**Morten Hjorth-Jensen Email morten.hjorth-jensen@fys.uio.no**, Department of Physics, University of Oslo and Department of Physics and Astronomy and National Superconducting Cyclotron Laboratory, Michigan State University

Date: **Jan 16, 2020**

Copyright 1999-2020, Morten Hjorth-Jensen Email morten.hjorth-jensen@fys.uio.no. Released under CC Attribution-NonCommercial 4.0 license

Be able to apply central many-particle methods like the Variational Monte Carlo method to properties of many-fermion systems and many-boson systems.

Understand how to simulate quantum mechanical systems with many interacting particles. The methods are relevant for atomic, molecular, solid state, materials science, nanotechnology, quantum chemistry and nuclear physics.

Learn to manage and structure larger projects, with unit tests, object orientation and writing clean code

Learn about a proper statistical analysis of large data sets

Parallelization and code optimizations

Lectures: Thursday (2.15pm-4pm), remotely. First time January 16.

Computerlab: Thursday (4.15pm-7pm), first time January 16, last lab session May 9.

Weekly plans and all other information are on the webpage of the course

Intensive week starts March 2 and ends March 6. Lectures Monday-Wednesday to be decided and regular session on Thursday February 7 (2.15pm-7pm).

**First project to be handed in March 23**.**Second and final project to be handed in June 1.**We use piazza for discussions. Sign up as soon as possible.

There is no final exam, only project work.

Two compulsory projects. Electronic reports only. You are free to choose your format. We use devilry to hand in the projects.

Evaluation and grading: The two projects count 1/2 each of the final mark. No exam.

The computer lab (room 397 in the Physics buidling) has no PCs, so please bring your own laptops. C/C++ is the default programming language, but Fortran2008 and Python are also used. All source codes discussed during the lectures can be found at the webpage of the course. We recommend either C/C++, Fortran2008 or Python as programming languages.

Parallelization (MPI and OpenMP), high-performance computing topics. Choose between Python, Fortran2008 and/or C++ as programming languages.

Algorithms for Monte Carlo Simulations (multidimensional integrals), Metropolis-Hastings and importance sampling algorithms. Improved Monte Carlo methods.

Statistical analysis of data from Monte Carlo calculations, bootstrapping, jackknife and blocking methods.

Eigenvalue solvers

For project 2 there will be three variants:

a. Variational Monte Carlo for fermions

b. Hartree-Fock theory for fermions

c. Coupled cluster theory for fermions (iterative methods)

d. Neural networks and Machine Learning to solve the same problems as in project 1

e. Eigenvalue problems with deep learning methods

Search for minima in multidimensional spaces (conjugate gradient method, steepest descent method, quasi-Newton-Raphson, Broyden-Jacobian). Convex optimization, gradient methods

Iterative methods for solutions of non-linear equations.

Object orientation

Data analysis and resampling techniques

Variational Monte Carlo (VMC) for 'ab initio' studies of quantum mechanical many-body systems.

Simulation of two-dimensional systems like quantum dots or atoms and molecules or systems from solid state physics

**Simulation of trapped bosons using VMC (project 1, default)****Machine learning and neural networks (project 2, default, system same as in project 1)**Extension of project 1 to fermionic systems (project 2)

Coupled cluster theory (project 2, depends on interest)

Other quantum-mechanical methods and systems can be tailored to one's interests (Hartree-Fock Theory, Many-body perturbation theory, time-dependent theories and more).

Most quantum mechanical problems of interest in for example atomic, molecular, nuclear and solid state physics consist of a large number of interacting electrons and ions or nucleons.

The total number of particles $N$ is usually sufficiently large that an exact solution cannot be found.

Typically, the expectation value for a chosen hamiltonian for a system of $N$ particles is

$$
\langle H \rangle =
\frac{\int d\boldsymbol{R}_1d\boldsymbol{R}_2\dots d\boldsymbol{R}_N
\Psi^{\ast}(\boldsymbol{R_1},\boldsymbol{R}_2,\dots,\boldsymbol{R}_N)
H(\boldsymbol{R_1},\boldsymbol{R}_2,\dots,\boldsymbol{R}_N)
\Psi(\boldsymbol{R_1},\boldsymbol{R}_2,\dots,\boldsymbol{R}_N)}
{\int d\boldsymbol{R}_1d\boldsymbol{R}_2\dots d\boldsymbol{R}_N
\Psi^{\ast}(\boldsymbol{R_1},\boldsymbol{R}_2,\dots,\boldsymbol{R}_N)
\Psi(\boldsymbol{R_1},\boldsymbol{R}_2,\dots,\boldsymbol{R}_N)},
$$

an in general intractable problem.

This integral is actually the starting point in a Variational Monte Carlo calculation. **Gaussian quadrature: Forget it**! Given 10 particles and 10 mesh points for each degree of freedom
and an
ideal 1 Tflops machine (all operations take the same time), how long will it take to compute the above integral? The lifetime of the universe is of the order of $10^{17}$ s.

As an example from the nuclear many-body problem, we have Schroedinger's equation as a differential equation

$$
\hat{H}\Psi(\boldsymbol{r}_1,..,\boldsymbol{r}_A,\alpha_1,..,\alpha_A)=E\Psi(\boldsymbol{r}_1,..,\boldsymbol{r}_A,\alpha_1,..,\alpha_A)
$$

where

$$
\boldsymbol{r}_1,..,\boldsymbol{r}_A,
$$

are the coordinates and

$$
\alpha_1,..,\alpha_A,
$$

$$
2^A\times \left(\begin{array}{c} A\\ Z\end{array}\right)
$$

coupled second-order differential equations in $3A$ dimensions.

For a nucleus like beryllium-10 this number is **215040**.
This is a truely challenging many-body problem.

Methods like partial differential equations can at most be used for 2-3 particles.

Monte-Carlo methods

Renormalization group (RG) methods, in particular density matrix RG

Large-scale diagonalization (Iterative methods, Lanczo's method, dimensionalities $10^{10}$ states)

Coupled cluster theory, favoured method in quantum chemistry, molecular and atomic physics. Applications to ab initio calculations in nuclear physics as well for large nuclei.

Perturbative many-body methods

Green's function methods

Density functional theory/Mean-field theory and Hartree-Fock theory

The physics of the system hints at which many-body methods to use.

**Pros and Cons of Monte Carlo.**

Is physically intuitive.

Allows one to study systems with many degrees of freedom. Diffusion Monte Carlo (DMC) and Green's function Monte Carlo (GFMC) yield in principle the exact solution to Schroedinger's equation.

Variational Monte Carlo (VMC) is easy to implement but needs a reliable trial wave function, can be difficult to obtain. This is where we will use Hartree-Fock theory to construct an optimal basis.

DMC/GFMC for fermions (spin with half-integer values, electrons, baryons, neutrinos, quarks) has a sign problem. Nature prefers an anti-symmetric wave function. PDF in this case given distribution of random walkers ($p\ge 0$).

The solution has a statistical error, which can be large.

There is a limit for how large systems one can study, DMC needs a huge number of random walkers in order to achieve stable results.

Obtain only the lowest-lying states with a given symmetry. Can get excited states with extra labor.

**Where and why do we use Monte Carlo Methods in Quantum Physics.**

Quantum systems with many particles at finite temperature: Path Integral Monte Carlo with applications to dense matter and quantum liquids (phase transitions from normal fluid to superfluid). Strong correlations.

Bose-Einstein condensation of dilute gases, method transition from non-linear PDE to Diffusion Monte Carlo as density increases.

Light atoms, molecules, solids and nuclei.

Lattice Quantum-Chromo Dynamics. Impossible to solve without MC calculations.

Simulations of systems in solid state physics, from semiconductors to spin systems. Many electrons active and possibly strong correlations.