Condensed Matter

Machine learning has recently been applied to many problems in condensed matter physics. A common point of many proposals is to save computational cost by training the machine with data from a simple example and then using the machine to make predictions for a more complicated example. Convolutional neural networks (CNN), which are one of the tools of machine learning, have proved to work well for assessing eigenfunctions in disordered systems. Here we apply a CNN to assess Kohn-Sham eigenfunctions obtained in density functional theory (DFT) simulations of the metal-insulator transition of a doped semiconductor. We demonstrate that a CNN that has been trained using eigenfunctions from a simulation of a doped semiconductor that neglects electron spin successfully predicts the critical concentration when presented with eigenfunctions from simulations that include spin.

We use record dynamics (RD), a coarse-grained description of the ubiquitous relaxation phenomenology known as "aging", as a diagnostic tool to find universal features that distinguish between the energy landscapes of Ising spin models and the ferromagnet. According to RD, a non-equilibrium system after a quench relies on fluctuations that randomly generate a sequence of irreversible record-sized events (quakes or avalanches) that allow the system to escape ever-higher barriers of meta-stable states within a complex, hierarchical energy landscape. Once these record events allow the system to overcome such barriers, the system relaxes by tumbling into the following meta-stable state that is marginally more stable. Within this framework, a clear distinction can be drawn between the coarsening dynamics of an Ising ferromagnet and the aging of the spin glass, which are often put in the same category. To that end, we interpolate between the spin glass and ferromagnet by varying the admixture p of ferromagnetic over anti-ferromagnetic bonds from the glassy state (at 50% each) to wherever clear ferromagnetic behavior emerges. The accumulation of record events grows logarithmic with time in the glassy regime, with a sharp transition at a specific admixture into the ferromagnetic regime where such activations saturate quickly. We show this effect both for the Edwards-Anderson model on a cubic lattice as well as the Sherrington-Kirkpatrick (mean-field) spin glass. While this transition coincides with a previously observed zero-temperature equilibrium transition in the former, that transition has not yet been described for the latter.

We introduce a powerful analytic method to study the statistics of the number N A (γ) of eigenvalues inside any contour γ∈C for infinitely large non-Hermitian random matrices A . Our generic approach can be applied to different random matrix ensembles, even when the analytic expression for the joint distribution of eigenvalues is not known. We illustrate the method on the adjacency matrices of weighted random graphs with asymmetric couplings, for which standard random-matrix tools are inapplicable. The main outcome is an effective theory that determines the cumulant generating function of N A via a path integral along γ , with the path probability distribution following from the solution of a self-consistent equation. We derive the expressions for the mean and the variance of N A as well as for the rate function governing rare fluctuations of N A (γ) . All theoretical results are compared with direct diagonalization of finite random matrices, exhibiting an excellent agreement.

We present analytical results for the distribution of first hitting times of random walks (RWs) on random regular graphs (RRGs) of degree c?? and a finite size N . Starting from a random initial node at time t=1 , at each time step t?? an RW hops randomly into one of the c neighbors of its previous node. In some of the time steps the RW may hop into a yet-unvisited node while in other time steps it may revisit a node that has already been visited before. The first time at which the RW enters a node that has already been visited before is called the first hitting time or the first intersection length. The first hitting event may take place either by backtracking (BT) to the previous node or by retracing (RET), namely stepping into a node which has been visited two or more time steps earlier. We calculate the tail distribution P( T FH >t) of first hitting (FH) times as well as its mean ??T FH ??and variance Var( T FH ) . We also calculate the probabilities P BT and P RET that the first hitting event will occur via the backtracking scenario or via the retracing scenario, respectively. We show that in dilute networks the dominant first hitting scenario is backtracking while in dense networks the dominant scenario is retracing and calculate the conditional distributions P( T FH =t|BT) and P( T FH =t|RET) , for the two scenarios. The analytical results are in excellent agreement with the results obtained from computer simulations. Considering the first hitting event as a termination mechanism of the RW trajectories, these results provide useful insight into the general problem of survival analysis and the statistics of mortality rates when two or more termination scenarios coexist.

Motivated by the recently discovered localization in disordered dipolar quantum systems, which is robust against effective power-law hopping r −a of dipole-flip excitations induced by dipole-dipole interactions, we consider 2d dipolar system, d=2 , with the generalized dipole-dipole interaction ∼ r −a , actual for experiments on trapped ions. We show that the homogeneous tilt β of the dipoles giving rise to the anisotropic dipole exchange leads to the non-trivial delocalization and reentrant localization. We find that the Anderson transitions occur at the finite values of the tilt parameter β=a , 0<a<d , keeping the localization to be robust at small anisotropy values. Moreover, the localized phase is shown to extend to smaller values of a= a AT (β)<d with respect to the standard resonance counting, a=d=2 , and this extension is non-monotonic with respect of the tilt angle. Both extensive numerical calculations and analytical methods show power-law localized eigenstates in the bulk of the spectrum, obeying recently discovered duality a↔2d−a of their spatial decay rate, on the localized side of the transition, a> a AT . This localization emerges due to the presence of the ergodic extended states at either spectral edge, which constitute a zero fraction of states in the thermodynamic limit, decaying though extremely slowly with the system size. The extended phase, a< a AT , is characterized by the finite-size multifractality going towards ergodicity.

The spectral dimension is a generalization of the Euclidean dimension and quantifies the propensity of a network to transmit and diffuse information. We show that, in hierarchical-modular network models of the brain, dynamics are anomalously slow and the spectral dimension is not defined. Inspired by Anderson localization in quantum systems, we relate the localization of neural activity - essential to embed brain functionality - to the network spectrum and to the existence of an anomalous "Lifshitz dimension". In a broader context, our results help shedding light on the relationship between structure and function in biological information-processing complex networks.

Anomalous transport in a circular comb is considered. The circular motion takes place for a fixed radius, while radii are continuously distributed along the circle. Two scenarios of the anomalous transport, related to the reflecting and periodic angular boundary conditions, are studied. The first scenario with the reflection boundary conditions for the circular diffusion corresponds to the conformal mapping of a 2D comb Fokker-Planck equation on the circular comb. This topologically constraint motion is named umbrella comb model. In this case, the reflecting boundary conditions are imposed on the circular (rotator) motion, while the radial motion corresponds to geometric Brownian motion with vanishing to zero boundary conditions on infinity. The radial diffusion is described by the log-normal distribution, which corresponds to exponentially fast motion with the mean squared displacement (MSD) of the order of e t . The second scenario corresponds to the circular diffusion with periodic boundary conditions and the outward radial diffusion with vanishing to zero boundary conditions at infinity. In this case the radial motion corresponds to normal diffusion. The circular motion in both scenarios is a superposition of cosine functions that results in the stationary Bernoulli polynomials for the probability distributions.

Using the non-interacting Anderson tight-binding model on the Bethe lattice as a toy model for the many-body quantum dynamics, we propose a novel and transparent theoretical explanation of the anomalously slow dynamics that emerges in the bad metal phase preceding the Many-Body Localization transition. By mapping the time-decorrelation of many-body wave-functions onto Directed Polymers in Random Media, we show the existence of a glass transition within the extended regime separating a metallic-like phase at small disorder, where delocalization occurs on an exponential number of paths, from a bad metal-like phase at intermediate disorder, where resonances are formed on rare, specific, disorder dependent site orbitals on very distant generations. The physical interpretation of subdiffusion and non-exponential relaxation emerging from this picture is complementary to the Griffiths one, although both scenarios rely on the presence of heavy-tailed distribution of the escape times. We relate the dynamical evolution in the glassy phase to the depinning transition of Directed Polymers, which results in macroscopic and abrupt jumps of the preferred delocalizing paths when a parameter like the energy is varied, and produce a singular behavior of the overlap correlation function between eigenstates at different energies. By comparing the quantum dynamics on loop-less Cayley trees and Random Regular Graphs we discuss the effect of loops, showing that in the latter slow dynamics and apparent power-laws extend on a very large time-window but are eventually cut-off on a time-scale that diverges at the MBL transition.

Tunneling two-level systems (TLSs), generic to amorphous solids, dictate the low-temperature properties of amorphous solids and dominate noise and decoherence in quantum nano-devices. The properties of the TLSs are generally described by the phenomenological standard tunneling model. Yet, significant deviations from the predictions of this model found experimentally suggest the need for a more precise model in describing TLSs. Here we show that the temperature dependence of the sound velocity, dielectric constant, specific heat, and thermal conductivity, can be explained using an energy-dependent TLS density of states reduced at low energies due to TLS-TLS interactions. This reduction is determined by the ratio between the strengths of the TLS-TLS interactions and the random potential, which is enhanced in systems with dominant electric dipolar interactions.

In the absence of frustration, interacting bosons in the ground state exist either in the superfluid or insulating phases. Superfluidity corresponds to frictionless flow of the matter field, and in optical conductivity is revealed through a distinct δ -functional peak at zero frequency with the amplitude known as the Drude weight. This characteristic low-frequency feature is instead absent in insulating phases, defined by zero static optical conductivity. Here we demonstrate that bosonic particles in disordered one dimensional, d=1 , systems can also exist in a conducting, non-superfluid, phase when their hopping is of the dipolar type, often viewed as short-ranged in d=1 . This phase is characterized by finite static optical conductivity, followed by a broad anti-Drude peak at finite frequencies. Off-diagonal correlations are also unconventional: they feature an integrable algebraic decay for arbitrarily large values of disorder. These results do not fit the description of any known quantum phase and strongly suggest the existence of a novel conducting state of bosonic matter in the ground state.