We first report new results on the classical Ising chain in magnetic field, where an infinite cascade of transitions(``disorder lines") is found [1]. The transition lacks a local order parameter (OP), and is signalled by a change ofasymptotic behavior of the string-string correlation functions. These transitions occur for m-periodic strings (m odd).The origin of the cascades are the Lee-Yang zeros of the partition function of the Ising model in the m-periodic complex magneticfield. These complex zeros are potentially measurable quantities [2].
We will also discuss the cascades of transitions in the well-known classical kinetic 1+1)-dimensional models [3]. We demonstrate that their active phases possess numerous hidden orders characterized by distinct backbones. The latter are subsets of percolative patterns, quantified by string-like OPs. From Monte-Carlo simulations we show the emergence of such backbones at specific critical points as a result of continuous phase transitions. These geometric transitions belong to the directed percolation universality class and their OPs are the capacities of corresponding percolative backbones. The multitude of conceivable percolative patterns implies the existence of infinite cascades of such transitions. It is shown that similar cascades exist in simple isotropic percolation. We conjectured that such cascades is a generic feature of transitions with nonlocal OPs.
A particular interesting direction to explore is the search for such hidden percolative patterns and transitions in (growing) networks. This is a very active field, since the network models apply to many biological,  neural, social and technological systems, emerging quantum space-time, and cosmology, showing amazing universality of scaling properties [4]. Another interesting problem is to relate the cascades of percolative transitions in various models we found to the complex zeros of some generalized ``partition functions" with non-Hermitian model Hamiltionians.

References:

1. P.N Timonin and G.Y. Chitov, Phys. Rev. E  96, 062123 (2017).

2. A. Krishnan, M. Schmitt, R. Moessner, and M. Hey, arXiv:1902.07155.

3. P.N Timonin and G.Y. Chitov, Phys. Rev. E  93, 012102 (2016); J. Phys. A: Math. Theor. 48, 135003 (2015).

4. D. Krioukov, et al, Sci. Rep. 2, 793 (2012); G. Bianconi, et al, Sci. Rep. 5, 10073 (2015).