Entanglement transitions in translation-invariant tensor networks

Entanglement transitions in random tensor networks and monitored random circuits are typically understood from effective statistical mechanics models. Randomness is crucial in this mapping, but it is unclear whether randomness is necessary for an entanglement transition. In this work we address the mechanisms for entanglement transitions in translation-invariant tensor networks. We consider the evolution of qubit chains under the repeated action of fixed non-unitary 'transfer matrices', and show numerically that there are purification and entanglement transitions as the degree of nonunitarity is varied. In this setting, purification is controlled by the singular values of powers of the transfer matrix and, at late times, these singular values are themselves fixed by the eigenvalue spectrum of the transfer matrix. A purification time which increases with system size then suggests a corresponding decrease in the gap between the magnitudes of the leading complex eigenvalues. We uncover the general mechanism which acts to suppress these gaps: eigenvalue attraction along the radial direction in the complex plane. To substantiate this picture of the non-purifying phase, we present exact results for spectral properties in a nonunitary model with global Haar invariance. We show also that, in the purifying phase, the evolution operator has a spectral gap that is independent of the overall system size. Based on these results we conjecture that purification transitions in translation-invariant tensor networks are accompanied by transitions in the spectral gaps of their transfer matrices.