Entanglement Entropy of Non-Hermitian Eigenstates and the Ginibre Ensemble

Nonorthogonal Eigenvectors, Fluctuation-Dissipation Relations, and Entropy Production

Celebrated fluctuation-dissipation theorem (FDT) linking the response function to time dependent correlations of observables measured in the reference unperturbed state is one of the central results in equilibrium statistical mechanics. In this Letter we discuss an extension of the standard FDT to the case when multidimensional matrix representing transition probabilities is strictly non-normal. This feature dramatically modifies the dynamics, by incorporating the effect of eigenvector nonorthogonality via the associated overlap matrix of Chalker-Mehlig type. In particular, the rate of entropy production per unit time is strongly enhanced by that matrix. We suggest, that this mechanism has an impact on the studies of collective phenomena in neural matrix models, leading, via transient behavior, to such phenomena as synchronization and emergence of the memory. We also expect, that the described mechanism generating the entropy production is generic for wide class of phenomena, where dynamics is driven by non-normal operators. For the case of driving by a large Ginibre matrix the entropy production rate is evaluated analytically, as well as for the Rajan-Abbott model for neural networks.

Measuring entanglement entropy and its topological signature for phononic systems

Entanglement entropy is a fundamental concept with rising importance in various fields ranging from quantum information science, black holes to materials science. In complex materials and systems, entanglement entropy provides insight into the collective degrees of freedom that underlie the systems' complex behaviours. As well-known predictions, the entanglement entropy exhibits area laws for systems with gapped excitations, whereas it follows the Gioev-Klich-Widom scaling law in gapless fermion systems. However, many of these fundamental predictions have not yet been confirmed in experiments due to the difficulties in measuring entanglement entropy in physical systems. Here, we report the experimental verification of the above predictions by probing the nonlocal correlations in phononic systems. We obtain the entanglement entropy and entanglement spectrum for phononic systems with the fermion filling analog. With these measurements, we verify the Gioev-Klich-Widom scaling law. We further observe the salient signatures of topological phases in entanglement entropy and entanglement spectrum.