2. Tutorial

• 2.1. Introduction
  • 2.1.1. Outline of the tutorial
  • 2.1.2. References
• 2.2. Getting started: a simple example with a one-dimensional chain
• 2.3. Second example: Fabry-Perot interferometer
  • 2.3.1. References
• 2.4. Time-dependent potentials and pulses
  • 2.4.1. Time dependent onsite elements
  • 2.4.2. Time dependent coupling elements
  • 2.4.3. Voltage pulses through a lead
• 2.5. Solving one-body problems
  • 2.5.1. Finite systems
    • Defining the Hamiltonian matrix directly
    • Defining the system using Kwant
  • 2.5.2. Infinite systems
    • Defining the system using Kwant
    • Infinite systems with initial scattering states
  • 2.5.3. References
• 2.6. Solving the many-body problem
  • 2.6.1. High-level automatic approach
    • Chemical potential and temperature of the leads
    • Adaptive refinement and error estimate
  • 2.6.2. Low-level manual approach
  • 2.6.3. Summary
• 2.7. Green functions
  • 2.7.1. Introduction
  • 2.7.2. The class \(\texttt{tkwant.manybody.Greenfunction}\)
  • 2.7.3. Example I: A two sites impurity on an infinite one-dimensional chain (double quantum dot)
    • Numerical calculation of \(G^<\) and \(G^R\) using Tkwant
      • Building the Kwant system
      • Time evolution and evaluation
      • Comparison to the analytical solution
    • Calculating equal time diagonal and off-diagonal Green functions related to density and current
    • Various kinds of Green functions
  • 2.7.4. Example II: Single impurity site: quench of the coupling to the leads
    • Transient dynamics
  • 2.7.5. References
  • 2.7.6. Appendix
• 2.8. Accounting for possible boundstates present in the system
  • 2.8.1. Model
  • 2.8.2. Analytical calculation of the boundstate
  • 2.8.3. Observables in presence of boundstates
  • 2.8.4. Numerical solution
    • Characterization of the boundstate and comparison to the analytical result
      • Boundstate energy
      • Boundstate wave function
    • Wrong equilibrium density in Tkwant due to a missing boundstate
    • Detecting boundstates with Tkwant
    • Including boundstates in Tkwant
  • 2.8.5. References
• 2.9. Self-consistent Tkwant: a generic solver for time-dependent mean field calculations
  • 2.9.1. Problem definition
  • 2.9.2. Toy example for a self-consistent Tkwant simulation
    • (I) Setting up the non-interacting wave-function \(\Psi_0\)
    • (II) Implementing the class to calculate \(y(t)\)
    • (III) Implementing the class to calculate the mean-field potential \(\mathbf{Q}[t, y]\)
    • (IV) Setting up the self-consistent solver
  • 2.9.3. API of the self-consistent solver and its helper classes
    • The class to calculate \(y(t)\)
    • The class to calculate the mean-field potential \(\mathbf{Q}[t, y]\)
    • Setting up the self-consistent solver
  • 2.9.4. Behind the scene: Timescale decoupling
  • 2.9.5. Real-live examples of self-consistent Tkwant simulations:
    • 1) Luttinger liquid physics from time-dependent Hartree approximation
      • (I) Non-interacting wavefunction \(\Psi_0\)
      • (II) Operator for the onsite density
      • (III) Class to compute the Hartree potential \(\mathbf{Q}(t)\)
      • (IV) Setting up the self-consistent solver
    • 2) The self-consistent Bogoliubov-deGennes / classical circuit problem
      • Modeling of the Josephson junction (quantum part)
      • Electrical environment (Classical part)
      • Full problem
      • Parameters
      • (I) Non-interacting wavefunction \(\Psi_0\)
      • (II) Operator for the current \(I(t)\) trough the junction
      • (III) Class to compute the mean-field term \(\mathbf{Q}(t)\)
      • (IV) Setting up the self-consistent solver
    • 3) Landau-Lifshitz-Gilbert equation study for spin dynamics
      • Parameters
      • (I) Non-interacting wavefunction \(\Psi_0\)
      • (II) Operator to compute the spin density
      • (III) Class to compute the mean-field term \(\mathbf{Q}(t)\)
      • (IV) Setting up the self-consistent solver
  • 2.9.6. Advanced settings
    • Accuracy of the self-consistent updates and adaptive stepsize control
    • Mean-field extrapolation
      • 0. order extrapolation
      • 1. order extrapolation
      • Changing the extrapolation order
  • 2.9.7. References
• 2.10. Advanced onebody settings
  • 2.10.1. Boundary conditions
  • 2.10.2. Time integration
  • 2.10.3. Time-dependent perturbation
  • 2.10.4. Saving and restarting states
• 2.11. Advanced manybody settings
  • 2.11.1. Lead occupation
    • Chemical potential
    • Temperature
    • Include / exclude bands
    • Include / exclude leads
    • Occupation data format
  • 2.11.2. Numerical integration
    • Quadrature order
    • Interval subdivision
    • Quadrature rule
    • Quadrature error
    • Energy vs. momentum integration
    • Interval data format
    • Tasks data format
    • Time integration
    • Time-dependent perturbation
  • 2.11.3. Retrieve one-body states
    • State identifier
    • State information
    • One-body states
    • Evolution time
  • 2.11.4. Miscellaneous
    • Band structure
    • Boundary conditions
    • Initial state
• 2.12. Boundary conditions
  • 2.12.1. Basic formalism
  • 2.12.2. A minimal example
    • Automatic boundary conditions
    • Changing default values
  • 2.12.3. How do tkwant boundary conditions work
    • Simple boundary conditions
    • Monomial absorbing boundary conditions
    • General absorbing boundary conditions
  • 2.12.4. Lead reflection analysis
    • Basic analysis
    • Advanced analysis
  • 2.12.5. An advanced example with “nasty modes”
    • Estimate optimal monomial parameters
    • Reflection of the nasty modes
    • Comparison of the reflection from the numerical exact and the monominal approximation
    • Comparison of the lead length for simple or absorbing boundary conditions
  • 2.12.6. References
• 2.13. Parallelization with MPI
  • 2.13.1. Running code in parallel with MPI
  • 2.13.2. Enabling output only from the MPI root rank
  • 2.13.3. MPI communicator
  • 2.13.4. Multi-threading
  • 2.13.5. Examples
    • References
• 2.14. Logging
  • 2.14.1. Level
  • 2.14.2. Handlers
  • 2.14.3. Filters
  • 2.14.4. Example
    • References
• 2.15. More examples
• 2.16. Frequent pitfalls encountered when doing Tkwant simulations
  • 2.16.1. Convergence of the manybody integral (1/2)
    • Problem
    • Diagnostic
    • Solution
  • 2.16.2. Convergence of the many-body integral (2/2)
    • Problem
    • Diagnostic
    • Solution
  • 2.16.3. Presence of unincluded boundstates
    • Problem
    • Diagnostic
    • Solution
  • 2.16.4. Error in Hamiltonian construction or unphysical parameter regime
    • Problem
    • Diagnostic
    • Solution
  • 2.16.5. Observables
    • Problem
    • Diagnostic
    • Solution
  • 2.16.6. Unsufficient accuracy in perturbation interpolation
    • Problem
    • Diagnostic
    • Solution
  • 2.16.7. Spurious reflections on the leads
    • Problem
    • Diagnostic
    • Solution
  • 2.16.8. Unsufficient time-stepping accuracy
    • Problem
    • Diagnostic
    • Solution
  • 2.16.9. Miscellaneous problems
  • 2.16.10. References
• 2.17. Frequently asked questions