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Register a new userIntegrable Matrix Models in Discrete Space-Time
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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.
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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.
We introduce a deterministic SO(3) invariant dynamics of classical spins on a discrete space-time lattice and prove its complete integrability by explicitly finding a related non-constant (baxterized) solution of the set-theoretic quantum Yang-Baxter equation over the 2-sphere. Equipping the algebraic structure with the corresponding Lax operator we derive an infinite sequence of conserved quantities with local densities. The dynamics depend on a single continuous spectral parameter and reduce to a (lattice) Landau-Lifshitz model in the limit of a small parameter which corresponds to the continuous time limit. Using quasi-exact numerical simulations of deterministic dynamics and Monte Carlo sampling of initial conditions corresponding to a maximum entropy equilibrium state we determine spin-spin spatio-temporal (dynamical) correlation functions with relative accuracy of three orders of magnitude. We demonstrate that in the equilibrium state with a vanishing total magnetization the correlation function precisely follow Kardar-Parisi-Zhang scaling hence the spin transport belongs to the universality class with dynamical exponent z=3/2, in accordance to recent related simulations in discrete and continuous time quantum Heisenberg spin 1/2 chains.
A physical-mathematical approach to anomalous diffusion may be based on generalized diffusion equations (containing derivatives of fractional order in space or/and time) and related random walk models. The fundamental solution (for the {Cauchy} problem) of the fractional diffusion equations can be interpreted as a probability density evolving in time of a peculiar self-similar stochastic process that we view as a generalized diffusion process. By adopting appropriate finite-difference schemes of solution, we generate models of random walk discrete in space and time suitable for simulating random variables whose spatial probability density evolves in time according to a given fractional diffusion equation.
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.
How much time does it take two molecules to react? If a reaction occurs upon contact, the answer to this question boils down to the classic first-passage time problem: find the random time it takes the two molecules to meet. However, this is not always the case as molecules switch stochastically between reactive and non-reactive states. In such cases, the reaction is said to be ``gated by the internal states of the molecules involved which could have a dramatic influence on kinetics. A unified, continuous-time, approach to gated reactions on networks was presented in [Phys. Rev. Lett. 127, 018301, 2021]. Here, we build on this recent advancement and develop an analogous discrete-time version of the theory. Similar to continuous-time, we employ a renewal approach to show that the gated reaction time can always be expressed in terms of the corresponding ungated first-passage and return times; which yields formulas for the generating function of the gated reaction-time distribution and its corresponding mean and variance. In cases where the mean reaction time diverges, we show that the long-time asymptotics of the gated problem is inherited from its ungated counterpart, where only the pre-factor of the power law tail changes. The discretization of time also gives rise to new phenomena that do not exist in the continuous-time analogue. Crucially, when the internal gating dynamics is in, or out of, phase with the spatial process governing molecular encounters resonance and anti-resonance phenomena emerge. These phenomena are illustrated using two case studies which also serve to show how the general approach presented herein greatly simplifies the analysis of gated reactions.
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