Entanglement entropy scaling laws and eigenstate thermalization in free fermion systems

Ken K. W., Kun Yang

We consider free fermion systems in arbitrary dimensions and represent the occupation pattern of each eigenstate as a classical binary string. We find that the Kolmogorov complexity of the string correctly captures the scaling behavior of its entanglement entropy (EE). In particular, the logarithmically enhanced area law for EE in the ground state and the volume law for EE in typical highly excited states are reproduced. Since our approach does not require bipartitioning the system, it allows us to distinguish typical and atypical eigenstates directly by their intrinsic complexity. We reveal that the fraction of atypical eigenstates which do not thermalize in the free fermion system vanishes exponentially in the thermodynamic limit. Our results illustrate explicitly the connection between complexity and EE of individual pure states in quantum systems.

Lucas Hackl, Lev Vidmar, Eugenio Bianchi A +1 more

Ken K. W., Alexander Volya, Kun Yang

The recent discovery of quantum many-body scar states has revealed the possibility of having states with low entanglement that violate the eigenstate thermalization hypothesis in nonintegrable systems. Eigenstates with low entanglement entropy are rare but naturally exist in the integrable system of free fermions. Here, we demonstrate analytically that these atypical states would always be eliminated when an arbitrary weak interaction is introduced between the fermions. In particular, we show that the probability of having a many-body scar state with entanglement entropy satisfying a subvolume scaling law decreases double exponentially as the system size. Thus, our results provide a quantitative argument for the disappearance of scar states in interacting fermion systems.

Pasquale Calabrese

Alexander Elgart, Leonid Andreevich Pastur, Mariya V Shcherbina

Hsin-Hua Lai, Kun Yang

We demonstrate that the entanglement entropy area law for free fermion ground states and the corresponding volume law for highly excited states are related by a position-momentum duality, and thus are of the same origin. For a typical excited state in the thermodynamic limit, we further show that the reduced density matrix of a subsystem approaches the thermal density matrix, provided the subsystem's linear size is small compared to that of the whole system in all directions, a property we dub eigenstate typicality. This provides an explicit example of thermalization via entanglement, and reveals how statistical physics emerges from a single eigenstate by tracing out a large number of degrees of freedom.