Condensed Matter

We provide some analytical tests of the density of states estimation from the "localization landscape" approach of Ref. [Phys. Rev. Lett. 116, 056602 (2016)]. We consider two different solvable models for which we obtain the distribution of the landscape function and argue that the precise spectral singularities are not reproduced by the estimation of the landscape approach.

In this work we perform computational and analytical studies of the Randić index R(G) in Erdös-Rényi models G(n,p) characterized by n vertices connected independently with probability p∈(0,1) . First, from a detailed scaling analysis, we show that ⟨ R ¯ ¯ ¯ ¯ (G)⟩=⟨R(G)⟩/(n/2) scales with the product ξ≈np , so we can define three regimes: a regime of mostly isolated vertices when ξ<0.01 ( R(G)≈0 ), a transition regime for 0.01<ξ<10 (where 0<R(G)<n/2 ), and a regime of almost complete graphs for ξ>10 ( R(G)≈n/2 ). Then, motivated by the scaling of ⟨ R ¯ ¯ ¯ ¯ (G)⟩ , we analytically (i) obtain new relations connecting R(G) with other topological indices and characterize graphs which are extremal with respect to the relations obtained and (ii) apply these results in order to obtain inequalities on R(G) for graphs in Erdös-Rényi models.

While Anderson localisation is largely well-understood, its description has traditionally been rather cumbersome. A recently-developed theory -- Localisation Landscape Theory (LLT) -- has unparalleled strengths and advantages, both computational and conceptual, over alternative methods. To begin with, we demonstrate that the localisation length cannot be conveniently computed starting directly from the exact eigenstates, thus motivating the need for the LLT approach. Then, we confirm that the Hamiltonian with the effective potential of LLT has very similar low energy eigenstates to that with the physical potential, justifying the crucial role the effective potential plays in our new method. We proceed to use LLT to calculate the localisation length for very low-energy, maximally localised eigenstates, as defined by the length-scale of exponential decay of the eigenstates, (manually) testing our findings against exact diagonalisation. We then describe several mechanisms by which the eigenstates spread out at higher energies where the tunnelling-in-the-effective-potential picture breaks down, and explicitly demonstrate that our method is no longer applicable in this regime. We place our computational scheme in context by explaining the connection to the more general problem of multidimensional tunnelling and discussing the approximations involved. Our method of calculating the localisation length can be applied to (nearly) arbitrary disordered, continuous potentials at very low energies.

Connectedness percolation phenomena in two-dimensional packings of elongated particles (discorectangles) were studied numerically. The packings were produced using random sequential adsorption (RSA) off-lattice model with preferential orientations of particles along a given direction. The partial ordering was characterized by order parameter S , with S=0 for completely disordered films (random orientation of particles) and S=1 for completely aligned particles along the horizontal direction x . The aspect ratio (length-to-width ratio) for the particles was varied within the range ε∈[1;100] . Analysis of connectivity was performed assuming a core-shell structure of particles. The value of S affected the structure of packings, formation of long-range connectivity and electrical conductivity behavior. The effects were explained accounting for the competition between the particles' orientational degrees of freedom and the excluded volume effects. For aligned deposition, the anisotropy in electrical conductivity was observed and the values along alignment direction, σ x , were larger than the values in perpendicular direction, σ y . The anisotropy in localization of percolation threshold was also observed in finite sized packings, but it disappeared in the limit of infinitely large systems.

We propose a novel model for a glass-forming liquid which allows to switch in a continuous manner from a standard three-dimensional liquid to a fully connected mean-field model. This is achieved by introducing k additional particle-particle interactions which thus augments the effective number of neighbors of each particle. Our computer simulations of this system show that the structure of the liquid does not change with the introduction of these pseudo neighbours and by means of analytical calculations, we determine the structural properties related to these additional neighbors. We show that the relaxation dynamics of the system slows down very quickly with increasing k and that the onset and the mode-coupling temperatures increase. The systems with high values of k follow the MCT power law behaviour for a larger temperature range compared to the ones with lower values of k. The dynamic susceptibility indicates that the dynamic heterogeneity decreases with increasing k whereas the non-Gaussian parameter is independent of it. Thus we conclude that with the increase in the number of pseudo neighbours the system becomes more mean-field like. By comparing our results with previous studies on mean-field like system we come to the conclusion that the details of how the mean-field limit is approached are important since they can lead to different dynamical behavior in this limit.

We develop deep autoregressive networks with multi channels to compute many-body systems with \emph{continuous} spin degrees of freedom directly. As a concrete example, we embed the two-dimensional XY model into the continuous-mixture networks and rediscover the Kosterlitz-Thouless (KT) phase transition on a periodic square lattice. Vortices characterizing the quasi-long range order are accurately detected by the autoregressive neural networks. By learning the microscopic probability distributions from the macroscopic thermal distribution, the neural networks compute the free energy directly and find that free vortices and anti-vortices emerge in the high-temperature regime. As a more precise evaluation, we compute the helicity modulus to determine the KT transition temperature. Although the training process becomes more time-consuming with larger lattice sizes, the training time remains unchanged around the KT transition temperature. The continuous-mixture autoregressive networks we developed thus can be potentially used to study other many-body systems with continuous degrees of freedom.

We analyse the correlation function of the quantum curvature in complex quantum systems, using a random matrix model to provide an exemplar of a universal correlation function. We show that the correlation function diverges as the inverse of the distance at small separations. We also define and analyse a correlation function of mixed states, showing that it is finite but singular at small separations. A scaling hypothesis on a universal form for both types of correlations is supported by Monte-Carlo simulations. We relate the correlation function of the curvature to the variance of Chern integers which can describe quantised Hall conductance.

Random Matrix Theory (RMT) provides a tool to understand physical systems in which spectral properties can be changed from Poissonian (integrable) to Wigner-Dyson (chaotic). Such transitions can be seen in Rosenzweig-Porter ensemble (RPE) by tuning the fluctuations in the random matrix elements. We show that integrable or chaotic regimes in any 2-level system can be uniquely controlled by the symmetry-breaking properties. We compute the Nearest Neighbour Spacing (NNS) distributions of these matrix ensembles and find that they exactly match with that of RPE. Our study indicates that the loss of integrability can be exactly mapped to the extent of disorder in 2-level systems.

The thermally activated creep motion of an elastic interface weakly driven on a disordered landscape is one of the best examples of glassy universal dynamics. Its understanding has evolved over the last 30 years thanks to a fruitful interplay between elegant scaling arguments, sophisticated analytical calculations, efficient optimization algorithms and creative experiments. In this article, starting from the pioneer arguments, we review the main theoretical and experimental results that lead to the current physical picture of the creep regime. In particular, we discuss recent works unveiling the collective nature of such ultra-slow motion in terms of elementary activated events. We show that these events control the mean velocity of the interface and cluster into "creep avalanches" statistically similar to the deterministic avalanches observed at the depinning critical threshold. The associated spatio-temporal patterns of activated events have been recently observed in experiments with magnetic domain walls. The emergent physical picture is expected to be relevant for a large family of disordered systems presenting thermally activated dynamics.

We study the critical level statistics at the many-body localization (MBL) transition region in random spin systems. By employing the inter-sample randomness as indicator, we manage to locate the MBL transition point in both orthogonal and unitary models. We further count the n -th order gap ratio distributions at the transition region up to n=4 , and find they fit well with the short-range plasma model (SRPM) with inverse temperature β=1 for orthogonal model and β=2 for unitary. These critical level statistics are argued to be universal by comparing results from systems both with and without total S z conservation. We also point out that these critical distributions can emerge from the spectrum of a Poisson ensemble, which indicates the thermal-MBL transition point is more affected by the MBL phase rather than thermal phase.