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We discuss several classes of integrable Floquet systems, i.e. systems which do not exhibit chaotic behavior even under a time dependent perturbation. The first class is associated with finite-dimensional Lie groups and infinite-dimensional generalization thereof. The second class is related to the row transfer matrices of the 2D statistical mechanics models. The third class of models, called here boost models, is constructed as a periodic interchange of two Hamiltonians - one is the integrable lattice model Hamiltonian, while the second is the boost operator. The latter for known cases coincides with the entanglement Hamiltonian and is closely related to the corner transfer matrix of the corresponding 2D statistical models. We present several explicit examples. As an interesting application of the boost models we discuss a possibility of generating periodically oscillating states with the period different from that of the driving field. In particular, one can realize an oscillating state by performing a static quench to a boost operator. We term this state a Quantum Boost Clock. All analyzed setups can be readily realized experimentally, for example in cod atoms.
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We introduce a class of integrable dynamical systems of interacting classical matrix-valued fields propagating on a discrete space-time lattice, realized as many-body circuits built from elementary symplectic two-body maps. The models provide an efficient integrable Trotterization of non-relativistic $sigma$-models with complex Grassmannian manifolds as target spaces, including, as special cases, the higher-rank analogues of the Landau-Lifshitz field theory on complex projective spaces. As an application, we study transport of Noether charges in canonical local equilibrium states. We find a clear signature of superdiffusive behavior in the Kardar-Parisi-Zhang universality class, irrespectively of the chosen underlying global unitary symmetry group and the quotient structure of the compact phase space, providing a strong indication of superuniversal physics.
Our review covers microscopic foundations of generalized hydrodynamics (GHD). As one generic approach we develop form factor expansions, for ground states and generalized Gibbs ensembles (GGE), and compare the so obtained results with predictions from GHD. One cornerstone of GHD is the GGE averaged microscopic currents. They can be obtained using form factors. Discussed is also a second, completely orthogonal approach based on the availability of a self-conserved current.
Using generalized hydrodynamics (GHD), we develop the Euler hydrodynamics of classical integrable field theory. Classical field GHD is based on a known formalism for Gibbs ensembles of classical fields, that resembles the thermodynamic Bethe ansatz of quantum models, which we extend to generalized Gibbs ensembles (GGEs). In general, GHD must take into account both solitonic and radiative modes of classical fields. We observe that the quasi-particle formulation of GHD remains valid for radiative modes, even though these do not display particle-like properties in their precise dynamics. We point out that because of a UV catastrophe similar to that of black body radiation, radiative modes suffer from divergences that restrict the set of finite-average observables; this set is larger for GGEs with higher conserved charges. We concentrate on the sinh-Gordon model, which only has radiative modes, and study transport in the domain-wall initial problem as well as Euler-scale correlations in GGEs. We confirm a variety of exact GHD predictions, including those coming from hydrodynamic projection theory, by comparing with Metropolis numerical evaluations.
We develop a comprehensive Renormalization Group (RG) approach to criticality in open Floquet systems, where dissipation enables the system to reach a well-defined Floquet steady state of finite entropy, and all observables are synchronized with the drive. We provide a detailed description of how to combine Keldysh and Floquet formalisms to account for the critical fluctuations in the weakly and rapidly driven regime. A key insight is that a reduction to the time-averaged, static sector, is not possible close to the critical point. This guides the design of a perturbative dynamic RG approach, which treats the time-dependent, dynamic sector associated to higher harmonics of the drive, on an equal footing with the time-averaged sector. Within this framework, we develop a weak drive expansion scheme, which enables to systematically truncate the RG flow equations in powers of the inverse drive frequency $Omega^{-1}$. This allows us to show how a periodic drive inhibits scale invariance and critical fluctuations of second order phase transitions in rapidly driven open systems: Although criticality emerges in the limit $Omega^{-1}=0$, any finite drive frequency produces a scale that remains finite all through the phase transition. This is a universal mechanism that relies on the competition of the critical fluctuations within the static and dynamic sectors of the problem.
For quantum integrable systems the currents averaged with respect to a generalized Gibbs ensemble are revisited. An exact formula is known, which we call collision rate ansatz. While there is considerable work to confirm this ansatz in various models, our approach uses the symmetry of the current-charge susceptibility matrix, which holds in great generality. Besides some technical assumptions, the main input is the availability of a self-conserved current, i.e. some current which is itself conserved. The collision rate ansatz is then derived. The argument is carried out in detail for the Lieb-Liniger model and the Heisenberg XXZ chain. The Fermi-Hubbard model is not covered, since no self-conserved current seems to exist. It is also explained how from the existence of a boost operator a self-conserved current can be deduced.
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