# Course Identification

## Lecturers and Teaching Assistants

## Course Schedule and Location

## Field of Study, Course Type and Credit Points

Chemical Sciences: 2.00 points

## Comments

July 28th - August 8th, 2024 (Sunday-Thursday + Sunday-Thursday, 09:00-17:00 every day). 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, though everyone is encouraged to apply.

Lecturers:

Prof. Edan Lerner (classical part, University of Amsterdam, see https://staff.fnwi.uva.nl/e.lerner/)

Prof. Natalia Chepiga (quantum part, Delft University of Technology, see https://nchepiga.github.io/homepage/)

Coordinator: Prof. Eran Bouchbinder (Ben May Center, https://centers.weizmann.ac.il/ben-may-chemical-computation/)

See additional details below

## 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)

## Restrictions

## Language of Instruction

## Attendance and participation

## Grade Type

## Grade Breakdown (in %)

## Evaluation Type

**No final exam or assignment**

## Scheduled date 1

## Estimated Weekly Independent Workload (in hours)

## Syllabus

__Detailed plan__

__Classical condensed matter (July 28th until and including Sunday, August 4th):__ 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__

- Research projects

*Exercise 6*:

- Oscillatory shear above and below jamming
- Computational projects

__Quantum condensed matter (August 5th until August 8th):__ 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

## Learning Outcomes

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.

## Reading List

- “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