Chapter
The semiclassical Green's function
The semiclassical Green's function
The short-time contribution to G
The short-time contribution to G
Manipulations on Van Vleck's determinant
Manipulations on Van Vleck's determinant
From action to monodromy matrix
From action to monodromy matrix
Examples of singularity-dominated strong fluctuations
Examples of singularity-dominated strong fluctuations
Smells in random winds, and the sex life of moths
Smells in random winds, and the sex life of moths
van Hove singularities and kin
van Hove singularities and kin
Spectral twinkling for integrable systems: superpoisson fluctuations
Spectral twinkling for integrable systems: superpoisson fluctuations
Spectral twinkling for mixed systems
Spectral twinkling for mixed systems
Chaos, dissipation and quantal Brownian motion
Chaos, dissipation and quantal Brownian motion
Definition of the problem
Definition of the problem
Restricted versions of the problem
Restricted versions of the problem
Fluctuations: intensity and correlation time
Fluctuations: intensity and correlation time
Fluctuations: time-dependent Hamiltonian
Fluctuations: time-dependent Hamiltonian
Actual, parametric and reduced energy changes
Actual, parametric and reduced energy changes
The sudden and the adiabatic approximations
The sudden and the adiabatic approximations
Ballistic and diffusive energy spreading
Ballistic and diffusive energy spreading
Energy spreading and dissipation
Energy spreading and dissipation
Application to the "piston" example
Application to the "piston" example
The route to stochastic behavior
The route to stochastic behavior
The transition probability kernel
The transition probability kernel
Limitations on quantal-classical correspondence (QCC)
Limitations on quantal-classical correspondence (QCC)
The parametric evolution of P(n|m)
The parametric evolution of P(n|m)
The time evolution of P_t(n|m)
The time evolution of P_t(n|m)
Actual and parametric dynamics
Actual and parametric dynamics
The over-simplified RMT picture
The over-simplified RMT picture
The perturbative core-tail spreading profile
The perturbative core-tail spreading profile
An improved perturbation theory
An improved perturbation theory
Consequences of the improved perturbative treatment
Consequences of the improved perturbative treatment
The quantum-mechanical sudden approximation
The quantum-mechanical sudden approximation
The quantum-mechanical adiabatic approximation
The quantum-mechanical adiabatic approximation
Classical Brownian motion
Classical Brownian motion
The white-noise approximation (WNA)
The white-noise approximation (WNA)
Brownian motion and dephasing
Brownian motion and dephasing
Quantum chaos in extended systems: Spreading wave packets and avoided band crossings
Quantum chaos in extended systems: Spreading wave packets and avoided band crossings
Modeling avoided band crossings
Modeling avoided band crossings
A perturbation calculation for the three-band model
A perturbation calculation for the three-band model
Dynamics of quantum systems with fractal spectrum
Dynamics of quantum systems with fractal spectrum
A brief introduction to random matrices
A brief introduction to random matrices
Gaussian ensembles of Hermitian matrices
Gaussian ensembles of Hermitian matrices
Why universality of spectral fluctuations?
Why universality of spectral fluctuations?
Dyson's circular ensembles
Dyson's circular ensembles
Two-point correlator and form factor
Two-point correlator and form factor
CUE averages of symmetric functions
CUE averages of symmetric functions
CUE form factor and pair correlator
CUE form factor and pair correlator
How to process a single spectrum into a two-point correlator
How to process a single spectrum into a two-point correlator
Gaussian statistics and independence of finite-time traces
Gaussian statistics and independence of finite-time traces
Ergodicity of the form factor
Ergodicity of the form factor
Universality of spectral fluctuations
Universality of spectral fluctuations
Autocorrelations of secular determinants
Autocorrelations of secular determinants
The semiclassical tool in mesoscopic physics
The semiclassical tool in mesoscopic physics
Quantum transport through classically chaotic cavities
Quantum transport through classically chaotic cavities
The scattering approach to the conductance
The scattering approach to the conductance
Quantum interference in ballistic cavities
Quantum interference in ballistic cavities
Semiclassical transmission amplitudes
Semiclassical transmission amplitudes
Transmission coefficients and average values
Transmission coefficients and average values
Weak localization in the ballistic regime
Weak localization in the ballistic regime
Scattering and integrability in quantum transport
Scattering and integrability in quantum transport
Scattering through a rectangular cavity
Scattering through a rectangular cavity
Semiclassical transmission amplitudes for square cavities
Semiclassical transmission amplitudes for square cavities
Mean conductance in a square cavity
Mean conductance in a square cavity
Conductance fluctuations in a square cavity
Conductance fluctuations in a square cavity
Circular billiards, diffraction and tunneling
Circular billiards, diffraction and tunneling
Experiments on ballistic transport and other aspects of the theory
Experiments on ballistic transport and other aspects of the theory
Conductance fluctuations and weak localization in ballistic microstructures
Conductance fluctuations and weak localization in ballistic microstructures
Semiclassics vs. random matrix theory
Semiclassics vs. random matrix theory
Semiclassical approach to bulk conductivity
Semiclassical approach to bulk conductivity
Orbital magnetism in clean systems
Orbital magnetism in clean systems
Semiclassical treatment of susceptibilities
Semiclassical treatment of susceptibilities
Integrable vs. chaotic behavior
Integrable vs. chaotic behavior
Semiclassical approach to weak disorder
Semiclassical approach to weak disorder
Single-particle Green function
Single-particle Green function
Two-particle Green function
Two-particle Green function
Fixed-size impurity average of the magnetic susceptibility
Fixed-size impurity average of the magnetic susceptibility
Combined impurity and energy average of the susceptibility
Combined impurity and energy average of the susceptibility
Relation to experiment and other theories
Relation to experiment and other theories
Electron-electron interactions in the ballistic regime
Electron-electron interactions in the ballistic regime
Screened Coulomb interaction in two dimensions
Screened Coulomb interaction in two dimensions
Thermodynamics and semiclassics of small interacting systems
Thermodynamics and semiclassics of small interacting systems
First-order perturbation, diagonal and non-diagonal contributions
First-order perturbation, diagonal and non-diagonal contributions
Statistics of energy levels and eigenfunctions in disordered and chaotic systems: Supersymmetry approach
Statistics of energy levels and eigenfunctions in disordered and chaotic systems: Supersymmetry approach
Introduction to the supersymmetry method and application to RMT
Introduction to the supersymmetry method and application to RMT
Green's function approach
Green's function approach
Average DOS from supersymmetry
Average DOS from supersymmetry
Comments and generalizations
Comments and generalizations
Structure of the saddle point manifold
Structure of the saddle point manifold
Gaussian ensembles of different symmetry
Gaussian ensembles of different symmetry
Ensembles with non-Gaussian distributions of matrix elements
Ensembles with non-Gaussian distributions of matrix elements
Parametric level statistics
Parametric level statistics
Level statistics in a disordered sample: Diffusive sigma-model
Level statistics in a disordered sample: Diffusive sigma-model
Derivation of the diffusive sigma-model
Derivation of the diffusive sigma-model
Reduction to the 0D sigma-model: Universal limit
Reduction to the 0D sigma-model: Universal limit
Deviations from universality
Deviations from universality
Deviations from universality at omega << E_c
Deviations from universality at omega << E_c
Spectral characteristics related to R_2(s)
Spectral characteristics related to R_2(s)
Eigenfunction statistics in terms of the supersymmetric sigma-model
Eigenfunction statistics in terms of the supersymmetric sigma-model
Quasi-one-dimensional geometry
Quasi-one-dimensional geometry
Exact solution of the sigma-model
Exact solution of the sigma-model
Metallic regime (arbitrary d)
Metallic regime (arbitrary d)
2D: Weak multifractality of eigenfunctions
2D: Weak multifractality of eigenfunctions
Spatial correlations of eigenfunction amplitudes
Spatial correlations of eigenfunction amplitudes
Metallic regime (arbitrary d)
Metallic regime (arbitrary d)
Anomalously localized states and long-time relaxation
Anomalously localized states and long-time relaxation
Supersymmetry approach to the quantum chaos
Supersymmetry approach to the quantum chaos
Introduction: What we have learned from the diffusive problem
Introduction: What we have learned from the diffusive problem
Ballistic sigma-model from disorder averaging
Ballistic sigma-model from disorder averaging
sigma-model from energy averaging
sigma-model from energy averaging
Non-universal corrections and statistical noise
Non-universal corrections and statistical noise
Billiard with diffuse surface scattering
Billiard with diffuse surface scattering
The level number variance
The level number variance
sigma-model for the kicked rotor
sigma-model for the kicked rotor
Experimental study of quantum chaos with cold atoms
Experimental study of quantum chaos with cold atoms
A two-level atom in a standing-wave potential
A two-level atom in a standing-wave potential
Current and future directions
Current and future directions
Quantum stochasticity and the many-body problem
Quantum stochasticity and the many-body problem
Symmetries and the semiclassical approximation
Symmetries and the semiclassical approximation
Invariant chaotic manifolds and collective motion
Invariant chaotic manifolds and collective motion
Invariant manifolds in interacting many-body systems
Invariant manifolds in interacting many-body systems
Classical model for the giant dipole resonance
Classical model for the giant dipole resonance
Localization in Hilbert space
Localization in Hilbert space
Localization in polyatomic molecules
Localization in polyatomic molecules
Fock-space localization in quantum dots
Fock-space localization in quantum dots
Fock-space localization in self-bound fermionic many-body systems
Fock-space localization in self-bound fermionic many-body systems
Spectral statistics and periodic orbits
Spectral statistics and periodic orbits
Criterion of applicability of diagonal approximation
Criterion of applicability of diagonal approximation
Beyond the diagonal approximation
Beyond the diagonal approximation
The Hardy-Littlewood conjecture
The Hardy-Littlewood conjecture
Construction of the density of states from a finite number of periodic orbits
Construction of the density of states from a finite number of periodic orbits
Off-diagonal terms for the Riemann zeta-function
Off-diagonal terms for the Riemann zeta-function
Off-diagonal contribution for dynamical systems
Off-diagonal contribution for dynamical systems
Random matrix universality
Random matrix universality
Riemann-Siegel form of density of states
Riemann-Siegel form of density of states
Quantum chaos and thermalization for interacting particles
Quantum chaos and thermalization for interacting particles
Two-body random interaction model
Two-body random interaction model
Structure of the Hamiltonian matrix
Structure of the Hamiltonian matrix
Correlations in off-diagonal matrix elements
Correlations in off-diagonal matrix elements
Density of states and spectrum statistics
Density of states and spectrum statistics
Structure of exact eigenstates
Structure of exact eigenstates
Analytical solution for the LDOS
Analytical solution for the LDOS
Non-statistical properties of the TBRI-model
Non-statistical properties of the TBRI-model
Thermalization and onset of chaos
Thermalization and onset of chaos
Distribution of occupation numbers
Distribution of occupation numbers
Microcanonical vs. canonical distribution
Microcanonical vs. canonical distribution
Transition to the Fermi-Dirac distribution
Transition to the Fermi-Dirac distribution
Analytical approach to the n_s-distribution
Analytical approach to the n_s-distribution
Effective Fermi-Dirac distribution for finite systems
Effective Fermi-Dirac distribution for finite systems
Transition to chaos and statistical equilibrium
Transition to chaos and statistical equilibrium
Comparison between quantum and classical dynamics in the effective action formalism
Comparison between quantum and classical dynamics in the effective action formalism
Effective action in quantum mechanics
Effective action in quantum mechanics
Loop expansion of the effective action
Loop expansion of the effective action
Derivative expansion of the effective action
Derivative expansion of the effective action
Derivative expansion: Euclidean approach
Derivative expansion: Euclidean approach
Derivative expansion as a WKB-like approximation
Derivative expansion as a WKB-like approximation
2D anharmonic oscillator: classical
2D anharmonic oscillator: classical
2D anharmonic oscillator: quantum
2D anharmonic oscillator: quantum
Torus maps, symmetries, and spectral statistics
Torus maps, symmetries, and spectral statistics
Spectral statistics and random matrix theory
Spectral statistics and random matrix theory
Time-reversal and quantum phases
Time-reversal and quantum phases
Quantum symmetries with no classical limit
Quantum symmetries with no classical limit
Localization in billiards
Localization in billiards
Dynamical localization in conservative systems
Dynamical localization in conservative systems
Localization in 2D quantum billiards
Localization in 2D quantum billiards
Generic (KAM-type) billiard with perfect classical barriers (tori)
Generic (KAM-type) billiard with perfect classical barriers (tori)
Rough billiards, and stadium billiard: partial classical barriers, slow classical diffusion, and cantori
Rough billiards, and stadium billiard: partial classical barriers, slow classical diffusion, and cantori
Quantum dynamical localization
Quantum dynamical localization
Localization in 3D quantum billiards
Localization in 3D quantum billiards
Scaling method for numerical solution of 2D Helmholz equation
Scaling method for numerical solution of 2D Helmholz equation
New Directions in Quantum Chaos
( International School of Physics “Enrico Fermi” )
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