Condensed Matter

The statistics of gap ratios between consecutive energy levels is a widely used tool, in particular in the context of many-body physics, to distinguish between chaotic and integrable systems, described respectively by Gaussian ensembles of random matrices and Poisson statistics. In this work we extend the study of the gap ratio distribution P(r) to the case where discrete symmetries are present. This is important, since in certain situations it may be very impractical, or impossible, to split the model into symmetry sectors, let alone in cases where the symmetry is not known in the first place. Starting from the known expressions for surmises in the Gaussian ensembles, we derive analytical surmises for random matrices comprised of several independent blocks. We check our formulae against simulations from large random matrices, showing excellent agreement. We then present a large set of applications in many-body physics, ranging from quantum clock models and anyonic chains to periodically-driven spin systems. In all these models the existence of a (sometimes hidden) symmetry can be diagnosed through the study of the spectral gap ratios, and our approach furnishes an efficient way to characterize the number and size of independent symmetry subspaces. We finally discuss the relevance of our analysis for existing results in the literature, as well as point out possible future applications and extensions.

The extreme-value statistics of the entanglement spectrum in disordered spin chains possessing a many-body localization transition is examined. It is expected that eigenstates in the metallic or ergodic phase, behave as random states and hence the eigenvalues of the reduced density matrix, commonly referred to as the entanglement spectrum, are expected to follow the eigenvalue statistics of a trace normalized Wishart ensemble. In particular, the density of eigenvalues is supposed to follow the universal Marchenko-Pastur distribution. We find deviations in the tails both for the disordered XXZ with total S z conserved in the half-filled sector as well as in a model that breaks this conservation. A sensitive measure of deviations is provided by the largest eigenvalue, which in the case of the Wishart ensemble after appropriate shift and scaling follows the universal Tracy-Widom distribution. We show that for the models considered, in the metallic phase, the largest eigenvalue of the reduced density matrix of eigenvector, instead follows the generalized extreme-value statistics bordering on the Fisher-Tipett-Gumbel distribution indicating that the correlations between eigenvalues are much weaker compared to the Wishart ensemble. We show by means of distributions conditional on the high entropy and normalized participation ratio of eigenstates that the conditional entanglement spectrum still follows generalized extreme value distribution. In the deeply localized phase we find heavy tailed distributions and Lévy stable laws in an appropriately scaled function of the largest and second largest eigenvalues. The scaling is motivated by a recently developed perturbation theory of weakly coupled chaotic systems.

In this letter we study a reference model in theoretical ecology, the disordered Lotka-Volterra model for ecological communities, in the presence of finite demographic noise. Our theoretical analysis, which takes advantage of a mapping to an equilibrium disordered system, proves that for sufficiently heterogeneous interactions and low demographic noise the system displays a multiple equilibria phase, which we fully characterize. In particular, we show that in this phase the number of stable equilibria is exponential in the number of species. Upon further decreasing the demographic noise, we unveil a "Gardner" transition to a marginally stable phase, similar to that observed in jamming of amorphous materials. We confirm and complement our analytical results by numerical simulations. Furthermore, we extend their relevance by showing that they hold for others interacting random dynamical systems, such as the Random Replicant Model. Finally, we discuss their extension to the case of asymmetric couplings.

The strain load Δγ that triggers consecutive avalanches is a key observable in the slow deformation of amorphous solids. Its temporally averaged value ⟨Δγ⟩ displays a non-trivial system-size dependence that constitutes one of the distinguishing features of the yielding transition. Details of this dependence are not yet fully understood. We address this problem by means of theoretical analysis and simulations of elastoplastic models for amorphous solids. An accurate determination of the size dependence of ⟨Δγ⟩ leads to a precise evaluation of the steady-state distribution of local distances to instability x . We find that the usually assumed form P(x)∼ x θ (with θ being the so-called pseudo-gap exponent) is not accurate at low x and that in general P(x) tends to a system-size-dependent \textit{finite} limit as x→0 . We work out the consequences of this finite-size dependence standing on exact results for random-walks and disclosing an alternative interpretation of the mechanical noise felt by a reference site. We test our predictions in two- and three-dimensional elastoplastic models, showing the crucial influence of the saturation of P(x) at small x on the size dependence of ⟨Δγ⟩ and related scalings.

The simplest Tree-Tensor-States (TTS) respecting the Parity and the Time-Reversal symmetries are studied in order to describe the ground states of Long-Ranged Quantum Spin Chains with or without disorder. Explicit formulas are given for the one-point and two-point reduced density matrices that allow to compute any one-spin and two-spin observable. For Hamiltonians containing only one-body and two-body contributions, the energy of the TTS can be then evaluated and minimized in order to obtain the optimal parameters of the TTS. This variational optimization of the TTS parameters is compared with the traditional block-spin renormalization procedure based on the diagonalization of some intra-block renormalized Hamiltonian.

While there are several methods, e.g., anharmonic lattice dynamics and normal mode decomposition, to compute the modal lattice vibrational information in perfect crystals, the modal information of vibrations, e.g., vibrational relaxation time, group velocity and mean free path, in amorphous solids are still challenge to be captured. By systematically analyzing the normal mode decomposition and structure factor methods, we conclude that the vibrational dispersion can be calculated by applying effective wave vectors in the structure factor method, while the vibrational relaxation time calculated by the normal mode decomposition method is questionable since the group velocity cannot be defined on the Gamma point. We also show that the anharmonicity caused by the system temperature has little effect on the relaxation times of the propagating modes in amorphous materials, and therefore, the corresponding modal and total thermal conductivity is temperature independent when all the vibrations are assumed to be excited. The non-propagating modes, i.e., diffusons, conduct heat via thermal coupling between different vibrational modes, and can be calculated by harmonic lattice dynamics using Allen-Feldman theory. As a result, the thermal conductivity contributed from diffusons is also temperature independent when all the vibrational modes are activated which is the situation in molecular dynamics simulations. The total thermal conductivity concerning both propagons (50%) and diffusons (50%) agree quite well with the results computed using Green-Kubo equilibrium molecular dynamics. By correcting the excitation state of the vibrations in amorphous solids, the thermal conductivity calculated by the structure factor method and Allen-Feldman theory can fully capture the experimentally measured temperature-dependent thermal conductivity.

We study the Coulomb glass emerging from the interplay of strong interactions and disorder in a model of spinless fermions on the Bethe lattice. In the infinite coordination number limit, strong interactions induce a metallic Coulomb glass phase with a pseudogap structure at the Fermi energy. Quantum and thermal fluctuations both melt this glass and induce a disordered quantum liquid phase. We combine self-consistent diagrammatic perturbation theory with continuous time quantum Monte-Carlo simulations to obtain the complete phase diagram of the electron glass, and to characterize its dynamical properties in the quantum liquid, as well as in the replica symmetry broken glassy phase. Tunneling spectra display an Efros-Shklovskii pseudogap upon decreasing temperatures, but the density of states remains finite at the Fermi energy due to residual quantum fluctuations. Our results bear relevance to the metallic glass phase observed in Si inversion layers.

Network models for equilibrium integer quantum Hall (IQH) transitions are described by unitary scattering matrices, that can also be viewed as representing non-equilibrium Floquet systems. The resulting Floquet bands have zero Chern number, and are instead characterized by a chiral Floquet (CF) winding number. This begs the question: How can a model without Chern number describe IQH systems? We resolve this apparent paradox by showing that non-zero Chern number is recovered from the network model via the energy dependence of network model scattering parameters. This relationship shows that, despite their topologically distinct origins, IQH and CF topology-changing transitions share identical universal scaling properties.

In this Letter, we analyze the quantum dynamics of the perceptron model: a particle is constrained on a N -dimensional sphere, with N→∞ , and subjected to a set of randomly placed hard-wall potentials. This model has several applications, ranging from learning protocols to the effective description of the dynamics of an ensemble of infinite-dimensional hard spheres in Euclidean space. We find that the jamming transition with quantum dynamics shows critical exponents different from the classical case. We also find that the quantum jamming transition, unlike the typical quantum critical points, is not confined to the zero-temperature axis, and the classical results are recovered only at T=∞ . Our findings have implications for the theory of glasses at ultra-low temperatures and for the study of quantum machine-learning algorithms.

When a quantum particle is launched with a finite velocity in a disordered potential, it may surprisingly come back to its initial position at long times and remain there forever. This phenomenon, dubbed ``quantum boomerang effect'', was introduced in [Phys. Rev. A 99, 023629 (2019)]. Interactions between particles, treated within the mean-field approximation, are shown to partially destroy the boomerang effect: the center of mass of the wave packet makes a U-turn, but does not completely come back to its initial position. We show that this phenomenon can be quantitatively interpreted using a single parameter, the average interaction energy.