Selected Topics in Computational Quantum Physics

• Numerical methods are playing an increasingly important role in the study of quantum many-body physics. 

• This course introduces modern numerical computational techniques that go beyond traditional analytical and perturbative approaches. 

• The content covers, but is not limited to, exact diagonalization (ED), density matrix renormalization group (DMRG), quantum Monte Carlo (QMC), and tensor network (TN) methods. 

• We will discuss the theoretical background, implementation, significant advancements, and key results of these methods. 

• Students are encouraged to apply advanced numerical methods to their own scientific research.

I. Introduction

• Quantum many-body problems and effective lattice models

• Quantum magnetic systems and quantum phase transitions

• Applications of linear algebra and matrix computation in computational physics

• Quickstart guide to the Python language

II. Exact Diagonalization (ED) Method

• Advantages, disadvantages, and algorithmic framework

• Constructing the matrix representation of the Hamiltonian

• Block diagonalization using symmetry

• Considerations for fermionic systems

• Exact diagonalization in momentum space and its applications

• Lanczos method for sparse matrix diagonalization

• Green's function theory and spectral function calculations

• Time evolution based on exact diagonalization

• Applications in cutting-edge research

III. Matrix Product States (MPS) and Density Matrix Renormalization Group (DMRG) Methods

• Origins and basic principles of the density matrix renormalization group method

• Graphical representation of tensors

• Reduced density matrices and entanglement entropy

• Traditional density matrix renormalization group methods

• Many-body entanglement and the area law

• Representation and canonical form of matrix product states

• AKLT states and parent Hamiltonians

• Constructing matrix product operator representations of quasi-one-dimensional Hamiltonians

• Variational matrix product state algorithm for one-dimensional systems

• Time-evolving block decimation algorithm for one-dimensional systems

• Transfer matrices and correlation lengths

• Density matrix renormalization group methods for quasi-one-dimensional systems

• Tangent space methods for matrix product states

• Applications in cutting-edge research

IV. Quantum Monte Carlo (QMC) Methods

• Probability theory, statistical methods, and importance sampling

• Markov chains and detailed balance

• The Metropolis method in classical Monte Carlo

• Phase transitions, critical exponents, and finite-size scaling analysis

• Applications of classical Monte Carlo methods

• Worldline quantum Monte Carlo methods and the continuous-time limit

• Stochastic series expansion (SSE) methods

• Fermion determinant quantum Monte Carlo methods

• The sign problem in quantum Monte Carlo

• Majorana quantum Monte Carlo methods

• An overview of other quantum Monte Carlo methods

• Applications in cutting-edge research

V. Tensor Network Methods

• From matrix product states to projected entangled-pair states (PEPS)

• Kitaev's toric code model and topological order

• Tensor network representation of the degenerate ground states for Kitaev's toric code model

• Tensor symmetries and their connection to topological order

• Bulk-edge correspondence of projected entangled-pair states

• Contraction methods for two-dimensional tensor networks

• Tensor network renormalization methods and applications

• Numerical solutions for two-dimensional projected entangled-pair states

• Multiscale entanglement renormalization ansatz (MERA) methods

• Combining various numerical methods

• Applications in cutting-edge research

VI. An Overview of Other Numerical Methods in Many-Body Physics

• Numerical renormalization group (NRG) methods and applications

• Applications of machine learning (ML) in many-body physics

VII. Conclusion and Discussion

• Selection of computational methods and analysis of numerical results

• Opportunities and challenges in quantum many-body computational physics

Courses co-lectured

• Lab. of Physics A/B, Undergraduate Course, Sep, 2019 - Jan. 2022, Feb. 2024 - Mar. 2024

• Introduction to Solid State Theory, Graduate Course, Feb. 2023 - Mar. 2023