Systematization of the world-line quantum Monte Carlo algorithm
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Resumen en inglés
The World-Line Quantum Monte Carlo method is a stochastic numerical analysis tool, the implementation of which is of particular relevance in the study of quantum many-body systems. These systems are often difficult to study using analytical techniques or conventional numerical methods. The objective of this study is to simulate a reticular bosonic spin 1 system in the Mott insulator phase, with the aim of characterizing the quantum phase transition between the XY-Ferro and Large D magnetic phases by means of the World-Lines Quantum Monte Carlo. The study highlights the importance of characterizing the magnetic phases using local and non-local observables, which provide information about the magnetic properties of the system in the z-axis. In addition, the transition between the Large D and XY-Ferro phases is analyzed and compared with the results obtained by the MPS algorithm and the field theory. The document begins by establishing the physics of the Bose-Hubbard model, which is a theoretical model for an ultracold system of bosonic spin-1 particles in an optical lattice. The inclusion of the spin degree of freedom in the model gives rise to two types of interactions between the bosons: one that preserves the spin projection after the interaction and one that modifies it. To study the magnetic properties of the system, a quadratic Zeeman field is incorporated into the model, which partially lifts the degeneracy and minimizes the energy of the system depending on the field strength and the magnetic projections of the bosons. Subsequently, Quasi-Degenerate Perturbation Theory is applied, considering the particles’ hopping between lattice sites as the perturbative term, to obtain an effective model of the system. The Bilinear-Biquadratic Heisenberg model is obtained, which restricts the system to the Mott insulator phase and is parametrized by the boson’s interaction strength θ and the field strength D. Finally, the condition of a fixed number of bosons in the lattice is imposed on the canonical partition function of the system. After performing algebraic manipulations to facilitate its calculation, the functional form of a partition function for a grand canonical ensemble in the presence of a linear Zeeman field is retrieved. The algorithm implementation involves the transformation of the partition function of the quantum system to another partition function corresponding to that of a classical system with an additional dimension, the imaginary time, which stores the information about the thermalization of the system ground-state. The Metropolis algorithm is addressed, this method implements the importance sampling and allows the measurement of local and non-local observables through estimators. Specifically, it is possible to measure the internal energy, the magnetization, the total number of bosons, the quadrupole density and two types of spin correlations. In addition, the magnetic susceptibility of the system was measured, which allows the construction of the phase diagram for the transition between the XY-Ferro and Large D phases.
Resumen en español
El método Monte Carlo cuántico de Líneas de Mundo constituye una herramienta de análisis numérico estocástico, cuya implementación resulta de particular relevancia en el estudio de sistemas cuánticos de muchos cuerpos. Estos sistemas suelen ser difíciles de abordar mediante técnicas analíticas o métodos numéricos convencionales. El propósito de este estudio es simular un sistema reticular de partículas bosónicas de espín 1 en la fase de aislante de Mott, con el propósito de caracterizar la transición de fase cuántica entre las fases magnéticas XY-Ferro y Large D mediante el Monte Carlo cuántico de Líneas de Mundo. Se destaca la relevancia de caracterizar las fases magnéticas a partir de observables locales y no locales, que proporcionan información sobre las propiedades magnéticas del sistema en el eje z. Además, se analiza la transición entre las fases Large D y XY-Ferro y se compara con los resultados obtenidos por el algoritmo MPS y la teoría de campos, técnicas numéricas y analíticas, respectivamente.