We study via renormalization group (RG), numerics, exact bounds, and qualitative arguments the equilibrium Gibbs measure of a particle in a d-dimensional Gaussian random potential with translationally invariant logarithmic spatial correlations. We show that for any d>/=1 it exhibits a transition at T=T(c)>0. The low-temperature glass phase has a nontrivial structure, being dominated by a few distant states (with replica symmetry breaking phenomenology). In finite dimension this transition exists only in this "marginal glass" case (energy fluctuation exponent straight theta=0) and disappears if correlations grow faster (single ground-state dominance straight theta>0) or slower (high-temperature phase). The associated extremal statistics problem for correlated energy landscapes exhibits universal features which we describe using a nonlinear Kolmogorov (KPP) RG equation. These include the tails of the distribution of the minimal energy (or free energy) and the finite-size corrections, which are universal. The glass transition is closely related to Derrida's random energy models. In d=2, the connection between this problem and Liouville and sinh-Gordon models is discussed. The glass transition of the particle exhibits interesting similarities with the weak- to strong-coupling transition in Liouville (c=1 barrier) and with a transition that we conjecture for the sinh-Gordon model, with correspondence in some exact results and RG analysis. Glassy freezing of the particle is associated with the generation under RG of new local operators and of nonsmooth configurations in Liouville. Applications to Dirac fermions in random magnetic fields at criticality reveal a peculiar "quasilocalized" regime (corresponding to the glass phase for the particle), where eigenfunctions are concentrated over a finite number of distant regions, and allow us to recover the multifractal spectrum in the delocalized regime.

We define a new Z2-valued index to characterize the topological properties of periodically driven two dimensional crystals when the time-reversal symmetry is enforced. This index is associated with a spectral gap of the evolution operator over one period of time. When two such gaps are present, the Kane-Mele index of the eigenstates with eigenvalues between the gaps is recovered as the difference of the gap indices. This leads to an expression for the Kane-Mele invariant in terms of the Wess-Zumino amplitude. We illustrate the relation of the new index to the edge states in finite geometries by numerically solving an explicit model on the square lattice that is periodically driven in a time-reversal invariant way. Introduction. -The recent discovery of the quantum spin Hall effect [1][2][3][4] renewed interest in topological insulating phases which were first encountered in the beginning of the 1980s with the discovery of the quantum Hall effect (QHE) [5][6][7][8]. In his seminal paper [9] of 1981, Laughlin related the quantized Hall conductance to a quantum pump adiabatically driven by the magnetic flux. As shown by Thouless [10], such pumps drive through the insulator an integer number of charges whose origin is topological. In deep analogy, several works interpreted topological insulators and their robust boundary states in terms of quantum adiabatic pumps [11][12][13]. Interestingly, quantum crystals can exhibit original topological features when periodically driven beyond the adiabatic regime. While such modulation was first proposed to trigger a topological phase transition [14][15][16], it can also yield specific topological properties which cannot be understood within the usual framework of topological band theory [17,18]. The search for these so-called Floquet topological states quickly became a very active field and has recently stimulated numerous experimental works. A realization of such phases in condensed matter is quite challenging [19,20]. Several alternative artificial systems have been proposed to simulate and probe analogous phases, such as lattices of photonic resonators periodically driven by electro-optic modulators [21], ring resonator lattices [22], or more recently, photons coupled to excitons in semiconductors [23]. Signatures of topological Floquet states have already been revealed in onedimensional quantum walks with photons [24], as well as in 2D waveguide lattices [25]. Shaken trapped cold atoms were also proposed as a good candidate [26][27][28] and non-trivial topological phases were recently observed there [29].

No abstract