eigenstate is gaussian with a modular hamiltonian, $h\equiv -\log \rho$, asymptotically proportional to the free fermion hamiltonian. In effect, the ET may be thought of as providing an independent route to establishing the Fermi-Dirac (FD) distribution for free fermions. We revisit this argument for fermions belonging to a superposition of degenerate eigenstates. Specifically, we show that the modular hamiltonian corresponding to a superposition of degenerate eigenstates in a 1-d free fermion lattice system is interacting, and therefore the reduced density matrix for the subsystem is not gaussian. Measurements of the single particle distribution of a small subsystem would then exhibit non-thermal deviations from the FD distribution. These features may be traced to a violation of Wick's theorem for the chosen excited state. We develop a perturbation theory for the modular hamiltonian corresponding to a weak mixture of degenerate states and study the modular hamiltonian (exactly) computationally. Finally, we speculate that initial conditions in a realistic fermi system evolving towards equilibrium might be encoded in the interaction terms of the modular hamiltonian and consequent correlations of fermions.

arXiv comment26 pages; 5 figures