1. Computational approaches for quantum many-body systems
HGSFP Graduate Days SS2019
Martin Gärttner

2. Organizational matters
90 min lecture + 90 min programming exercises
Materials: heidelberg.de/user/marting/teaching/ss19_hgsfp_graddays
Programming exercises: Python with Jupyter notebooks
→ Install Anaconda with Python 3 (
 Alternative:
→ log in with your uni-id
Active participation and feedback is essential!

3. Course overview Lecture 1: Introduction to many-body spin systems
Quantum Ising model, Bloch sphere, tensor structure, exact diagonalization
Lecture 2: Collective spin models
LMG model, symmetry, semi-classical methods, Monte Carlo
Lecture 3: Entanglement
Mixed states, partial trace, Schmidt decomposition
Lecture 4: Tensor network states
Area laws, matrix product states, tensor contraction, AKLT model
Lecture 5: DMRG and other variational approaches
Energy minimization, PEPS and MERA, neural quantum states

4. Learning goals After today you will be able to …
… interpret the evolution of a single spin in the Bloch sphere picture.
… explain the complexity problem of quantum many-body systems and understand many-body spin Hamiltonians.
… apply the spin toolbox to build and diagonalize many-body spin Hamiltonians.
… study a quantum phase transition in the transverse field Ising model.

5. What is a spin? https://answergarden.ch/910798
Intrinsic angular momentum
Electron spin
Nuclear spin
Polarizations of a photon
Ground and excited level of atom/ion…
States of a superconducting circuit…
States 0 and 1
Unit of quantum information
Physical spin
Pseudo spin
Two-level system
Qubit

6. Why care about spins?
Simple, but still shows fundamental physical phenomena
Analytically solvable many-body problems
Many condensed matter physics problems come in the form of spin models (magnetism, Hubbard models map so spin models in specific cases)
Quantum computers are just many-spin systems

7. Quantum simulation Special purpose quantum computers
Emulate (spin) model Hamiltonians in controlled experiments
Overcome problem of quantum complexity
Numerical methods for spin models
Benchmark quantum simulators in tractable regimes
Testing approximations using comparison to experiment
Examples:
Trapped ions (Bollinger, Monroe, Blatt)
Rydberg atoms (Lukin, Broways, Weidemüller)
Ultracold atoms in optical lattices (Greiner, Bloch)
Nature 484, (2012)
Nature 551,   (2017)
Nat. Phys. 8,  (2012)
Nature 551, (2017)
Nature 561, (2018)
Science  342, (2013) Nature 545, (2017)
Science 349, 842 (2015)