# Topological integrability, classical and quantum chaos, and the theory of dynamical systems in the physics of condensed matter

@article{Maltsev2019TopologicalIC,
title={Topological integrability, classical and quantum chaos, and the theory of dynamical systems in the physics of condensed matter},
author={A. Ya. Maltsev and Sergei Novikov},
journal={Russian Mathematical Surveys},
year={2019},
volume={74},
pages={141 - 173}
}
• Published 20 November 2018
• Physics, Mathematics
• Russian Mathematical Surveys
This survey is devoted to questions connected with the Novikov problem of describing the geometry of level curves of quasi-periodic functions on the plane with different numbers of quasi-periods. Considered here are the history of the question, the current state of research in this field, and a number of applications of this problem to various physical problems. The main focus is on applications of results obtained in this area to the theory of transport phenomena in electron systems…
5 Citations
Distinctive Features of Oscillatory Phenomena in Reconstructions of the Topological Structure of Electron Trajectories on Complex Fermi Surfaces
• A. Maltsev
• Physics
Journal of Experimental and Theoretical Physics
• 2021
We consider the behavior of classical and quantum oscillations in metals with complex Fermi surfaces near the directions of $\, {\bf B} \,$ corresponding to changes in the topological structure of
Reconstructions of the Electron Dynamics in Magnetic Field and the Geometry of Complex Fermi Surfaces
The paper considers the semiclassical dynamics of electrons on complex Fermi surfaces in the presence of strong magnetic fields. The reconstructions of the general topological structure of such
A bound on quantum chaos from Random Matrix Theory with Gaussian Unitary Ensemble
• Physics
Journal of High Energy Physics
• 2019
A bstractIn this article, using the principles of Random Matrix Theory (RMT) with Gaussian Unitary Ensemble (GUE), we give a measure of quantum chaos by quantifying Spectral From Factor (SFF)
Arnoux-Rauzy interval exchange transformations
• Mathematics
ANNALI SCUOLA NORMALE SUPERIORE - CLASSE DI SCIENZE
• 2021
The Arnoux-Rauzy systems are defined in [5], both as symbolic systems on three letters and exchanges of six intervals on the circle. In connection with a conjecture of S.P. Novikov, we investigate

## References

SHOWING 1-10 OF 68 REFERENCES
Quasiperiodic functions and dynamical systems in quantum solid state physics
• Physics
• 2003
Abstract.This is a survey article dedicated to the study of topological quantities in theory of normal metals discovered in the works of the authors during the last years. Our results are based on
The Theory of Closed 1-Forms, Levels of Quasiperiodic Functions and Transport Phenomena in Electron Systems
• Physics, Mathematics
Proceedings of the Steklov Institute of Mathematics
• 2018
The paper is devoted to the applications of the theory of dynamical systems to the theory of transport phenomena in metals in the presence of strong magnetic fields. More precisely, we consider the
Topology of quasi-periodic functions on the plane
• Mathematics, Physics
• 2004
In this paper the topological theory of quasi-periodic functions on the plane is presented. The development of this theory was started (in another terminology) by the Moscow topology group in the
Quasiperiodic functions theory and the superlattice potentials for a two-dimensional electron gas
We consider Novikov problem of the classification of level curves of quasiperiodic functions on the plane and its connection with the conductivity of two-dimensional electron gas in the presence of
Topological Phenomena in Normal Metals
• Mathematics, Physics
• 1997
This paper is devoted to topological phenomena in normal metals with rather complicated Fermi surface. The results of the article are based on the deep topological theorems concerning the geometry of
Characterization of the set of 'ergodic directions' in Novikov's problem of quasi-electron orbits in normal metals
In this paper we give a beautiful characterization of the ‘ergodic directions’ in Novikov’s problem of semiclassical orbits of quasi-electrons in a normal metal. Since the end of the 1950s physicists
The Hamiltonian formalism and a many-valued analogue of Morse theory
CONTENTS Introduction § 1. The Hamiltonian formalism. Simplest examples. Systems of Kirchhoff type. Factorization of the Hamiltonian formalism for the B-phase of 3He § 2. The Hamiltonian formalism of
Diffusion for chaotic plane sections of 3-periodic surfaces
• Mathematics
• 2014
We study chaotic plane sections of some particular family of triply periodic surfaces. The question about possible behavior of such sections was posed by S. P. Novikov. We prove some estimations on
Solitons, Geometry, and Topology: On the Crossroad
• Mathematics
• 1997
Hyperelliptic Kleinian functions and applications by V. M. Buchstaber, V. Z. Enolskii, and D. V. Leikin Functionals of the Peierls-Frohlich type and the variational principle for the Whitham
Existence and measure of ergodic leaves in Novikov's problem on the semiclassical motion of an electron
We show that ergodic regime'' appears for generic dispersion relations in the semiclassical motion of electrons in a metal and we prove that, in the fixed energy picture, the measure of the set of