In this Ph.D. thesis quantum Monte Carlo methods are applied to investigate the properties of a number of ultracold quantum systems. In Chapter 1 we discuss the analytical approaches and approximations used in the subsequent Chapters; also we describe the general concepts of the two-particle scattering problem as a tool to construct Jastrow terms in trial wave functions. Chapter 2 explains in details the Quantum Monte Carlo methods employed in our calculations from the theoretical and practical points of view. In Chapter 3 we explain the Ewald summation technique, applied to a power-law 1/|r|k interaction potential, and a generic approach to obtain the Ewald terms. The obtained expressions of this analytic work are implemented into simulations of different physically relevant systems (Rydberg atoms and Yukawa particles). Chapter 5 is devoted to the modelling of a system, governed by the model potential between Rydberg atoms $1/r^6$. The phase diagram of the system is obtained for a relevant range of densities and temperatures, combining quantum simulations at low temperature and classical treatment at higher temperature. A special attention is paid to the classical description of this system, composed of Rydberg atoms, and its comparison to the quantum system. In Chapter 4 we present the simulation of a system with the Yukawa interaction potential. The following Chapter 6 presents the results of the Quantum Monte Carlo simulations of molecular para-hydrogen at zero and finite temperatures, performed in our Group. Conclusions are drawn in Chapter 7.
In this Ph.D. thesis quantum Monte Carlo methods are applied to investigate the properties of a number of ultracold quantum systems. In Chapter 1 we discuss the analytical approaches and approximations used in the subsequent Chapters; also we describe the general concepts of the two-particle scattering problem as a tool to construct Jastrow terms in trial wave functions. Chapter 2 explains in details the Quantum Monte Carlo methods employed in our calculations from the theoretical and practical points of view. In Chapter 3 we explain the Ewald summation technique, applied to a power-law 1/|r|k interaction potential, and a generic approach to obtain the Ewald terms. The obtained expressions of this analytic work are implemented into simulations of different physically relevant systems (Rydberg atoms and Yukawa particles). Chapter 5 is devoted to the modelling of a system, governed by the model potential between Rydberg atoms 1/r6. The phase diagram of the system is obtained for a relevant range of densities and temperatures, combining quantum simulations at low temperature and classical treatment at higher temperature. A special attention is paid to the classical description of this system, composed of Rydberg atoms, and its comparison to the quantum system. In Chapter 4 we present the simulation of a system with the Yukawa interaction potential. The following Chapter 6 presents the results of the Quantum Monte Carlo simulations of molecular para-hydrogen at zero and finite temperatures, performed in our Group. Conclusions are drawn in Chapter 7.
The Thesis is devoted to simulations of quantum phase transitions by means of Quantum Monte Carlo techniques. Quantum phase transition is a transition between phases at zero or low enough temperature, where quantum effects play an important role. The recent advances in the field of ultracold atom manipulation and optical lattices allowed to produce the systems with unique properties. This opened a perspective to observe quantum phase transitions in many-body systems with non-trivial interparticle interactions in a wide range of the system's characteristic physical parameters and geometries. First, we develop the explicit expressions for the Ewald sums in systems with an interaction potential of a generic 1/r^k type, and in 3D, 2D and 1D geometry. These generalizations can be useful in simulating systems with important interaction potentials as the dipole-dipole, van der Waals interaction, etc. In this Thesis we give the functional forms for the terms of the Ewald sums, ready for implementation in a code. The derivation and the functional form of the results differ in the cases of short-ranged, long-ranged and "marginal" forces, and for a jellium model. It is argued that in the case of some short-range potentials the Ewald method can be advantageous with respect to a direct summation due to a faster convergence rate. We also give a discussion of the convergence properties of a quasi-neutral Coulomb system. We have obtained the zero-temperature phase diagram of bosons interacting through Yukawa forces. We have used a diffusion Monte Carlo simulation starting from a good approximation to the optimal variational ground-state wave function obtained by solving the corresponding Euler-Lagrange hypernetted chain equations. The phase diagram shows that any fermionic mixture of pure elements will always be seen in gaseous form, as the mass ratios required for crystallization of weakly bound fermionic molecules are far beyond the ones that can be achieved in nature. We investigate an alternative mechanism based on the confinement of one of the species to a deep optical lattice which increases its effective mass. The resulting mass ratio of the mixture created in this way can then be tuned at will and could be used to check experimentally the predicted phase diagram both in the gas and crystal (superlattice) phases. We performed a QMC study of the system, comrised of Rydberg atoms. The applied QMC techniques allowed to parametrize a model with isotropic van der Waals interactions into a universal phase diagram. We have characterized the phase diagram of Rydberg atoms by considering a model of bosons with repulsive van der Waals 1/r^6 interaction, and determined solidification and Bose-Einstein condensation conditions. Relaxation mechanisms other than thermal motion should be considered if one considers Rydberg systems on timescales of several tenths of microseconds. We have also studied the excitation spectrum within the approximation of a classical harmonic crystal. We also discuss that interactions between Rydberg excitations open a possibility of new supersolid scenarios. In the last Chapter of the Thesis I present a study of the system of para-hydrogen atoms at low temperatures below the point of crystallization by means of QMC methods. The zero-temperature simulation was performed in order to investigate the properties of a metastable liquid phase and to find the fraction of the Bose-Einstein condensate in the relevant range of densities. The methods of choice for the zero-temperature simulations of the para-H2 system were VMC and DMC techniques. The results of the zero-temperature simulations suggest that the metastable liquid para-hydrogen is a strongly correlated liquid, which again serves as an evidence of high instability of this hypothetical system. The calculation of the Bose-Einstein condensate shows that the condensate fraction is substantially lower than in the liquid helium He4. Los avances recientes en manipulación de átomos ultrafrios y retículos ópticos abrieron la posibilidad de observar las transiciones de fase en sistemas de muchos cuerpos con las interacciones interparticulares no triviales para un amplio rango de los parámetros físicos característicos y geometrías del sistema. En principio desarrollamos las expresiones explicitas para las sumas de Ewald en el caso del potencial de interacción genérico 1/r^k, y en las geometrías arbitrarias: 3D, 2D y 1D. Dichas generalizaciones pueden ser útiles para simular sistemas con los potenciales importantes como dipolo-dipolo, interacción de van der Waals, etc. En la Tesis presentamos las formas funcionales para los términos de las sumas de Ewald, listas para la implementación actual. La derivación y las formas funcionales cambian en función de la potencial de corto, largo o alcance "marginal", y en particular para el modelo de jellium. Argüimos que en el caso del potencial de corto alcance el método de Ewald puede ser ventajoso respecto a la sumatorio directo gracias a la convergencia más rápida. También presentamos la discusión sobre las propiedades de convergencia del sistema de Coulomb сuasi-neutro. Hemos obtenido el diagrama de fase a temperatura cero de los bosones interactuando mediante a las fuerzas de Yukawa. Hemos usado la simulación de Monte Carlo difusivo empezando de una buena aproximación a la función de onda óptima del estado de base obtenida a través de la solución de las ecuaciones de Euler-Lagrange del método HNC. El diagrama de fase demuestra que la mezcla fermiónica de los elementos puros siempre aparece en la forma gaseosa, como los parametros requeridos para la cristalización de estas moléculas fermiónicas están fuera de lo que puede ser visto en la naturaleza. Investigamos el mecanismo alternativo basado en el confinamiento de una de las especies en el retículo óptico, que aumenta su masa efectiva. El cociente de masas de la mezcla creada de esta manera puede ser ajustada arbitrariamente y usada para comprobar el diagrama de fase predicha en el estudio tanto en fase liquida como en la cristalina. Hemos hecho el estudio QMC del sistema de los átomos de Rydberg. Las técnicas de Monte Carlo cuánticas aplicadas nos permitieron parametrizar el modelo mediante la interacción de van der Waals isotrópica y así obtener el diagrama de fase universal. Caracterizamos el diagrama de fase de los átomos de Rydberg considerando el modelo de bosones con la interacción repulsiva 1/r^6, y determinamos las condiciones de solidificación y condensación de Bose-Einstein. Los mecanismos de relajación aparte del movimiento térmico deben de ser tenidos en cuenta a escala de tiempo de decenas de microsegundos. Estudiamos también el espectro de excitaciones dentro de la aproximación de cristal clásico harmónico. Finalmente, discutimos que las interacciones entre las excitaciones de Rydberg abren la posibilidad de los escenarios nuevos del supersólido.
This thesis is composed of two parts. In part one we report results of two quantum Monte Carlo methods - variational Monte Carlo and diffusion Monte Carlo - on the potential energy curve of the helium dimer, the prototypical van der Waals system. In contrast to previous quantum Monte Carlo calculations on this system, we have employed trial wave functions of the Slater-Jastrow form and used the fixed node approximation for the fermion nodal surface. We find both methods to be in excellent agreement with the best theoretical results at short range. In addition, the diffusion Monte Carlo results give very good agreement across the whole potential energy curve, while the Slater-Jastrow wave function fails to bind the dimer at all. In part two we switch to investigations of many-body systems at finite temperature. We use the path integral representation of statistical mechanics to investigate the symmetry properties of the canonical ensemble partition function and find a representation in terms of irreducible representations of the symmetric group. We use this as a foundation to propose a novel technique for the stochastic sampling of the bosonic partition function for quantum gases. It is shown that in principle we are able to use two operators which enable us to construct a Markov chain through a graph of the irreducible representation of the symmetric group. As an illustration of this method, a test calculation of four particles in a harmonic trap is performed.
Quantum Monte Carlo methods are a powerful suite of techniques for solving the quantum many-body problem. By using random numbers to stochastically sample quantum properties, QMC methods are capable of studying low-temperature quantum systems well beyond the reach of conventional deterministic techniques. QMC techniques have likewise been indispensible tools for augmenting our current knowledge of superfluidity and superconductivity. In this thesis, I present two new quantum Monte Carlo techniques, the Monte Carlo Power Method and Bose-Fermi Auxiliary-Field Quantum Monte Carlo, and apply previously developed Path Integral Monte Carlo methods to explore two new phases of quantum hard spheres and hydrogen. I lay the foundation for a subsequent description of my research by first reviewing the physics of quantum liquids in Chapter One and the mathematics behind Quantum Monte Carlo algorithms in Chapter Two. I then discuss the Monte Carlo Power Method, a stochastic way of computing the first several extremal eigenvalues of a matrix too memory-intensive to be stored and therefore diagonalized. As an illustration of the technique, I demonstrate how it can be used to determine the second eigenvalues of the transition matrices of several popular Monte Carlo algorithms. This information may be used to quantify how rapidly a Monte Carlo algorithm is converging to the equilibrium probability distribution it is sampling. I next present the Bose-Fermi Auxiliary-Field Quantum Monte Carlo algorithm. This algorithm generalizes the well-known Auxiliary-Field Quantum Monte Carlo algorithm for fermions to bosons and Bose-Fermi mixtures. Despite some shortcomings, the Bose-Fermi Auxiliary-Field Quantum Monte Carlo algorithm represents the first exact technique capable of studying Bose-Fermi mixtures of any size in any dimension. In Chapter Six, I describe a new Constant Stress Path Integral Monte Carlo algorithm for the study of quantum mechanical systems under high pressures. While the eventual hope is to apply this algorithm to the exploration of yet unidentified high-pressure, low-temperature phases of hydrogen, I employ this algorithm to determine whether or not quantum hard spheres can form a low-temperature bcc solid if exchange is not taken into account. In the final chapter of this thesis, I use Path Integral Monte Carlo once again to explore whether glassy para-hydrogen exhibits superfluidity. Physicists have long searched for ways to coax hydrogen into becoming a superfluid. I present evidence that, while glassy hydrogen does not crystallize at the temperatures at which hydrogen might become a superfluid, it nevertheless does not exhibit superfluidity. This is because the average binding energy per p-H2 molecule poses a severe barrier to exchange regardless of whether the system is crystalline. All in all, this work extends the reach of Quantum Monte Carlo methods to new systems and brings the power of existing methods to bear on new problems.
This thesis details four research projects related to zero temperature quantum Monte Carlo. Chapters 2-4 focus on continuum quantum Monte Carlo, and specifically its application to molecular systems; whereas Chapter 5 focuses on quantum Monte Carlo in a discrete space. Chapter 2 focuses on improving upon the single-particle basis functions employed in quantum Monte Carlo calculations for molecular systems. For calculations requiring non-diverging pseudopotentials, a class of functions is introduced that is capable of producing the short- and long-range asymptotic behavior of the exact wavefunction. It is demonstrated that this form of basis function produces superior accuracy and efficiency when compared to the basis sets typically employed in quantum Monte Carlo. Although the basis functions introduced in Chapter 2 are capable of producing superior results, it is necessary that the parameters of the functional form are near-optimal for the full potential of the functions to be realized. Chapter 3 introduces a simple yet general method for constructing basis sets of a desired functional form appropriate for molecular electronic structure calculations. A standard basis set is created for each of the elements from hydrogen to argon. Chapter 4 explores the effect of different aspects of the trial wavefunction on the accuracy of quantum Monte Carlo. By systematically testing the effect of the basis size, orbital quality, and determinant expansion quality, this work offers guidance to quantum Monte Carlo practitioners for achieving results to within chemical accuracy of experiment. In Chapter 5, semistochastic projection, a hybrid of deterministic and stochastic projection, is introduced for finding the dominant eigenvalue and eigenvector of a matrix. This method, like stochastic projection, is applicable to matrices well beyond the size that can be handled by deterministic methods. Semistochastic projection improves over stochastic projection by significantly reducing the computational time required to obtain the eigenvalue within a specified statistical uncertainty. After the semistochastic projection method is introduced, it is applied to determine the ground state energy of the Hamiltonian in a discrete basis. This special case of semistochastic projection, dubbed semistochastic quantum Monte Carlo, is shown to be orders of magnitude more efficient than stochastic quantum Monte Carlo.
This doctoral thesis studies low dimensional quantum systems that can be realized in recent cold atom experiments. From the viewpoint of quantum statistical mechanics, the main emphasis is on the detailed study of the different quantum and thermal phases and their transitions using numerical methods, such as quantum Monte Carlo and the Tensor Network Renormalization Group. The first part of this work deals with a lattice Boson model subject to strong three-body losses. In a quantum-Zeno like manner, a dynamical three-body constraint arises that restricts the evolution of the quantum system to the subspace with at most two particles per site i.e., three-body losses will be suppressed. Its low-energy description by a Feshbach model leads to the appearance of a pair condensate phase with intriguing quantum critical properties controlled by the Coleman-Weinberg mechanism. A comprehensive numerical study, presented in Chapter 3, identifies the pair phase and provides evidence for an unconventional Berezinksii-Kosterlitz-Thouless transition due to the unbinding of half-vortices. The study of the pair phase with worm-like algorithms turns out to be challenging and an extension of the directed loop algorithm is presented that overcomes the sampling limitations posed by the appearance of fat-tail distributions. In the following chapters, attention is drawn to lattice systems of polar molecules in optical lattices. Their intriguing dipolar interaction and sensitivity towards external fields make them ideal candidates to realize new states of matter. Particularly, it has been shown that one can drive these systems into a regime where the interaction is dominated by competing two- and three-body potentials. First, the triangular lattice supersolid is considered in Chapter 4. An extensive numerical study combined with a detailed symmetry analysis answers the question of the nature of the supersolid quantum nucleation transition. Following this discussion, Chapter 5 investigates whether the characteristic features of this model persist in the presence of interactions for a realistic system with polar molecules. The three-body interaction regime in a honeycomb Bose-Hubbard model turns out to give rise to a set of highly complex phases in terms of valence bond solids and leads to frustration. Chapter 6 studies the phase diagram and shows how the competition between two- and three-body interactions exhibiting a cascade of phases in the regime of intermediate couplings. In addition, it is demonstrated how the Tensor Network Renormalization Group can be applied to systems with multibody interactions and generic algorithmic limitations are described. Finally, preliminary results concerning the effect of quantum disorder on the triangular supersolid and recent calculations for the entropy of one-dimensional trapped SU(N) Hubbard models are discussed in Chapter 7.
In this thesis we present a detailed investigation of the role played by quantum and thermal fluctuations in ultracold Bose gases. We begin with a review of several important concepts and analytical tools within a functional integration formalism. We first focus on the so-called zero-range approximation for the interaction potential, by recovering the Bogoliubov results and the Landau two-fluid model from a field-theory perspective. In deriving the beyond-mean-field equation of state, we are going to show that a crucial point concerns the proper regularization of the divergent zero-point energy. Among the alternative approaches to investigate finite-temperatures Bose gases, we apply the kinetic theory to explain some recent results on the sound propagation in two-dimensional Bose gases. We then move to consider the eventual corrections to the thermodynamics of Bose gases due to the finite-range character of the interaction potential. The coupling constants of the finite-range model are related to measurable scattering parameters through an effective-field-theory approach. The role of finite-range corrections is considered not only in three spatial dimensions but also in systems with lower dimensionalities. Our analytical predictions are in good agreement with available Monte Carlo simulations and consistent with other theoretical frames, as the Lieb-Lininger theory for one-dimensional systems. In the third chapter, the relevance of fluctuations is investigated from an alternative point of view. Indeed, for a single-component Bose gas we have actually considered their effect as deviations of thermodynamic quantities from the mean-field and zero-range picture. In the case of collapsing Bose mixtures, we are going to show that zero-temperature fluctuations play a crucial stabilizing role against the collapse instability. Because of this peculiar mechanism, ultracold mixtures can display finite-density configurations also in free space. Inspired by recent experiments, we characterize this novel self-bound state by comparison with bright solitons, following a variational scheme. We also consider the case of binary mixtures where a coherent internal coupling is turned on. In the last chapter, we move to deal with dipolar condensates. In particular, we are interested in beyond-mean-field effects leading to the formation of inhomogeneous ground states. In order to provide a reliable answer to the open issue of superfluid properties of these structures, we present our recent numerical investigation on the phase diagram of dipolar bosons.