We have obtained major new results under all three pillars, and in particular pillar I and III. I would emphasize the most important results here:
1) Pillar I:
We have discovered several new classes of integrable systems, and in particular new dynamical (to some extend universal, or even what we dub as super-universal - explained below) properties of integrable systems. This we have achieved in several fronts (all mentioned in the proposal for the actions):
a) We have found exact solutions for interacting reversible classical cellular automata.
b) We have proposed a new paradigm of Yang-Baxter integrable Floquet spin chains within brickwork circuit architecture, the so-called Integrable Trotterization, and in particular in both setups, of closed (unitary) and open (dissipative, non-unitary) systems.
c) PI and his group have discovered a completely unexpected Kardar-Parisi-Zhang (KPZ) universality in high-temperature equilibrium spin transport in Heisenberg spin-1/2 chain.
d) We have discovered new integrable classical spin chains in discrete space time, which provide the natural classical limit for the integrable trotterizations. In particular, we have proposed a class of integrable matrix models with arbitrary SU(N) symmetry, and found again the emergence of KPZ physics. This lead us to state the conjecture on super-universality of KPZ spin transport in integrable chains with non-abelian symmetry.
2) Pillar III:
The objective of this pillar was to look for exactly solvable instances of quantum many-body chaos. We have found three major groundbreaking results:
a) In 2017/2018, PI et al. proposed a simple model, a kicked Ising chain with long-range interaction and transverse field, for which SFF could be derived in a manner analogous to periodic orbit theory. Each term in the expansion of SFF is then computed analytically in terms of an 1D Ising-type partition sum, and it is shown how random matrix theory result is obtain in the leading order in time, as well as the famous Sieber-Richter correction in the subleading order.
b) The previous result is nice, but it requires still a sort of random-phase assumption and is hence not fully rigorous. Hence, in 2018 PI et al. provided an exact analytical computation of the SFF in kicked Ising model at the so-called self-dual points, in terms of a cute algebraic problem which they could solve exactly. This provides the first closed form solution of SFF in locally interacting quantum many-body system which shows agreement with Random matrix theory. It thus provide a minimal model of quantum many-body chaos.
c) Finally, in 2019, PI et al. defined the paradigm of dual unitary quantum circuits, for which two-point space-time correlation functions of local observables could be solved exactly, in particular, in terms of a single qubit/qudit quantum channels. These dual unitary circuits naturally generalize the self-dual kicked Ising model, and PI et al. later also extended their proof of RMT SFF to almost arbitrary dual unitary circuits.
3) Pillar II:
Finally, the pillar II, whose objective was to considered perturbative expansion and perturbative stability of integrable/exactly solvable systems turned out to be the most challenging. Yet, we found two remarkable fundamental results along this line as well:
a) We have analytically shown a perturbative stability of dual unitary circuits, in particular in the limit when the density of perturbed gates is small. We have found a good numerical evidence that such circuits are perturbatively stable even if every dual unitary gate is deformed, but by a sufficiently small amount. The result for the perturbed two-point correlation function has been expressed in terms of a simple path integral type formula, where the free propagator is given by the 2-point correlator of the unperturbed chaotic dual unitary circuit.
b) PI in collaboration with Lev Vidmar studied stability of many-body localization and found very clear numerical evidence which indicated a flow of the critical disorder for the putative transition with the system size. This result stimulated a new critical discussion within the field of MBL, which now largely changed the view of the community because of this publication (in Physical Review E, 2020).
