## Periodic Reporting for period 4 - OMNES (Open Many-body Non-Equilibrium Systems)

Reporting period: 2021-04-01 to 2022-09-30

With OMNES we are striving to deepen our understanding of non-equilibrium dynamics of systems of many interacting particles, both in classical and quantum dynamics, mainly through exact and rigorous analysis of exactly solvable models. We focus on lattice systems in low (one or two) spatial dimensions and having local (say, nearest neighbour) interactions. Exactly solvable paradigmatic models which are representatives of their universality classes are of crucial importance in theoretical physics since they give us the basic understanding of complex collective phenomena. Within OMNES we are deriving such models for dynamics and statistical mechanics far from thermal equilibrium. For reaching that goals we have to as well develop new mathematical methods, which as equally important objective of OMNES. Both aspects, namely that of obtaining new exactly solved models and developing new mathematical methods for non-equilibrium dynamics, are of fundamental importance for expanding human knowledge and could have potential future applications in developing nanoscale devices that manipulate quantum or classical information.

The project is structured into the three main pillars: I) Construction of new integrable systems, and in particular new integrable dynamics (time dependent exact solutions) in the realm of classical and quantum dynamics, both open (noisy/stochastic) and closed (deterministic), II) Perturbative analysis around exact solutions of non-equilibrium dynamics, III) Many-body quantum chaos, with the emphasis on the quest for exactly solvable models of many-body chaos.

We obtained major new results (one could characterise some as breakthrough) along all three Pillars within implementation of OMNES. Please see "Major achievements" for details.

The project is structured into the three main pillars: I) Construction of new integrable systems, and in particular new integrable dynamics (time dependent exact solutions) in the realm of classical and quantum dynamics, both open (noisy/stochastic) and closed (deterministic), II) Perturbative analysis around exact solutions of non-equilibrium dynamics, III) Many-body quantum chaos, with the emphasis on the quest for exactly solvable models of many-body chaos.

We obtained major new results (one could characterise some as breakthrough) along all three Pillars within implementation of OMNES. Please see "Major achievements" for details.

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).

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).

The results described above all go substantially beyond the state of the art. Several of the described results, in particular under Pillars I and III, can be considered as breakthroughs, in particular invention of new integrable models, the first integrable models of interacting Floquet quantum circuits, both closed and open (noisy), discovery of Kardar-Parisi-Zhang physics in integrable systems with non-abelian symmetries, and last and foremost, exact solutions to minimal models of quantum many-body chaos of spin-1/2 systems, and in particular, the invention of dual-unitary systems which is currently one of the most hot topics in field, broadly defined. It becomes among the main topics of major conferences in the field.