Protected zero modes in quantum physics traditionally arise in the context of ground states of many-body Hamiltonians. Here we study the case where zero modes exist in the center of a reflection-symmetric many-body spectrum, giving rise to the notion of a protected infinite-temperature degeneracy. For a certain class of nonintegrable spin chains, we show that the number of zero modes is determined by a chiral index that grows exponentially with system size. We propose a dynamical protocol, feasible in ongoing experiments in Rydberg atom quantum simulators, to detect these many-body zero modes and their protecting spectral reflection symmetry. Finally, we consider whether the zero energy states obey the eigenstate thermalization hypothesis, as is expected of states in the middle of the many-body spectrum. We find intriguing differences in their eigenstate properties relative to those of nearby nonzero-energy eigenstates at finite system sizes.
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