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Vladimir E. Kravtsov

Publications and source records attributed to Vladimir E. Kravtsov.

17 recordsLinked to original sources
arXiv2026

Semi-localized ground state in a 1D system with long-range hopping

We study the localization of a quantum particle in a one-dimensional disordered system with long-range hopping amplitudes $t(r)\propto r^{-a}$. In contrast to the standard one-dimensional Anderson model ($a\to\infty$), in which all states are localized and the localization length is minimal at the band edge, the long-range model with $1<a<3/2$ exhibits a disorder-driven transition at the band edge, while high-energy states remain localized at arbitrary disorder strength. We investigate this transition for the ground state in momentum space. In the weak-disorder regime, we derive perturbative expressions for the characteristic functions and moments of the momentum-space wave function, as well as for its fractal dimensions. Our results demonstrate that the ground state exhibits $\it semilocalization$ rather than conventional localization, thereby extending the class of models displaying the unusual $\it semifractality$ of wave functions.

cond-mat.dis-nn↗
arXiv2026

Semi-fractality and localization on a chiral Cayley tree

We study a quantum particle hopping on an infinite Cayley tree with nearest-neighbor hopping amplitudes drawn from a distribution singular as $|t|^{-a}$ near weak links and no on-site disorder. Because the graph is bipartite, the model has chiral symmetry, which strongly affects the statistics of eigenstates at the center of the spectrum. Using population dynamics to solve the cavity equations for the propagator, we analyze the distribution of the local density of states and show that it develops broad power-law tails. These tails imply an unusual form of wave-function statistics, which we call semi-fractality: the eigenstates occupy an extensive fraction of the system, but their higher moments behave as in a multifractal state. We find that the symmetry properties of the local-density-of-states distribution are not fixed only by the symmetry class, but vary continuously with the exponent controlling the power-law hopping distribution. As this exponent is changed, the system crosses from a semi-fractal regime to a localized one. At the transition, the wave functions realize an extreme intermediate form that we call semi-localized, simultaneously extended in their support but localized according to higher moments.

cond-mat.dis-nn↗
arXiv2024

Renormalization group for Anderson localization on high-dimensional lattices

We discuss the dependence of the critical properties of the Anderson model on the dimension $d$ in the language of $β$-function and renormalization group recently introduced in Ref.[arXiv:2306.14965] in the context of Anderson transition on random regular graphs. We show how in the delocalized region, including the transition point, the one-parameter scaling part of the $β$-function for the fractal dimension $D_{1}$ evolves smoothly from its $d=2$ form, in which $β_2\leq 0$, to its $β_\infty\geq 0$ form, which is represented by the regular random graph (RRG) result. We show how the $ε=d-2$ expansion and the $1/d$ expansion around the RRG result can be reconciled and how the initial part of a renormalization group trajectory governed by the irrelevant exponent $y$ depends on dimensionality. We also show how the irrelevant exponent emerges out of the high-gradient terms of expansion in the nonlinear sigma-model and put forward a conjecture about a lower bound for the fractal dimension. The framework introduced here may serve as a basis for investigations of disordered many-body systems and of more general non-equilibrium quantum systems.

cond-mat.dis-nn↗
arXiv2024

Renormalization Group Analysis of the Anderson Model on Random Regular Graphs

We present a renormalization group analysis of the problem of Anderson localization on a Random Regular Graph (RRG) which generalizes the renormalization group of Abrahams, Anderson, Licciardello, and Ramakrishnan to infinite-dimensional graphs. The renormalization group equations necessarily involve two parameters (one being the changing connectivity of sub-trees), but we show that the one-parameter scaling hypothesis is recovered for sufficiently large system sizes for both eigenstates and spectrum observables. We also explain the non-monotonic behavior of dynamical and spectral quantities as a function of the system size for values of disorder close to the transition, by identifying two terms in the beta function of the running fractal dimension of different signs and functional dependence. Our theory provides a simple and coherent explanation for the unusual scaling behavior observed in numerical data of the Anderson model on RRG and of Many-Body Localization.

cond-mat.dis-nn↗
arXiv2022

Sensitivity of (multi)fractal eigenstates to a perturbation of the Hamiltonian

We study the response of an isolated quantum system governed by the Hamiltonian drawn from the Gaussian Rosenzweig-Porter random matrix ensemble to a perturbation controlled by a small parameter. We focus on the density of states, local density of states and the eigenfunction amplitude overlap correlation functions which are calculated exactly using the mapping to the supersymmetric nonlinear sigma model. We show that the susceptibility of eigenfunction fidelity to the parameter of perturbation can be expressed in terms of these correlation functions and is strongly peaked at the localization transition: It is independent of the effective disorder strength in the ergodic phase, grows exponentially with increasing disorder in the fractal phase and decreases exponentially in the localized phase. As a function of the matrix size, the fidelity susceptibility remains constant in the ergodic phase and increases in the fractal and in the localized phases at modestly strong disorder. We show that there is a critical disorder strength inside the insulating phase such that for disorder stronger than the critical the fidelity susceptibility decreases with increasing the system size. The overall behavior is very similar to the one observed numerically in a recent work by Sels and Polkovnikov [Phys. Rev. E 104, 054105 (2021)] for the normalized fidelity susceptibility in a disordered XXZ spin chain.

cond-mat.dis-nn↗
arXiv2019

Electron-phonon cooling power in Anderson insulators

First microscopic theory for electron-phonon energy exchange in Anderson insulators is developed. The major contribution to the cooling power as a function of electron temperature is shown to be directly related to the correlation function of the local density of electron states at small energy difference argument. In Anderson insulators not far from localization transition, this correlation function is strongly enhanced by wave-function's multi-fractality and, additionally, by the presence of Mott's resonant pairs of localized states. The theory we develop explains huge enhancement of the cooling power observed in insulating Indium Oxide films as compared to predictions of the theory previously developed for disordered metals. Our results open the way to predict the conditions appropriate for the observation of Many Body Localization transition those presence in electronic insulators was advocated in the seminal paper by Basko, Aleiner and Altshuler (2006) but have not been convincingly demonstrated yet.

cond-mat.mes-hall↗
arXiv2016

Conduction in quasi-periodic and quasi-random lattices: Fibonacci, Riemann, and Anderson models

We study the ground state conduction properties of noninteracting electrons in aperiodic but non-random one-dimensional models with chiral symmetry, and make comparisons against Anderson models with non-deterministic disorder. The first model we consider is the Fibonacci lattice, which is a paradigmatic model of quasicrystals; the second is the Riemann lattice, which we define inspired by Dyson's proposal on the possible connection between the Riemann hypothesis and a suitably defined quasicrystal. Our analysis is based on Kohn's many-particle localization tensor defined within the modern theory of the insulating state. In the Fibonacci quasicrystal, where all single-particle eigenstates are critical (i.e., intermediate between ergodic and localized), the noninteracting electron gas is found to be a conductor at most electron densities, including the half-filled case; however, at various specific fillings $ρ$, including the values $ρ= 1/g^n$, where $g$ is the golden ratio and $n$ is any integer, the gas turns into an insulator due to spectral gaps. Metallic behaviour is found at half-filling in the Riemann lattice as well; however, in contrast to the Fibonacci quasicrystal, the Riemann lattice is generically an insulator due to single-particle eigenstate localization, likely at all other fillings. Its behaviour turns out to be alike that of the off-diagonal Anderson model, albeit with different system-size scaling of the band-centre anomalies. The advantages of analysing the Kohn's localization tensor instead of other measures of localization familiar from the theory of Anderson insulators (such as the participation ratio or the Lyapunov exponent) are highlighted.

cond-mat.dis-nn↗
arXiv2013

Support set of random wave-functions on the Bethe lattice

We introduce a new measure of ergodicity, the support set $S_\varepsilon$, for random wave functions on disordered lattices. It is more sensitive than the traditional inverse participation ratios and their moments in the cases where the extended state is very sparse. We express the typical support set $S_{\varepsilon}$ in terms of the distribution function of the wave function amplitudes and illustrate the scaling of $S_{\varepsilon}\propto N^α$ with $N$ (the lattice size) for the most general case of the multi-fractal distribution. A number of relationships between the new exponent $α$ and the conventional spectrum of multi-fractal dimensions is established. These relationships are tested by numerical study of statistics of wave functions on disordered Bethe lattices. We also obtain numerically the finite-size spectrum of fractal dimensions on the Bethe lattice which shows two apparent fixed points as $N$ increases. The results allow us to conjecture that extended states on the Bethe lattice at all strengths of disorder below the localization transition are non-ergodic with a clear multifractal structure that evolves towards almost ergodic behavior in the clean limit.

cond-mat.stat-mech↗
arXiv2012

Anderson localization of one-dimensional hybrid particles

We solve the Anderson localization problem on a two-leg ladder by the Fokker-Planck equation approach. The solution is exact in the weak disorder limit at a fixed inter-chain coupling. The study is motivated by progress in investigating the hybrid particles such as cavity polaritons. This application corresponds to parametrically different intra-chain hopping integrals (a "fast" chain coupled to a "slow" chain). We show that the canonical Dorokhov-Mello-Pereyra-Kumar (DMPK) equation is insufficient for this problem. Indeed, the angular variables describing the eigenvectors of the transmission matrix enter into an extended DMPK equation in a non-trivial way, being entangled with the two transmission eigenvalues. This extended DMPK equation is solved analytically and the two Lyapunov exponents are obtained as functions of the parameters of the disordered ladder. The main result of the paper is that near the resonance energy, where the dispersion curves of the two decoupled and disorder-free chains intersect, the localization properties of the ladder are dominated by those of the slow chain. Away from the resonance they are dominated by the fast chain: a local excitation on the slow chain may travel a distance of the order of the localization length of the fast chain.

cond-mat.dis-nn↗
arXiv2011

Level compressibility in a critical random matrix ensemble: The second virial coefficient

We study spectral statistics of a Gaussian unitary critical ensemble of almost diagonal Hermitian random matrices with off-diagonal entries $<|H_{ij}|^{2} > \sim b^{2} |i-j|^{-2}$ small compared to diagonal ones $<|H_{ii}|^{2} > \sim 1$. Using the recently suggested method of {\it virial expansion} in the number of interacting energy levels (J.Phys.A {\bf 36},8265 (2003)), we calculate a coefficient $\propto b^{2}\ll 1$ in the level compressibility $χ(b)$. We demonstrate that only the leading terms in $χ(b)$ coincide for this model and for an exactly solvable model suggested by Moshe, Neuberger and Shapiro (Phys.Rev.Lett. {\bf 73}, 1497 (1994)), the sub-leading terms $\sim b^{2}$ being different. Numerical data confirms our analytical calculation.

cond-mat.dis-nn↗
arXiv2009

Horizon in Random Matrix Theory, Hawking Radiation and Flow of Cold Atoms

We propose a Gaussian scalar field theory in a curved 2D metric with an event horizon as the low-energy effective theory for a weakly confined, invariant Random Matrix ensemble (RME). The presence of an event horizon naturally generates a bath of Hawking radiation, which introduces a finite temperature in the model in a non-trivial way. A similar mapping with a gravitational analogue model has been constructed for a Bose-Einstein condensate (BEC) pushed to flow at a velocity higher than its speed of sound, with Hawking radiation as sound waves propagating over the cold atoms. Our work suggests a three-fold connection between a moving BEC system, black-hole physics and unconventional RMEs with possible experimental applications.

cond-mat.str-el↗
arXiv2007

Multiphoton Processes in Driven Mesoscopic Systems

We study the statistics of multi-photon absorption/emission processes in a mesoscopic ring threaded by an harmonic time-dependent flux $Φ(t)$. For this sake, we demonstrate a useful analogy between the Keldysh quantum kinetic equation for the electrons distribution function and a Continuous Time Random Walk in energy space with corrections due to interference effects. Studying the probability to absorb/emit $n$ quanta $\hbarω$ per scattering event, we explore the crossover between ultra-quantum/low-intensity limit and quasi-classical/high-intensity regime, and the role of multiphoton processes in driving it.

cond-mat.mes-hall↗
arXiv2003

Density of states for almost diagonal random matrices

We study the density of states (DOS) for disordered systems whose spectral statistics can be described by a Gaussian ensemble of almost diagonal Hermitian random matrices. The matrices have independent random entries $ H_{i \geq j} $ with small off-diagonal elements: $ <|H_{i \neq j}|^{2} > \ll <|H_{ii}|^{2} > \sim 1 $. Using the recently suggested method of a {\it virial expansion in the number of interacting energy levels} (Journ.Phys.A {\bf 36}, 8265), we calculate the leading correction to the Poissonian DOS in the cases of the Gaussian Orthogonal and Unitary Ensembles. We apply the general formula to the critical power-law banded random matrices and the unitary Moshe-Neuberger-Shapiro model and compare DOS of these models.

cond-mat.dis-nn↗
arXiv1998

Topological Spectral Correlations in 2D Disordered Systems

It is shown that the tail in the two-level spectral correlation function R(s) for particles on 2D closed disordered surfaces is determined entirely by surface topology: $R(s)=-χ/(6π^2βs^2)$, where $β$ = 1,2 or 4 for the orthogonal, unitary and symplectic ensembles, and $χ$ = 2(1-p) is the Euler characteristics of the surface with p "handles" (holes). The result is valid for g << s << g^2 for $β$=1,4 and for g << s << g^3 for $β$=2, where g >> 1 is the dimensionless conductance.

cond-mat↗
arXiv1995

SPECTRAL CORRELATIONS IN DISORDERED ELECTRONIC SYSTEMS: CROSSOVER FROM METAL TO INSULATOR REGIME

We use the semiclassical approach combined with the scaling results for the diffusion coefficient to consider the two-level correlation function $R(\varepsilon)$ for a disordered electron system in the crossover region, characterized by the appearance of a macroscopic correlation or localization length, $ξ$, that diverges at the metal-insulator transition. We show new critical statistics, characterized by a nontrivial asymptotic behavior of $R(\varepsilon)$, to emerge on both sides of the transition at higher energies, and to expand to all energies larger than mean level spacing when $ξ$ exceeds the system size.

cond-mat↗
arXiv1995

LEVEL CORRELATIONS DRIVEN BY WEAK LOCALIZATION IN 2-D SYSTEMS

We consider the two-level correlation function in two-dimensional disordered systems. In the non-ergodic diffusive regime, at energy $ε>E_{c}$ ($E_{c}$ is the Thouless energy), it is shown to be completely determined by the weak localization effects, thus being extremely sensitive to time-reversal and spin symmetry breaking: it decreases drastically in the presence of magnetic field or magnetic impurities and changes its sign in the presence of a spin-orbit interaction. In contrast to this, the variance of the levels number fluctuations is shown to be almost unaffected by the weak localization effects.

cond-mat↗