Chiral defect fermions on the lattice in 4+1 dimensions are analyzed using mean field theory. The fermion propagator has a localized chiral mode in weak coupling but loses it when the coupling in the unphysical fifth direction becomes too large. A layered phaseà la Fu-Nielsen appears where the theory is vector-like in every layer.

We propose a finite discretization for the black hole, near horizon, geometry and dynamics. We realize our proposal, in the case of extremal black holes, for which the radial and temporal near horizon geometry is known to be AdS 2 = SL(2, R)/SO(1, 1, R). We implement its discretization by replacing the set of real numbers R with the set of integers modulo N with AdS 2 going over to the finite geometry AdS 2 [N ] = SL(2, Z N )/SO(1, 1, Z N ). We model the dynamics of the microscopic degrees of freedom by generalized Arnol'd cat maps, A ∈ SL(2, Z N ) which are isometries of the geometry, at both the classical and quantum levels. These are well known to exhibit fast quantum information processing through the well studied properties of strong arithmetic chaos, dynamical entropy, nonlocality and factorization in the cutoff discretization N . We construct, finally, a new kind of unitary and holographic correspondence, for AdS 2 [N ]/CFT 1 [N ], via coherent states of the bulk and boundary geometries.

We construct explicitly the quantization of classical linear maps of SL(2, R) on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that Finite Quantum Mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of SL(2, Z) to the subgroup SL(2, Z)/Γ l , Γ l being the principal congruent subgroup mod l, on a finite dimensional Hilbert space. The generators of the "rotation group" mod l, O l (2) ⊂ SL(2, l), for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates. †

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We develop number theoretic tools that allow to perform computations relevant for the quantum mechanics over finite fields of arbitrary, odd size, with the same speedup that is enjoyed by the Fast Fourier Transform. †

At short time scales, the inertia term becomes relevant for the magnetization dynamics of ferromagnets and leads to nutation for the magnetization vector. For the case of spatially extended magnetic systems, for instance, Heisenberg spin chains with the isotropic spin-exchange interaction, this leads to the appearance of a collective excitation, the “nutation wave,” whose properties are elucidated by analytical arguments and numerical studies. The one-particle excitations can be identified as relativistic massive particles. These particles, the “nutatons,” acquire their mass via the Brout–Englert–Higgs mechanism, through the interaction of the wave with an emergent topological gauge field. This spin excitation would appear as a peak in the spectrum of the scattering structure factor in inelastic neutron scattering experiments. The high frequency and speed of the nutation wave can open paths for realizing ultrafast spin dynamics.

We establish the phase diagram of the five-dimensional anisotropic Abelian Higgs model by mean field techniques and Monte Carlo simulations. The anisotropy is encoded in the gauge couplings as well as in the Higgs couplings. In addition to the usual bulk phases (confining, Coulomb and Higgs) we find four-dimensional "layered" phases (3-branes) at weak gauge coupling, where the layers may be in either the Coulomb or the Higgs phase, while the transverse directions are confining. *