The identification, description, and classification of topological features is an engine of discovery and innovation in several fields of physics. This research encompasses a broad variety of systems, from the integer and fractional Chern insulators in condensed matter, to protected states in complex photonic lattices in optics, and the structure of the QCD vacuum. Here, we introduce another playground for topology: the dissipative dynamics of the Sachdev-Ye-Kitaev (SYK) model, $N$ fermions in zero dimensions with strong $q$-body interactions coupled to a Markovian bath. For $q = 4, 8, \ldots$ and certain choices of $N$ and bath details, involving pseudo-Hermiticity, we find a rectangular block representation of the vectorized Liouvillian that is directly related to the existence of an anomalous trace of the unitary operator implementing fermionic exchange. As a consequence of this rectangularization, the Liouvillian has purely real modes for any coupling to the bath. Some of them are demonstrated to be topological by an explicit calculation of the spectral flow, leading to a symmetry-dependent topological index $\nu$. Topological properties have universal features: they are robust to changes in the Liouvillian provided that the symmetries are respected and they are also observed if the SYK model is replaced by a quantum chaotic dephasing spin chain in the same symmetry class. Moreover, the topological symmetry class can be robustly characterized by the level statistics of the corresponding random matrix ensemble. In the limit of weak coupling to the bath, topological modes govern the approach to equilibrium, which may enable a direct path for experimental confirmation of topology in dissipative many-body quantum chaotic systems.
Comment: 11 pages, 9 figures. v2: references added