We study analytically and numerically the properties of one-dimensional chain of cold ions placed in a periodic potential of optical lattice and global harmonic potential of a trap. In close similarity with the Frenkel-Kontorova model, a transition from sliding to pinned phase takes place with the increase of the optical lattice potential for the density of ions incommensurate with the lattice period. Quantum fluctuations lead to a quantum phase transition and melting of pinned instanton glass phase at large values of dimensional Planck constant. The obtained results are also relevant for a Wigner crystal placed in a periodic potential. PACS numbers: 32.80.Lg, 32.80.Pj, 63.70.+h, 61.44.Fw Nowadays experimental techniques allow to store thousands of cold ions and observe various ordered structures formed by Coulomb repulsion in ion traps [1]. These structures include the one-dimensional (1D) Wigner crystal, zig-zag and helical structures in three dimensions. The Cirac-Zoller proposal of quantum computations with cold trapped ions [2] generated an enormous experimental progress in this field with implementations of quantum algorithms and quantum state preparation with up to 8 qubits [3,4]. In these experiments [3,4] ions form a 1D chain placed in a global harmonic potential which frequency ω determines the eigenfrequencies of chain oscillations being independent of ion charge [5,6]. Highly accurate experimental control of the chain modes allows to perform quantum gates between internal levels of ions. In addition to ion traps, modern laser techniques allow to create periodic optical lattices and store in them thousands of cold atoms (see e.g. [7]). A single ion dynamics in an optical lattice has been also studied experimentally [8]. The combination of these two techniques makes possible to study experimentally the properties of a 1D chain of few tens of ions placed in an optical lattice and a global harmonic potential at ultra low temperatures.In this Letter we analyze the physical properties of such a system and show that it is closely related to the Frenkel-Kontorova Model (FKM) [9] which gives a mathematical description of various physical phenomena including crystal dislocations, commensurateincommensurate phase transitions, epitaxial monolayers on a crystal surface, magnetic chains and fluxon dynamics in Josephson junctions (see [10] and Refs. therein). As in the classical FKM the ion chain exhibits the Aubry analyticity breaking transition [11] when the amplitude of optical potential becomes larger than a certain critical value. Below the critical point the classical chain can slide (oscillate) in the incommensurate optical lattice while above the transition it becomes pinned by the lattice and a large gap opens in the spectrum of phonon excitations. Above the transition the positions of ions form a devil's staircase corresponding to a fractal Cantor set which replaces a continuous Kolmogorov-Arnold-Moser (KAM) curve in the phase space below the transition. According to [11] the FKM ground state is unique ...

We study the Anderson transition on a generic model of random graphs with a tunable branching parameter 1 < K ≤ 2, through large scale numerical simulations and finite-size scaling analysis. We find that a single transition separates a localized phase from an unusual delocalized phase which is ergodic at large scales but strongly non-ergodic at smaller scales. In the critical regime, multifractal wavefunctions are located on few branches of the graph. Different scaling laws apply on both sides of the transition: a scaling with the linear size of the system on the localized side, and an unusual volumic scaling on the delocalized side. The critical scalings and exponents are independent of the branching parameter, which strongly supports the universality of our results.Ergodicity properties of quantum states are crucial to assess transport properties and thermalization processes. They are at the heart of the eigenstate thermalization hypothesis which has attracted enormous attention lately [1]. A paramount example of non-ergodicity is Anderson localization where the interplay between disorder and interference leads to exponentially localized states [2]. In 3D, a critical value of disorder separates a localized from an ergodic delocalized phase. At the critical point eigenfunctions are multifractal, another non trivial example of non-ergodicity [3,4]. Recently, those questions have been particularly highlighted in the problem of many-body localization [5][6][7][8][9]. Because Fock space has locally a tree-like structure, the problem of Anderson localization on different types of graphs [10][11][12][13][14][15] has attracted a renewed activity [16][17][18][19][20][21][22][23][24][25]. In particular, the existence of a delocalized phase with non-ergodic (multifractal) eigenfunctions lying on an algebraically vanishing fraction of the system sites is debated [19-21, 23, 25].The problem of non-ergodicity also arises in another context corresponding to glassy physics [26]. For directed polymers on the Bethe lattice [27], a glass transition leads to a phase where a few branches are explored among the exponential number available. As there is a mapping to directed polymer models in the Anderson-localized phase [10,[28][29][30], it has been recently proposed that this type of non-ergodicity (where the volume occupied by the states scales logarithmically with system volume) could also be relevant in the delocalized phase [18]. Note however that it has been envisioned that this picture could be valid only up to a finite but very large length scale [31].In this letter, we study the Anderson transition (AT) in a family of random graphs [32][33][34], where a tunable parameter p allows us to interpolate continuously between the 1D Anderson model and the random regular graph model of infinite dimensionality. Our main tool is the single parameter scaling theory of localization [35]. It has been used as a crucial tool to interpret the numerical simulations of Anderson localization in finite dimensions [3,[36][37][38] and to achieve...