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## Eigenstate correlations around the many-body localization transition

We explore correlations of eigenstates around the many-body localization (MBL) transition in their dependence on the energy difference (frequency) ω and disorder W. In addition to the genuine many-body problem, XXZ spin chain in random field, we consider localization on random regular graphs that serves as a toy model of the MBL transition. Both models show a very similar behavior. On the localized side of the transition, the eigenstate correlation function β(ω) shows a power-law enhancement of correlations with lowering ω; the corresponding exponent depends on W. The correlation between adjacent-in-energy eigenstates exhibits a maximum at the transition point W_c, visualizing the drift of W_c with increasing system size towards its thermodynamic-limit value. The correlation function β(ω) is related, via Fourier transformation, to the Hilbert-space return probability. We discuss measurement of such (and related) eigenstate correlation functionson state-of-the-art quantum computers and simulators.

A number of problems in statistical physics can be reformulated in terms of a two-state system evolving in a random field. The corresponding evolution operator can be written in the form of time-ordered operator exponential. Functional formalism allows us to rewrite the latter as a product of usual matrix exponentials using a nonlinear change of functional integration variables. In this review I present this formalism applied to two physical systems the quantum Heisenberg magnet and one-dimensional quantum mechanics in a spatially random potential. First, I derive a representation of the partition function of a quantum Heisenberg ferromagnet as a functional integral over number valued fields (a real one and a complex one) free of constraints. The fields of integration as functions of time obey initial conditions instead of the usual periodic boundary conditions. This is a manifestation of the finite-dimensionality of the space of spin states. In the subsequent sections I study the one-dimensional localization problem. The change of functional integration variables gives simultaneously explicit expressions for the averaging weight and for the Green function of the stationary Schrödinger equation. It allows to compute density correlators of arbitrary orders. The generalization to the case of different energy correlators (Berezinskii-Gor’kov equations) is considered too. In the present review such technical points as regularizations of functional integrals and transformations are discussed in more details than in the original papers.

start from the derivation of the Abrikosov-Ryzhkin model for the 1D random potential problem. In its framework I find closed functional representations for various physical quantities. The representation uses number-valued fields only. These functional integrals are calculated exactly without the use of any perturbative expansions. Expressions for the multipoint densities correlators are obtained. These correlators allow to compute the distribution function of inverse sizes of localized wave functions valid both for an infinite sample and for a sample with a finite length.

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.

Radiation conditions are described for various space regions, radiation-induced effects in spacecraft materials and equipment components are considered and information on theoretical, computational, and experimental methods for studying radiation effects are presented. The peculiarities of radiation effects on nanostructures and some problems related to modeling and radiation testing of such structures are considered.

The paper provides a number of proposed draft operational guidelines for technology measurement and includes a number of tentative technology definitions to be used for statistical purposes, principles for identification and classification of potentially growing technology areas, suggestions on the survey strategies and indicators. These are the key components of an internationally harmonized framework for collecting and interpreting technology data that would need to be further developed through a broader consultation process. A summary of definitions of technology already available in OECD manuals and the stocktaking results are provided in the Annex section.