Detailed plan
Classical condensed matter: In this module, we will focus on the atomistic-micromechanical world: we will ask and answer how systems of discrete interacting particles — representing bubbles, droplets or any other discrete mesoscopic entities — flow, jam, form solids and yield. In addition to delving into the physics of soft condensed matter, we will aim at constructing a broadly applicable computational platform that will be used to study the behavior of simple computer models for soft matter systems.
Day 1
Lecture 1:
- Overview: emergent phenomena in soft condensed matter
- The role of computer experiments in soft matter research
- Course overview – what are we going to learn this week?
- Introduction to molecular dynamics: complexity, integrators, thermostats and boundary conditions
- Quick and dirty thermostats and barostats
Exercise 1:
- Simple computer models of soft matter: inverse power law and harmonic interactions
- ‘My first MD’
- Is my code working? Tests via conservation laws
Day 2
Lecture 2:
- Equilibrium statistical physics and liquid state theory
- Metropolis Monte Carlo
- Diffusivity, viscosity and the Stokes-Einstein relation
- Supercooled liquids and the glass transition
Exercise 2:
- Cell-lists – reducing the computational complexity
- Measuring a liquid’s viscosity and diffusivity
- Equation-of-state and heat capacity
- Liquid structure and dynamics
Day 3
Lecture 3:
- Disordered solids: overview and open questions
- Continuum elasticity and Debye’s vibrational density of states
- Atomistic elasticity at finite temperature
- Athermal atomistic elasticity
- Viscoelasticity
Exercise 3:
- Thermal and athermal elasticity of disordered solids
Day 4
Lecture 4:
- Elastoplasticity – overview and open questions
- Theory of plastic instabilities in the zero-temperature limit
- Computational approaches to soft matter deformation and flow
Exercise 4:
- Lees-Edwards boundary conditions
- Stress-strain curves at finite temperatures
- Herschel-Bulkley rheology
Day 5
Lecture 5:
- ‘Jamming’ and ‘unjamming’, mean-field treatments
- Strain-stiffening
- Divergent viscosity of non-Brownian suspensions
Exercise 5:
- Elastic moduli across jamming
- The coordination-pressure relation
- The vibrational density of states near unjamming
- Finite-size scaling near the jamming point
- Memory formation, absorbing states
Day 6
Exercise 6:
- Oscillatory shear above and below jamming
- Computational projects
Quantum condensed matter: In this module, we will focus on collective phenomena in quantum many-body systems, whose modern understanding relies to a large extent on computational approaches. An overview of numerical tools for strongly correlated quantum systems, with a focus on Hilbert-space approaches, will be given. The module will provide a technical overview of main existing algorithms, hands-on tutorial on implementation of basic exact diagonalization and tensor network codes, and theoretical lectures on applications.
Day 7
Lecture 7 (Introduction):
- Quantum many-body problems
- Exact diagonalization
- Complexity
- Quantum-Classical correspondence
- Corner Transfer Matrix Renormalization Group (CTMRG) for classical 2D systems
- Entanglement
- Schmidt decomposition
- Area law
Exercise 7:
- Exact diagonalization (Ising + Heisenberg chains)
- Area law
Day 8
Lecture 8 (Basic Density-Matrix-Renormalization-Group – DMRG – algorithm):
- Matrix Product States (MPS) – tensor network representation of quantum state
- Matrix Product Operator (MPO) – tensor network representation of many-body Hamiltonian
- Variational optimization of networks
- Infinite-size DMRG
- Observables – how to make measurements with tensor networks
- Periodic boundary conditions
Exercise 8:
- Exact MPS
- Expressing a quantum state obtained with ED as a tensor network
- Constructing MPO for the simplest models (Ising, Heisenberg, ...)
Day 9
Lecture 9 (Applications):
- Study of quantum phase transitions:
powerful combination of DMRG and boundary conformal field theory
- Topological phases and entanglement spectra
- Time evolution/finite-temperature calculations
Exercise 9:
- Implementation of a finite-size DMRG (part-2)
- Contracting the network
Day 10
Lecture 10 (Beyond DMRG):
- MERA – tensor network ansatz for critical systems
- Sliced-DMRG – tensor networks for non-lattice Hamiltonians
- Tree- and comb-tensor networks
- iPEPS – tensor networks in 2D: Introduction + Applications
- Current developments of tensor networks in 3D
Exercise 10:
- Implementation of a finite-size DMRG (part-3)
- Variational optimization of the network