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...

Dielectric micro-cavities are widely used as laser resonators and characterizations of their spectra are of interest for various applications. We experimentally investigate micro-lasers of simple shapes (Fabry-Perot, square, pentagon, and disk). Their lasing spectra consist mainly of almost equidistant peaks and the distance between peaks reveals the length of a quantized periodic orbit. To measure this length with a good precision, it is necessary to take into account different sources of refractive index dispersion. Our experimental and numerical results agree with the superscar model describing the formation of long-lived states in polygonal cavities. The limitations of the two-dimensional approximation are briefly discussed in connection with micro-disks.

The construction of perturbation series for slightly deformed dielectric circular cavity is discussed in details. The obtained formulae are checked on the example of cut disks. A good agreement is found with direct numerical simulations and far-field experiments.Comment: 17 pages, 12 figure

The construction of the trace formula for open dielectric cavities is examined in detail. Using the Krein formula it is shown that the sum over cavity resonances can be written as a sum over classical periodic orbits for the motion inside the cavity. The contribution of each periodic orbit is the product of the two factors. The first is the same as in the standard trace formula and the second is connected with the product of reflection coefficients for all points of reflection with the cavity boundary. Two asymptotic terms of the smooth resonance counting function related with the area and the perimeter of a convex cavity are derived. The coefficient of the perimeter term differs from the one for closed cavities due to unusual high-energy asymptotics of the S matrix for the scattering on the cavity. Corrections to the leading semi-classical formula are briefly discussed. Obtained formulas agree well with numerical calculations for circular dielectric cavities.

We present a full description of the nonergodic properties of wavefunctions on random graphs without boundary in the localized and critical regimes of the Anderson transition. We find that they are characterized by two critical localization lengths: the largest one describes localization along rare branches and diverges with a critical exponent ν = 1 at the transition. The second length, which describes localization along typical branches, reaches at the transition a finite universal value (which depends only on the connectivity of the graph), with a singularity controlled by a new critical exponent ν ⊥ = 1/2. We show numerically that these two localization lengths control the finite-size scaling properties of key observables: wavefunction moments, correlation functions and spectral statistics. Our results are identical to the theoretical predictions for the typical localization length in the many-body localization transition, with the same critical exponent. This strongly suggests that the two transitions are in the same universality class and that our techniques could be directly applied in this context. arXiv:1904.08869v2 [cond-mat.dis-nn]

We derive a trace formula that expresses the level density of chaotic many-body systems as a smooth term plus a sum over contributions associated to solutions of the nonlinear Schrödinger (or Gross-Pitaevski) equation. Our formula applies to bosonic systems with discretised positions, such as the Bose-Hubbard model, in the semiclassical limit as well as in the limit where the number of particles is taken to infinity. We use the trace formula to investigate the spectral statistics of these systems, by studying interference between solutions of the nonlinear Schrödinger equation. We show that in the limits taken the statistics of fully chaotic many-particle systems becomes universal and agrees with predictions from the Wigner-Dyson ensembles of random matrix theory. The conditions for Wigner-Dyson statistics involve a gap in the spectrum of the Frobenius-Perron operator, leaving the possibility of different statistics for systems with weaker chaotic properties.

Dielectric resonators are open systems particularly interesting due to their wide range of applications in optics and photonics. In a recent paper [Phys. Rev. E 78, 056202 (2008)] the trace formula for both the smooth and the oscillating parts of the resonance density was proposed and checked for the circular cavity. The present paper deals with numerous shapes which would be integrable (square, rectangle, and ellipse), pseudointegrable (pentagon), and chaotic (stadium), if the cavities were closed (billiard case). A good agreement is found between the theoretical predictions, the numerical simulations, and experiments based on organic microlasers.

Abstract. It is well established numerically that spectral statistics of pseudointegrable models differs considerably from the reference statistics of integrable and chaotic systems. In [PRL 93 (2004) 254102] statistical properties of a certain quantized pseudo-integrable map had been calculated analytically but only for a special sequence of matrix dimensions. The purpose of this paper is to obtain the spectral statistics of the same quantum map for all matrix dimensions.