1. Introduction

Traditionally, there are two major paradigms in science: The theoretical and the experimental approach. But one might argue that with the advent of computers, a third paradigm, namely computation, has emerged. These “bicycles for our minds”, as Steve Jobs used to call computers[1] [1, 2], opened up completely new possibilities in research. With the development of computational tools, theoreticians have gained the ability to generate data themselves, an approach I utilized extensively in the research presented in this thesis.

Both research projects investigate the ground states of interacting quantum systems, but solving such systems is a hard problem. Generically, one needs to consider all possible configurations to solve the system.

Many common problems in computational science amount to summation over a high-dimensional space, which leads to a hard challenge: The volume of a space scales exponentially with the number of dimensions and therefore (when utilizing a direct approach) the computational effort as well. For bigger systems, this quickly becomes unfeasible, as the following example shows: Consider \(N_{\rm S}\) spins, each having two possible states, up or down. This amounts to the simplest \(N_{\rm S}\)-dimensional space, with size \(2^{N_{\rm S}}\). The method for directly solving a corresponding model numerically is called exact diagonalization (ED) and amounts to calculating the eigenvalues (and -vectors) of a \(2^{N_{\rm S}} \times 2^{N_{\rm S}}\) matrix[2] [3]. Doing this for \(N_S=14\) needs about \(2\,\rm GB\) of RAM and one minute on a current laptop, but becomes quickly very expensive. To my knowledge, the biggest spin system solved via ED comprises 50 spins and took \(15.5\,\rm TB\) of RAM to solve [4].

Rather than integrating over the whole space, one can average over a random set of its elements. This does not produce an absolutely exact result, but for a large number \(N\) of statistically independent samples in a “well-behaved” problem one can use the central limit theorem to show that the result is exact up to a statistical error proportional to \(1/\sqrt{N}\) [5]. As a result, many problems that would need an exponential amount of processing power can be solved with polynomial effort instead. This method, known as statistical sampling, is older than the Monte Carlo method, but due to the tedious work of obtaining the samples it has not had significant scientific impact prior to the existence of computers.

John von Neumann and Stanislaw Ulam were possibly the first to realize the possibilities arising from combining statistical sampling with the tireless work of computers. They used the approach to simulate neutron diffusion processes in fissionable material at Los Alamos National Laboratory in the late 1940s. The secret project needed a code name and they used Monte Carlo, after the Casino in Monaco, a label that stuck [6].

The initial and ongoing exponential growth of computational power, described by Gordon E. Moore in 1965 [7] that has since become known as “Moore’s law”, enables new simulations leveraging the Monte Carlo method that push the boundaries of what has previously been possible.

There exist countless algorithms based on the statistical sampling approach, the method I used throughout my research is the BSS algorithm [8, 9, 10], which is a specific auxiliary field QMC algorithm. Section 1.1 will give a brief introduction to the method. More specifically, all QMC results in this thesis are produced with the program package Algorithms for Lattice Fermions (ALF) [11, 12], which implements the BSS algorithm in a very generic way, making it easy to implement new models. During my doctoral work, I added some significant contributions to ALF, like support for HDF5, a better separation/encapsulation between model definitions and the QMC algorithm, and various usability improvements [13]. The source code for ALF is publicly available at https://git.physik.uni-wuerzburg.de/ALF/ALF.

My first doctoral research project is discussed in Chapter 2, consisting in a collaboration with Lukas Janssen, Kai Sun, Zi Yang Meng, Igor F. Herbut, Matthias Vojta, and Fakher F. Assaad whose results are published in [14]. We studied nematic quantum criticality of Dirac fermions, where the lattice rotation symmetry of the system is spontaneously broken, such that the Dirac points, which are pinned in the disordered phase, begin to meander at the critical point. Such transitions are of experimental importance, especially in the realm of \(d\)-wave superconductors [15, 16]. I provide the first numerical simulations that tackle this problem in form of QMC simulations of two distinct models which reveal that this kind of transition is continuous. One key aspect of the transition is the lack of Lorentz invariance encoded in a Fermi velocity anisotropy. Comparison with an \(\epsilon\)-expansion conducted by Lukas Janssen reveals that the flow to the fixed point is extremely slow. Hence numerics as well as experiments will be characterized by a crossover regime with drifting exponents.

Chapter 3 covers the second research project, consisting in a collaboration with Francesco Parisen Toldin and Fakher F. Assaad published in [17]. Here, we treat the long-standing problem of the ground state phase diagram of the SU(\(N\)) antiferromagnet of spin \(S\) on a square lattice [18]. We simulate a fermionic representation of an SU(\(N\))-symmetric antiferromagnetic spin model on a square lattice. Each site hosts an irreducible representation of SU(\(N\)) described by a square Young tableau of \(N/2\) rows and \(2S\) columns. For any \(S\) and even \(N\), our Quantum Monte Carlo (QMC) simulations are sign problem free, which enables us to generate the first ever exact ground state phase diagram for this model with \(N\in\{2,4,\dots,20\}\), \(S\in\{1/2,1,3/2,2\}\). In the large-\(N\) limit, the saddle point approximation favors a four-fold degenerate valence bond solid (VBS) phase. In the large \(S\)-limit, the semi-classical approximation points to Néel order. On a line set by \(N=8S + 2\) in the \(S\) versus \(N\) phase diagram, we observe a variety of phases proximate to the Néel state. At \(S = 1/2\) and \(3/2\) we observe the aforementioned four fold degenerate VBS state. At \(S=1\), a two fold degenerate spin nematic state, in which the C\(_4\) lattice symmetry is broken down to C\(_2\), emerges. Finally, at \(S=2\) we observe a unique ground state that pertains to a two-dimensional version of the Affleck-Kennedy-Lieb-Tasaki (AKLT) state [18, 19, 20, 21]. For our specific realization, this symmetry protected topological state is characterized by an SU(18), \(S=1/2\) boundary state, that has a dimerized ground state. These phases which are proximate to the Néel state are consistent with the notion of monopole condensation of the antiferromagnetic order parameter. In particular, one expects spin disordered states with degeneracy set by \(\text{mod}(4,2S)\).

While working on these two projects, I optimized my workflows of using ALF. This resulted both in significant contributions to ALF and a set of Python scripts to streamline in particular the post-processing of data produced by ALF. These scripts eventually lead to the development of pyALF, a Python package built on top of ALF that is meant to simplify the different steps of working with ALF. Chapter 4 contains the documentation for pyALF. The source code for pyALF is publicly available at https://git.physik.uni-wuerzburg.de/ALF/pyALF.

Note

This work is also available as a website found at https://purl.org/diss-jschwab. This might represent a more convenient read, especially for the pyALF documentation in Chapter 4. The document is built with Jupyter Book [22], aiming to achieve a more interactive experience.

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[1]

Which is because human locomotion on foot is not extraordinarily energy efficient, compared to other animals, but a human on a bicycle soars to the top [23]. In that sense, the computer is a tool for the mind, as the bicycle is for locomotion. Unfortunately, most people use much less efficient modes of transport.

[2]

One can usually use symmetries to split the matrix into smaller blocks, which helps immensely but does not solve the problem of exponential scaling.