Phases and phase transitions in SU(\(N\)) spin and Dirac systems: Auxiliary field quantum Monte Carlo studies

This thesis presents three interrelated projects centered around Quantum Monte Carlo (QMC) simulations and the development of computational tools to study strongly correlated quantum systems. Two of these projects leverage the QMC package *Algorithms for Lattice Fermions* (ALF) to investigate critical phenomena in Dirac fermions and SU($N$)-symmetric antiferromagnetic spin models, respectively. The third project introduces *pyALF*, a Python package that simplifies and enhances the use of ALF, making advanced simulations more accessible and efficient. The first project explores nematic quantum phase transitions in Dirac fermions. The focus lies on two models with either $C_{2v}$ or $C_{4v}$ lattice point-group symmetry, respectively, which are spontaneously broken in the ordered phase, allowing for meandering Dirac points. These models are specifically designed to be free of the negative sign problem, allowing for efficient QMC simulations. My numerically obtained results, complemented by a collaborator's $\epsilon$-expansion renormalization group study, show that both models undergo continuous phase transitions. In contrast to generic Gross-Neveu dynamical mass generation, the quantum critical regime is characterized by large velocity anisotropies, with fixed-point values being approached very slowly. Due to this slow renormalization group flow, both experimental and numerical investigations will not be representative of the infrared fixed point, but of a quasiuniversal regime where the drift of the exponents tracks the velocity anisotropy. Notably, even though the $\epsilon$-expansion finds qualitatively distinct fixed points for the two investigated models, the numerical investigation finds no distinction in the respective exponents. Therefore, it seems that the quasiuniversial regime is at least close to the ultraviolet beginning identical for both models, even though their infrared universality is different. The second project investigates the ground state phase diagram 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 with $N/2$ rows and $2S$ columns. Negative sign-free QMC simulations are feasible for all values of $S$ and even values of $N$, allowing for a comprehensive exploration of the phase diagram. In the large-$N$ limit, the saddle point approximation favors a four-fold degenerate valence bond solid (VBS) phase, while in the large-$S$ limit, semi-classical approximations predict Néel order. Along a line defined by $N=8S + 2$ in the $S$ versus $N$ phase diagram, we observe a rich variety of phases. For $S = 1/2$ and $3/2$, the system forms a four-fold degenerate VBS state, while for $S = 1$, we identify a two-fold degenerate spin nematic state that breaks the $C_4$ lattice symmetry down to $C_2$. At $S = 2$, we observe a unique symmetry-protected topological state, characterized by a dimerized SU(18) boundary state, reminiscent of the two-dimensional Affleck-Kennedy-Lieb-Tasaki (AKLT) state. These phases proximate to the Néel state align with the theoretical framework of monopole condensation of the antiferromagnetic order parameter, with degeneracies following a $\text{mod}(4,2S)$ rule. The third project documents the development of pyALF, a Python-based package designed to lower the barrier of entry for users new to ALF and enhance the productivity of experienced ALF users by providing a streamlined workflow for setting up and analyzing QMC simulations. Through easily reproducible examples, the documentation introduces key concepts such as preparing, executing and postprocessing simulations. The postprocessing tools in pyALF leverage Python's dynamic capabilities, enabling users to quickly define custom observables, such as correlation ratios and other complex order parameters. The package also allows for interactive checks of warmup and autocorrelation times, with a convenient way to adjust corresponding analysis parameters. Furthermore, the analysis tools allow researchers the implementation of improved estimators with minimal effort, leveraging e.g. lattice point-group symmetries to improve the accuracy of their results. By utilizing Python’s extensive libraries for data analysis and visualization, pyALF enhances the workflow, with data conveniently stored in *pandas DataFrames* for easy access. Together, these three projects offer new insights into the study of highly correlated quantum systems and provide powerful computational tools that significantly advance numerical approaches in condensed matter physics.