It is well known that typical pure states in the Hilbert space are (nearly) maximally entangled. In my talk I will discuss, from the perspective of bipartite entanglement entropy, how different are typical eigenstates of physical Hamiltonians from typical states in the Hilbert space.

In the first part, I will present tools to compute the average entanglement entropy of all eigenstates of translationally-invariant quadratic fermionic Hamiltonians, and derive exact bounds [1]. I will prove that (i) if the subsystem volume is a finite fraction of the system volume, then the average entanglement entropy is smaller than the result for typical pure states in the thermodynamic limit (the difference is extensive with system volume), and (ii) in the limit in which the subsystem volume is a vanishing fraction of the system volume, the average entanglement entropy is maximal; i.e., typical eigenstates of such Hamiltonians exhibit eigenstate thermalization.

In the second part, I will focus on eigenstates of quantum chaotic many-body Hamiltonians [2]. I will prove that, in a system that is away from half filling and divided in two equal halves, an upper bound for the average entanglement entropy of random pure states with a fixed particle number and normally distributed real coefficients exhibits a deviation from the maximal value that grows with the square root of the volume of the system. Exact numerical results for highly excited eigenstates of a particle number conserving quantum chaotic model indicate that the bound is saturated with increasing system volume.

* Lev Vidmar, F1 Department of Theoretical Physics, IJS

[1] Vidmar, Hackl, Bianchi and Rigol, Phys. Rev. Lett. 119, 020601 (2017)

[2] Vidmar and Rigol, Phys. Rev. Lett. 119, 220603 (2017)