Computational Condensed Matter: A Summer course
Offered by the Ben May Center for Theory & Computation
Goals: Students will acquire basic and hands-on computational skills in the fields of classical and quantum condensed matter. In addition, they will become familiar with outstanding open questions related to emergent phenomena in hard and soft condensed matter.
Schedule & Structure: August 3rd – August 14th, 2024 (Sunday-Thursday + Sunday-Thursday, 09:00-17:00 every day, WsoS, room B). The course will be structured so that frontal lectures are given in the mornings, followed by practical, hands-on sessions after lunch. The course will host 26 participants, priority will be given to early-stage PhD students.
Lecturers:
Prof. Edan Lerner (classical part, see https://staff.fnwi.uva.nl/e.lerner/)
Dr. Natalia Chepiga (quantum part, see https://nchepiga.github.io/homepage/)
Prerequisites:
A graduate-level course in Quantum Mechanics, A graduate-level course in Statistical Physics/Thermodynamics, Knowledge of a low-level programming language such as C or Fortran is preferable (for the classical part), Basic knowledge of Matlab (for the quantum part)
Detailed plan
Classical condensed matter (Sunday, August 3rd, until Thursday, August 7th): 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:
- Introduction
- Overview: emergent phenomena in soft condensed matter
- The role of computer experiments in soft matter research
- Course overview – what are we going to learn and do this week?
- Introduction to molecular dynamics (MD): complexity, integrators, thermostats, barostats, boundary conditions, Langevin dynamics
- Monte Carlo (MC) methods
Exercise 1:
- Kramer’s escape problem
- ‘My first MD’,
- MC benchmarking ‘My first MD’
Day 2
Lecture 2:
- Continuum fluid mechanics
- Quantifying the microscopic structure of liquids
- Quantifying liquids’ structural dynamics
- Viscosity and the Stokes-Einstein relation
- Dynamical arrest – supercooled liquids and the glass transition
Exercise 2:
- Diffusivity & its Green-Kubo relation
- Viscosity and Stokes-Einstein relation
Day 3
Lecture 3:
- Continuum solid mechanics
- Thermal and athermal Elastic moduli, Poisson’s ratio
- Debye’s theory of phonons
- Scaling theory of disorder-induced phonon-band widths
- Nonphononic vibrations in glasses
- Viscoelasticity
Exercise 3:
- The FIRE minimization algorithm
- Lees-Edwards boundary conditions
- Thermal and athermal elastic moduli
- Debye’s density of states
Day 4
Lecture 4:
- Rheology of soft matter – yield stress and the Herschel-Bulkley law
- Theory of plastic instabilities
- Nonlinear plastic modes
- Statistics of plasticity avalanches in sheared glasses
- Strain localization phenomena
Exercise 4:
- Herschel-Bulkley law
- Physics of oscillatory shear
Day 5
Lecture 5:
- Elastic jamming/unjamming: vibrations, lengthscales, elastic moduli
- Non-Brownian suspension rheology
- Strain-stiffening
- Jamming physics of hard spheres
Exercise 5:
- Unjamming of bead-spring networks: spectra & elastic moduli
- Pressure-coordination relation in harmonic packings
Quantum condensed matter (Sunday, August 10th, until Thursday, August 14th): 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 6
Lecture 6 (Motivation and Introduction):
- Quantum many-body problems
- Exact diagonalization
- Complexity
- Quantum-Classical correspondence
- Corner Transfer Matrix Renormalization Group (CTMRG) for classical 2D systems
Exercise 6:
- Exact diagonalization (Ising + Heisenberg chains)
Day 7
Lecture 7 (Basic Density-Matrix-Renormalization-Group – DMRG – algorithm):
- Area law
- Entanglement
- Matrix Product States (MPS) – tensor network representation of quantum states
- Matrix Product Operator (MPO) – tensor network representation of many-body Hamiltonian
- Variational optimization of networks and DMRG
Exercise 7:
- Area law
- Exact MPS
- Expressing a quantum state obtained with ED as a tensor network
- Constructing MPO for the simplest models (Ising, Heisenberg, AKLT)
Day 8
Lecture 8 (Top-up):
- Observables
- Infinite-size DMRG
- Periodic boundary conditions
- Time evolution/finite-temperature calculations
Exercise 8:
- Implementation of a finite-size DMRG (part-1): Environments
Day 9
Lecture 9 (Advanced DMRG):
- Symmetries and fragmentation of the Hilbert space in Tensor Networks
- Constrained tensor networks
- Excitations
- Boundaries and finite-size scaling
Exercise 9:
- Implementation of a finite-size DMRG (part-2): Variational optimization
Day 10
Lecture 9 (Beyond DMRG):
- MERA – tensor network ansatz for critical systems
- iPEPS – tensor networks in 2D: Introduction + Applications
- iPEPS in 3D and layered systems
- Sliced-DMRG – tensor networks for non-lattice Hamiltonians
- Tree- and comb-tensor networks
Exercise 10:
- Implementation of a finite-size DMRG (part-3): Measuring observables
Grading:
Pass/Fail, depending on 100% attendance (which is obligatory) and completion of all hands-on assignments
Reading:
- “Computer simulation of liquids”, M.P. Allen and D.J. Tildesley,
Oxford university press (2017)
- “The density-matrix renormalization group in the age of matrix product states”,
U. Schollwoeck, Annals of Physics 326, 96 (2011)
- Additional teaching materials will be distributed in due time