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Over the past decade, non-Hermitian, $\mathcal{PT}$-symmetric Hamiltonianshave been investigated as candidates for both, a fundamental, unitary, quantumtheory, and open systems with a non-unitary time evolution. In this paper, weinvestigate the implications of the former approach in the context of thelatter. Motivated by the invariance of the $\mathcal{PT}$ (inner) product undertime evolution, we discuss the dynamics of wave-function phases in a wide rangeof $\mathcal{PT}$-symmetric lattice models. In particular, we numerically showthat, starting with a random initial state, a universal, gain-site locationdependent locking between wave function phases at adjacent sites occurs in the$\mathcal{PT}$-symmetry broken region. Our results pave the way towardsunderstanding the physically observable implications of time-invariants in thenon-unitary dynamics produced by $\mathcal{PT}$-symmetric Hamiltonians.

Time invariant $\mathcal{PT}$-product and phase locking in $\mathcal{PT}$-symmetric lattice models