We consider non-equilibrium transport in disordered conductors. We calculate the interaction correction to the current for a short wire connected to electron reservoirs by resistive interfaces. In the absence of charging effects we find a universal current-voltage-characteristics. The relevance of our calculation for existing experiments is discussed as well as the connection with alternative theoretical approaches.
Graphene nanoribbons provide an opportunity to integrate phase-coherent transport phenomena with nanoelectromechanical systems (NEMS). Due to the strain induced by a deflection in a graphene nanoribbon resonator, coherent electron transport and mechanical deformations couple. As the electrons in graphene have a Fermi wavelength λ~ a_0 = 1.4 Å, this coupling can be used for sensitive displacement detection in both armchair and zigzag graphene nanoribbon NEMS. Here it is shown that for ordered as well as disordered ribbon systems of length L, a strain ε~ (w/L)^2 due to a deflection w leads to a …
This is a review of the properties of spectral fluctations in disordered metals, their relation with Random Matrix Theory and semiclassical picture. We also review the physics of persistent currents in mesoscopic isolated rings, the parametric correlations and curvature distributions.
Topological behavior can be masked when disorder is present. A topological insulator, either intrinsic or interaction induced, may turn gapless when sufficiently disordered. Nevertheless, the metallic phase that emerges once a topological gap closes retains several topological characteristics. By considering the self-consistent disorder-averaged Green function of a topological insulator, we derive the condition for gaplessness. We show that the edge states survive in the gapless phase as edge resonances and that, similar to a doped topological insulator, the disordered topological metal also h…
In realistic nanoelectronics, disordered impurities/defects are inevitable and play important roles in electron transport. However, due to the lack of effective quantum transport method to do disorder average, the important effects of disorders remain largely un-explored or poorly understood. Here, we report a generalized non-equilibrium vertex correction method with coherent potential approximation for the non-equilibrium quantum transport simulation of disordered nanoelectronics. In this method, the disorder average of various Green's functions are computed by a generalized coherent potentia…
We present a diagrammatic theory for coherent backscattering from disordered dilute media in the nonlinear regime. The approach is non-perturbative in the strength of the nonlinearity. We show that the coherent backscattering enhancement factor is strongly affected by the nonlinearity, and corroborate these results by numerical simulations. Our theory can be applied to several physical scenarios like scattering of light in nonlinear Kerr media, or propagation of matter waves in disordered potentials.
We derive a formula for the quantum corrections to the electrical current for a metal out of equilibrium. In the limit of linear current-voltage characteristics our formula reproduces the well known Altshuler-Aronov correction to the conductivity of a disordered metal. The current formula is obtained by a direct diagrammatic approach, and is shown to agree with what is obtained within the Keldysh formulation of the non-linear sigma model. As an application we calculate the current of a mesoscopic wire. We find a current-voltage characteristics that scales with $eV/kT$, and calculate the differ…
We examine the charge transport through disordered arrays of metallic dots using numerical simulations. We find power law scaling in the current-voltage curves for arrays containing no voids, while for void-filled arrays charge bottlenecks form and a single scaling is absent, in agreement with recent experiments. In the void-free case we also show that the scaling exponent depends on the effective dimensionality of the system. For increasing applied drives we find a transition from 2D disordered filamentary flow near threshold to a 1D smectic flow which can be identified experimentally using c…
We calculate the thermal conductivity of interacting electrons in disordered metals. In our analysis we point out that the interaction affects thermal transport through two distinct mechanims, associated with quantum interference corrections and energy exchange of the quasi particles with the electromagnetic environment, respectively. The latter is seen to lead to a violation of the Wiedemann-Franz law. Our theory predicts a strong enhancement of the Lorenz ratio $κ/σT$ over the value which is predicted by the Wiedemann-Franz law, when the electrons encounter a large environmental impedance.
The form factor for spectral correlations in a diffusive metal is calculated in the presence of several Aharonov-Bohm fluxes. When the fluxes $ϕ$ are equal, the correlations are universal functions of $n g^2 ϕ$ where $g$ is the dimensionless conductance and $n$ is the number of applied fluxes. This explains recent flux dependence of the correlations found numerically at the metal-insulator transition.
We report on magnetotransport measurements performed on a large metallic two-dimensional $\mathcal{T}_{3}$ network. Superimposed on the conventional Altshuler-Aronov-Spivak (AAS) oscillations of period $h/2e$, we observe clear $h/e$ oscillations in magnetic fields up to $8 T$. Different interpretations of this phenomenon are proposed.
We study the universality class for localization which arises from models of non-interacting quasiparticles in disordered superconductors that have neither time-reversal nor spin-rotation symmetries. Two-dimensional systems in this category, which is known as class D, can display phases with three different types of quasiparticle dynamics: metallic, localized, or with a quantized (thermal) Hall conductance. Correspondingly, they can show a variety of delocalization transitions. We illustrate this behavior by investigating numerically the phase diagrams of network models with the appropriate sy…
We calculate the distribution of the conductance P(g) for a quasi-one-dimensional system in the metal to insulator crossover regime, based on a recent analytical method valid for all strengths of disorder. We show the evolution of P(g) as a function of the disorder parameter from a insulator to a metal. Our results agree with numerical studies reported on this problem, and with analytical results for the average and variance of g.
Using a one-dimensional tight-binding Anderson model, we study a disordered nanowire in the presence of an external gate which can be used for depleting its carrier density (field effect transistor device configuration). In this first paper, we consider the low temperature coherent regime where the electron transmission through the nanowire remains elastic. In the limit where the nanowire length exceeds the electron localization length, we derive three analytical expressions for the typical value of the thermopower as a function of the gate potential, in the cases where the electron transport …
We model the effect of phase-breaking collisions on the coherent electron transport in a disordered one-dimensional single-channel wire. In our model the phase-breaking collisions break the wire into segments, where each segment is an independent series resistor with coherent electronic resistance and the segmentation is a stochastic process with Poisson distribution of phase-breaking scattering times. The wire resistance as a function of the wire length $L$, coherence length $L_ϕ$, and localisation length $ξ$ is calculated and the transition from coherent to incoherent transport is traced qua…
We consider the finkelstein action describing a system of spin polarized or spinless electrons near two dimensions, in the presence of disorder as well as the Coulomb interactions.We extend the renormalization group analysis of our previous work and evaluate the metal-insulator transition of the electron gas to second order in an (d-2) expansion. We obtain the complete scaling behavior of physical observables like the conductivity and the specific heat with varying frequency, temperature and/or electron density.
We present a microscopic theory of the anomalous Hall effect in metallic multi-band ferromagnets, which accounts for all scattering-independent contributions, i.e., both the intrinsic and the so-called side jump. For a model of Gaussian disorder, the anomalous Hall effect is expressed solely in terms of the electronic band structure of the host material. Our theory handles systematically the interband-scattering coherence effects. We demonstrate the method in the two-dimensional Rashba and three-dimensional ferromagnetic (III,Mn)V semiconductor models. Our formalism is directly amenable to ab …
We study electron transport through a domain wall in a ferromagnetic nanowire subject to spin-dependent scattering. A scattering matrix formalism is developed to address both coherent and incoherent transport properties. The coherent case corresponds to elastic scattering by static defects, which is dominant at low temperatures, while the incoherent case provides a phenomenological description of the inelastic scattering present in real physical systems at room temperature. It is found that disorder scattering increases the amount of spin-mixing of transmitted electrons, reducing the adiabatic…
A metal-insulator transition in two-dimensional electron gases at B=0 is found in Ga(Al)As heterostructures, where a high density of self-assembled InAs quantum dots is incorporated just 3 nm below the heterointerface. The transition occurs at resistances around h/e^2 and critical carrier densities of 1.2 10^11cm^-2. Effects of electron-electron interactions are expected to be rather weak in our samples, while disorder plays a crucial role.
We use the Lanczos method to calculate the variance of the number of energy levels in an energy window of width E below the Fermi energy for non-interacting disordered electrons on a thin three-dimensional ring threaded by an Aharonov-Bohm flux . We find that for small E the flux-dependent part of the variance is well described by a well-known Feynman diagram involving two Cooperons. However, this result cannot be extrapolated to energies E where the energy-dependence of the average density of states becomes significant. We discuss consequences for persistent currents.
The conductance of disordered wires with symplectic symmetry is studied by a random-matrix approach. It has been shown that the behavior of the conductance in the long-wire limit crucially depends on whether the number of conducting channels is even or odd. We focus on the metallic regime where the wire length is much smaller than the localization length, and calculate the ensemble-averaged conductance and its variance for both the even- and odd-channel cases. We find that the weak-antilocalization correction to the conductance in the odd-channel case is equivalent to that in the even-channel …
We calculate the magnetization dynamics induced by the inverse Faraday effect in disordered metals in THz regime by using the diagrammatic method. We find that the induced magnetization is proportional to the frequency of circularly polarized light.
The quantum phase diagram of disordered wires in a strong magnetic field is studied as a function of wire width and energy. The two-terminal conductance shows zero-temperature discontinuous transitions between exactly integer plateau values and zero. In the vicinity of this transition, the chiral metal-insulator transition (CMIT), states are identified that are superpositions of edge states with opposite chirality. The bulk contribution of such states is found to decrease with increasing wire width. Based on exact diagonalization results for the eigenstates and their participation ratios, we c…
We investigate the statistical properties of the scattering matrix $S$ describing the electron transport through quasi-one dimensional disordered systems. For weak disorder (metallic regime), the energy dependence of the phase shifts of $S$ is found to yield the same universal parametric correlations as those characterizing chaotic Hamiltonian eigenvalues driven by an external parameter. This is analyzed within a Brownian-motion model for $S$, which is directly related to the distribution of the Wigner-Smith delay time matrix. For large disorder (localized regime), transport is dominated by re…
We consider an antiferromagnetic insulator that is in contact with a metal. Spin accumulation in the metal can induce spin-transfer torques on the staggered field and on the magnetization in the antiferromagnet. These torques relate to spin pumping: the emission of spin currents into the metal by a precessing antiferromagnet. We investigate how the various components of the spin-transfer torque are affected by spin-independent disorder and spin-flip scattering in the metal. Spin-conserving disorder reduces the coupling between the spins in the antiferromagnet and the itinerant spins in the met…
We develop an effective medium theory for electromagnetic wave propagation through gapless non-uniform systems with dynamic chiral magnetic effect. The theory allows to calculate macroscopic-disorder-induced corrections to the values of optical, as well as chiral magnetic conductivities. In particular, we show that spatial fluctuations of optical conductivity induce corrections to the effective value of the chiral magnetic conductivity. Experimentally, these corrections can be observed as sharp features in the Faraday rotation angle near frequencies that correspond to the bulk plasmon resonanc…
We study Coulomb drag between two parallel disordered mesoscopic 1D-wires. By numerical ensemble averaging we calculate the statistical properties of the transconductance G_21 including its distribution. For wires with mutually uncorrelated disorder potentials we find that the mean value is finite, but with comparable fluctuations so that sign-reversal is possible. For identical disorder potentials the mean value and the fluctuations nare enhanced compared to the case of uncorrelated disorder.
It is believed, that a disordered one-dimensional (1D) wire with coherent electronic conduction is an insulator with the mean resistance <ρ> \simeq e^{2L/ξ} and resistance dispersion Δ_ρ \simeq e^{L/ξ}, where L is the wire length and ξis the electron localisation length. Here we show that this 1D insulator undergoes at full coherence the crossover to a 1D "metal", caused by thermal smearing and resonant tunnelling. As a result, Δ_ρ is smaller than unity and tends to be L/ξ- independent, while <ρ> grows with L/ξfirst nearly linearly and then polynomially, manifesting the so-called medium locali…
We measure the statistical distribution of the local density of optical states (LDOS) on disordered semi-continuous metal films. We show that LDOS fluctuations exhibit a maximum in a regime where fractal clusters dominate the film surface. These large fluctuations are a signature of surface-plasmon localization on the nanometer scale.
We propose a physical model based on disordered (a hole punched inside a material) monolayer transition metal dichalcogenides (TMDs) to demonstrate a large-gap quantum valley Hall insulator. We find an emergence of bound states lying inside the bulk gap of the TMDs. They are strongly affected by spin-valley coupling, rest- and kinetic- mass terms and the hole size. In addition, in the whole range of the hole size, at least two in-gap bound states with opposite angular momentum, circulating around the edge of the hole, exist. Their topological insulator (TI) feature is analyzed by the Chern num…
Temperature dependence of electron dephasing time $τ_ϕ(T)$ is calculated for a disordered metal with small concentration of superconductive grains. Above the macroscopic superconducting transition line, when electrons in the metal are normal, Andreev reflection from the grains leads to a nearly temperature-independent contribution to the dephasing rate. In a broad temperature range $1/τ_phi(T)$ strongly exceeds the prediction of the classical theory of dephasing in normal disordered conductors, whereas magnetoresistance is dominated (in two dimensions) by the Maki-Tompson correction and is pos…
We study coherent electron transport in a one-dimensional wire with disorder modeled as a chain of randomly positioned scatterers. We derive analytical expressions for all statistical moments of the wire resistance $ρ$. By means of these expressions we show analytically that the distribution $P(f)$ of the variable $f=\ln(1+ρ)$ is not exactly Gaussian even in the limit of weak disorder. In a strict mathematical sense, this conclusion is found to hold not only for the distribution tails but also for the bulk of the distribution $P(f)$.
Several aspects of interplay between Anderson localization and coherent amplification/absorption, and aspect of mirrorless laser for a laser-active (amplifying) disordered dielectric medium have been addressed. We have calculated the statistics of the reflection coefficient and the associated phase for a light wave reflected from a coherently amplifying/absorbing one-dimensional disordered dielectric medium for different lengths, different disorder strengths and different "amplification"/"absorption" parameters of the disordered sample. Problems with modeling coherent amplification/absorption …
We compute energy level correlations in weakly disordered metallic grains using the fermionic replica method. We use the standard sigma-model approach and show that non--trivial saddle points, which break replica symmetry, must be included in the calculation to reproduce the oscillatory behavior of the correlations. We calculate the correlation functions in all three classical ensembles GOE, GUE and GSE including the finite-dimensional gradient corrections. Our results coincides with those obtained by the supersymmetric sigma-model and the semi-classical trace formula.
The screening of magnetic moments in metals, the Kondo effect, is found to be quenched with a finite probability in the presence of nonmagnetic disorder. Numerical results for a disordered electron system show that the distribution of Kondo temperatures deviates strongly from the result expected from random matrix theory. A pronounced second peak emerges for small Kondo temperatures, showing that the probability that magnetic moments remain unscreened at low temperatures increases with disorder. Analytical calculations, taking into account correlations between eigenfunction intensities yield a…
These notes contain a rapid overview of the methods and results obtained in the field of propagation of waves in disordered media. The case of Schrödinger and Helmholtz equations are considered that describe respectively electrons in metals and scalar electromagnetic waves. The assumptions on the nature of disorder are discussed and perturbation methods in the weak disorder limit are presented. A central quantity, namely the probability of quantum diffusion is defined and calculated in the same limit. It is then shown that several relevant physical quantities are related to the return probabil…
We report on a novel non-linear electric field effect in the conductivity of disordered conductors. We find that an electric field gives rise to dephasing in the particle-hole channel, which depresses the interference effects due to disorder and interaction and leads to a non-linear conductivity. This non-linear effect introduces a field dependent temperature scale $T_E$ and provides a microscopic mechanism for electric field scaling at the metal-insulator transition. We also study the magnetic field dependence of the non-linear conductivity and suggest possible ways to experimentally verify o…
We calculate the orbital linear magnetic response of disordered metallic rings to an Aharonov-Bohm flux using the BCS model for attractive electron-electron interaction. The contribution of all levels including those up to a high energy cutoff results in a much larger value than previously obtained using the local interaction model. The possible relevance of our results to the resolution of the discrepancy between the experimental and theoretical values for the ensemble-averaged persistent currents in these systems is discussed.
A disordered array of metal grains with large and random intergrain conductances is studied within the one-loop accuracy renormalization group approach. While at low level of disorder the dependence of conductivity on log T is nonuniversal (it depends on details of the array's geometry), for strong disorder this dependence is described by a universal nonlinear function, which depends only on the array's dimensionality. In two dimensions this function is found numerically. The dimensional crossover in granular films is discussed.
We study localization properties of a one-dimensional disordered system characterized by a random non-hermitean hamiltonian where both the randomness and the non-hermiticity arises in the local site-potential; its real part being random, and a constant imaginary part implying the presence of either a coherent absorption or amplification at each site. While the two-probe transport properties behave seemingly very differently for the amplifying and the absorbing chains, the logarithmic resistance $u$ = ln$(1+R_4)$ where $R_4$ is the 4-probe resistance gives a unified description of both the case…
A transfer matrix technique is used to model phase coherent spin transport in the weakly disordered quasi one-dimensional channel of a gate-controlled electron spin interferometer [Datta and Das, Appl. Phys. Lett., 56, 665 (1990)]. It includes the effects of an axial magnetic field in the channel of the interferometer (caused by the ferromagnetic contacts), a Rashba spin-orbit interaction, and elastic (non-magnetic) impurity scattering. We show that in the presence of the axial magnetic field, non-magnetic impurities can cause spin relaxation in a manner similar to the Elliott-Yafet mechanism.…
The Kondo temperature of a magnetic impurity in a weakly disordered metal is distributed due to the randomness in the local exchange coupling, and the local electronic density of states (LDOS). We show that in a closed, phase coherent metal particle the resulting distribution of $T_K$ is strongly asymmetric and scales with the mean level spacing $Δ$. Its width is $2 \sqrt{Δ/βT_K}$, where $Δ$ is the mean level spacing, and $β=1,2$, with, without time reversal symmetry, respectively. Increasing the density of magnetic impurities, the distribution of Kondo temperatures (DKT) is found to become mo…
Photonic crystals with a two-dimensional triangular lattice have a conical singularity in the spectrum. Close to this so-called Dirac point, Maxwell's equations reduce to the Dirac equation for an ultrarelativistic spin-1/2 particle. Here we show that the half-integer spin and the associated Berry phase remain observable in the presence of disorder in the crystal. While constructive interference of a scalar (spin-zero) wave produces a coherent backscattering peak, consisting of a doubling of the disorder-averaged reflected photon flux, the destructive interference caused by the Berry phase sup…
A theoretical study of the coherent light scattering from disordered photonic crystal is presented. In addition to the conventional enhancement of the reflected light intensity into the backscattering direction, the so called coherent backscattering (CBS), the periodic modulation of the dielectric function in photonic crystals gives rise to a qualitatively new effect: enhancement of the reflected light intensity in directions different from the backscattering direction. These additional coherent scattering processes, dubbed here {\em umklapp scattering} (CUS), result in peaks, which are most p…
We numerically investigate the properties of speckle patterns formed by nonlinear point scatterers. We show that, in the weak localization regime, dynamical instability appears, eventually leading to chaotic behavior of the system. Analysing the statistical properties of the instability thresholds for different values of the system size and disorder strength, a scaling law is emphasized. The later is also found to govern the smallest decay rate of the linear system, putting thus forward the crucial importance of interference effects. This is also underlined by the fact that coherent backscatte…
We investigate a disordered two-dimensional lattice model for noninteracting electrons with long-range power-law transfer terms and apply the method of level statistics for the calculation of the critical properties. The eigenvalues used are obtained numerically by direct diagonalization. We find a metal-insulator transition for a system with orthogonal symmetry. The exponent governing the divergence of the correlation length at the transition is extracted from a finite size scaling analysis and found to be $ν=2.6\pm 0.15$. The critical eigenstates are also analyzed and the distribution of the…
The interplay of interactions and disorder in two-dimensional (2D) electron systems has actively been studied for decades. The paradigmatic approach involves starting with a clean Fermi liquid and perturbing the system with both disorder and interactions. We instead start with a clean non-Fermi liquid near a 2D ferromagnetic quantum critical point and consider the effects of disorder. In contrast with the disordered Fermi liquid, we find that our model does not suffer from runaway flows to strong coupling and the system has a marginally stable fixed point with perfect conduction.
Atomistic simulations are used to study segregation-induced intergranular film formation in Cu-Zr and Cu-Nb alloys. While Cu-Zr forms structurally disordered or amorphous films, ordered films comprised of a second phase usually precipitate in Cu-Nb, with a critical nucleation size of ~1 nm below which the ordered phase cannot form. While the ordered film is retained at high temperature for a low energy Σ11 (113) boundary, a disordering transition is observed for a high energy Σ5 (310) boundary at low dopant concentrations. Finally, the effect of free surfaces on dopant segregation and intergra…
Persistent current is a small but perpetual electric current that flows in metallic rings in the absence of any applied source. We compute the persistent currents of one-dimensional disordered metallic rings of interacting electrons in the presence of impurity on lattices up to 8-sites at half-filling and also away from half-filling using the Lanczos algorithm. For the case of half-filling, we observe that both interaction and disorder suppress the amplitude of the persistent currents by localizing the electrons. However, in the presence of disorder and away from half-filling, the Coulomb inte…
We study the effect of electron-electron interaction on the one-particle density of states (\emph{DOS}) $ρ^{(d)}(ε,T)$ of low-dimensional disordered metals near Fermi energy within the framework of the finite temperature conventional impurity diagram technique. We consider only diffusive limit and by a geometric re-summation of the most singular first order self-energy corrections via the Dyson equation we obtain a non-divergent solution for the \emph{DOS} at low energies, while for higher energies the well-known Altshuler-Aronov corrections are recovered. At the Fermi level $ρ^{(d)}(ε,T=0)\to…
We study the electronic densities of states (DOS) of strongly disordered superconducting thin films of TiN. We find, using Scanning Tunneling Microscopy (STM) that the DOS decreases towards the Fermi level in the normal phase obtained by applying magnetic fields. The DOS shows spatial fluctuations whose length scale is related to the energy dependent DOS and is similar in normal and superconducting phases. This suggests that Coulomb interactions lead to a spatially varying DOS in the normal phase of a disordered superconductor.
The proximity effect between a superconductor and a highly diffusive two-dimensional metal was revealed in a Scanning Tunneling Spectroscopy experiment. The in-situ elaborated samples consisted of superconducting single crystalline Pb islands interconnected by a non-superconducting atomically thin disordered Pb wetting layer. In the vicinity of each superconducting island the wetting layer acquires specific tunneling characteristics which reflect the interplay between the proximity-induced superconductivity and the inherent electron correlations of this ultimate diffusive two-dimensional metal…
We investigate the finite-size scaling of the boundary quantum geometric tensor (QGT) numerically close to the Anderson localization transition in the presence of small external magnetic fields. The QGT exhibits universal scaling and reveals the crossover between the orthogonal and unitary critical states in weak random magnetic fields. The flow of the QGT near the critical points determines the critical exponents. Critical distributions of the QGT are universal and exhibit a remarkable isotropy even in a homogeneous magnetic field. We predict universal and isotropic Hall conductance fluctuati…
We calculate self-consistently the mutual dependence of electron correlations and electron-defect scattering for a two dimensional electron gas at finite temperature. We employ an STLS approach to calculate the electron correlations while the electron scattering rate off Coulombic impurities and surface roughness is calculated using self-consistent current-relaxation theory. The methods are combined and self-consistently solved. We discuss a metal-insulator transition for a range of disorder levels and electron densities. Our results are in good agreement with recent experimental observations.
Attractive negative-$U$ interactions between electrons facilitated by strong electron-phonon interaction are common in highly polarizable and disordered materials such as amorphous chalcogenides, but there is no direct evidence for them in large-band-gap insulators. Here we report how such negative-$U$ centers are responsible for widespread insulator-to-metal transitions in amorphous HfO$_2$ and Al$_2$O$_3$ thin films with a 10$^9$-fold resistance drop. Triggered by a static hydraulic pressure or a 0.1 ps impulse of magnetic pressure, the transition can proceed at such low pressure that there …
The crossover between a free magnetic moment phase and a Kondo phase in low dimensional disordered metals with dilute magnetic impurities is studied. We perform a finite size scaling analysis of the distribution of the Kondo temperature as obtained from a numerical renormalization group calculation of the local magnetic susceptibility and from the solution of the self-consistent Nagaoka-Suhl equation. We find a sizable fraction of free (unscreened) magnetic moments when the exchange coupling falls below a disorder-dependent critical value $J_{\rm c}$. Our numerical results show that between th…
Quantum transport in disordered systems is studied using a polaron-based master equation. The polaron approach is capable of bridging the results from the coherent band-like transport regime governed by the Redfield equation to incoherent hopping transport in the classical regime. A non-monotonic dependence of the diffusion coefficient is observed both as a function of temperature and system-phonon coupling strength. In the band-like transport regime, the diffusion coefficient is shown to be linearly proportional to the system-phonon coupling strength, and vanishes at zero coupling due to Ande…
We demonstrate that a weakly disordered metal with short-range interactions exhibits a transition in the quantum chaotic dynamics when changing the temperature or the interaction strength. For weak interactions, the system displays exponential growth of the out-of-time-ordered correlator (OTOC) of the current operator. The Lyapunov exponent of this growth is temperature-independent in the limit of vanishing interaction. With increasing the temperature or the interaction strength, the system undergoes a transition to a non-chaotic behaviour, for which the exponential growth of the OTOC is absen…
