This thesis is divided into two parts. In the first part we study quantum phases in bosonic systems in an optical lattice by using numerical unbiased Quantum Monte Carlo (QMC) method. Ultracold atoms experiments in optical lattices provide a possibility to simulate the materials of condensed matter in a clean and well-controled way. We focus on two components bosonic systems where exotic phases such as pair-superfluid and pair-supersolid appear. In the second part we provide numerical methods based on QMC simulations to study the entanglement properties of strong correlated systems. By employing the replica trick, we reconstruct the entanglement spectrum from the Renyi entropies. Furthermore, we study the trace of the power of the partial transposed reduced density matrix, which is related to the entanglement measurement negativity for mixed states. In the following we briefly summarize each chapter. In Chapter 1 we very briefly introduce the cold atom systems and the connection with QMC simulations. In Chapter 2 we review the QMC method based on the path integral (world line) representation. We focus on the directed worm algorithm which is an efficient algorithm with global updtae for bosonic and spin systems. All the following methods in this thesis – the method of two component bosonic systems, the reconstruction of entanglement spectrum and the measuring of partial transposed quantities, are based on directed worm algorithm introduced in this chapter. In Chapter 3 we discuss two component bosons in a square lattice. We show that the interspecies attraction and nearest-neighbor intraspecies repulsion result in the pair-supersolid phase, where a diagonal solid order coexists with an off-diagonal pair-superfluid order. The quantum and thermal transitions out of the pair-supersolid phase are characterized. It is found that there is a direct first-order transition from the pair-supersolid phase to the double-superfluid phase without an intermediate region. Furthermore, the melting of the pair-supersolid occurs in two steps. Upon heating, first the pair-superfluid is destroyed via a Kosterlitz-Thouless transition, then the solid order melts via an Ising transition. In Chapter 4 we represent a new method to reconstruct a subset of the entanglement spectrum of quantum many body systems by QMC, where the method can in principle be applied to two or higher dimension. The approach builds on the replica trick to evaluate particle number resolved traces of the first n of powers of a reduced density matrix. From this information we reconstruct first n entanglement spectrum levels using a polynomial root solver. We illustrate the power and limitations of the method by an application to the extended Bose-Hubbard model in one dimension where we are able to resolve the quasidegeneracy of the entanglement spectrum in the Haldane-insulator phase. In general, the method is able to reconstruct the largest few eigenvalues in each symmetry sector. In Chapter 5 we devise a Quantum Monte Carlo (QMC) method to calculate the moments of the partially transposed reduced density matrix at finite temperature. These are used to construct scale invariant combinations that are related to the negativity, a true measure of entanglement for two intervals embedded in a chain. In particular, we study several scale invariant combinations of the moments for the 1D hard-core boson model. For two adjacent intervals unusual finite size corrections are present, showing parity effects that oscillate with a filling dependent period. For large chains we find perfect agreement with conformal field theory (CFT) calculations. Oppositely, for disjoint intervals corrections are more severe and CFT is recovered only asymptotically. Furthermore, we provide evidence that their exponent is the same as that governing the corrections of the mutual information. Additionally we study the 1D Bose-Hubbard model in the superfluid phase. The finite-size effects are smaller and QMC data are already in impressive agreement with CFT at moderate large sizes.
This thesis is divided into two parts. In the first part we study quantum phases in bosonic systems in an optical lattice by using numerical unbiased Quantum Monte Carlo (QMC) method. Ultracold atoms experiments in optical lattices provide a possibility to simulate the materials of condensed matter in a clean and well-controled way. We focus on two components bosonic systems where exotic phases such as pair-superfluid and pair-supersolid appear. In the second part we provide numerical methods based on QMC simulations to study the entanglement properties of strong correlated systems. By employing the replica trick, we reconstruct the entanglement spectrum from the Renyi entropies. Furthermore, we study the trace of the power of the partial transposed reduced density matrix, which is related to the entanglement measurement negativity for mixed states. In the following we briefly summarize each chapter. In Chapter 1 we very briefly introduce the cold atom systems and the connection with QMC simulations. In Chapter 2 we review the QMC method based on the path integral (world line) representation. We focus on the directed worm algorithm which is an efficient algorithm with global updtae for bosonic and spin systems. All the following methods in this thesis – the method of two component bosonic systems, the reconstruction of entanglement spectrum and the measuring of partial transposed quantities, are based on directed worm algorithm introduced in this chapter. In Chapter 3 we discuss two component bosons in a square lattice. We show that the interspecies attraction and nearest-neighbor intraspecies repulsion result in the pair-supersolid phase, where a diagonal solid order coexists with an off-diagonal pair-superfluid order. The quantum and thermal transitions out of the pair-supersolid phase are characterized. It is found that there is a direct first-order transition from the pair-supersolid phase to the double-superfluid phase without an intermediate region. Furthermore, the melting of the pair-supersolid occurs in two steps. Upon heating, first the pair-superfluid is destroyed via a Kosterlitz-Thouless transition, then the solid order melts via an Ising transition. In Chapter 4 we represent a new method to reconstruct a subset of the entanglement spectrum of quantum many body systems by QMC, where the method can in principle be applied to two or higher dimension. The approach builds on the replica trick to evaluate particle number resolved traces of the first n of powers of a reduced density matrix. From this information we reconstruct first n entanglement spectrum levels using a polynomial root solver. We illustrate the power and limitations of the method by an application to the extended Bose-Hubbard model in one dimension where we are able to resolve the quasidegeneracy of the entanglement spectrum in the Haldane-insulator phase. In general, the method is able to reconstruct the largest few eigenvalues in each symmetry sector. In Chapter 5 we devise a Quantum Monte Carlo (QMC) method to calculate the moments of the partially transposed reduced density matrix at finite temperature. These are used to construct scale invariant combinations that are related to the negativity, a true measure of entanglement for two intervals embedded in a chain. In particular, we study several scale invariant combinations of the moments for the 1D hard-core boson model. For two adjacent intervals unusual finite size corrections are present, showing parity effects that oscillate with a filling dependent period. For large chains we find perfect agreement with conformal field theory (CFT) calculations. Oppositely, for disjoint intervals corrections are more severe and CFT is recovered only asymptotically. Furthermore, we provide evidence that their exponent is the same as that governing the corrections of the mutual information. Additionally we study the 1D Bose-Hubbard model in the superfluid phase. The finite-size effects are smaller and QMC data are already in impressive agreement with CFT at moderate large sizes.
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.
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.
In my dissertation, I will discuss unconventional superfluid states of various bosonic systems and present benchmark calculations for phase diagrams. The method of study is based on quantum Monte Carlo simulations by the Worm algorithm. The two qualitatively different systems to be discussed in the thesis are: (i) superfluidity of lower dimensional objects, i.e dislocations and defects, embedded in solid structures both in optical lattices and in realistic systems such as 4He . (ii) the two-component Bose-Hubbard model. Besides the interesting phases and phase diagrams that they exhibit, bosonic optical lattice systems are also interesting for studies of quantum magnetism where under certain conditions they can be mapped onto various spin Hamiltonians. On the other hand, superfluid dislocations in 4He is an important and interesting subject for it's relevance to supersolid behavior observed in solid 4 He . I will also present technical details of the path integral Monte Carlo and the Worm algorithm and generalization of the algorithm to the two-component bosonic systems.
The bosonic t-J model is a strong-on-site repulsion limit of the two-component Bose-Hubbard model and is expected to be realized by experiments of cold atoms in an optical lattice. In previous papers, we studied the bosonic t-J model by both analytical methods and numerical Monte - Carlo (MC) simulations. However, in the case of finite Jz, where Jz is the z-component coupling constant of the pseudospin interaction, the phase diagram of the model was investigated by assuming the checkerboard type of boson densities. In this study, we shall continue our previous study of the bosonic t-J model using both the Gross-Pitaevskii (GP) theory and MC simulations without assuming any pattern of boson densities. These two methods complement each other and give reliable results. We show that as Jz is increased, the superfluid state evolves into a supersolid (SS), and furthermore into a genuine solid with the checkerboard symmetry. In the present study, we propose a method identifying quantum phase transitions in the GP theory. We also study finite-temperature phase transitions of the superfluidity and the diagonal solid order of the SS by MC simulations.
We study the supercounterfluid (SCF) states in the two-component hard-core Bose-Hubbard model on a square lattice, using the quantum Monte Carlo method based on the worm (directed-loop) algorithm. Since the SCF state is a state of a pair condensation characterized by $\ensuremath{\langle}{a}^{\ifmmode\dagger\else\textdagger\fi{}}b\ensuremath{\rangle}\ensuremath{\ne}0,\ensuremath{\langle}a\ensuremath{\rangle}=0$, and $\ensuremath{\langle}b\ensuremath{\rangle}=0$, where a and b are the order parameters of the two components, it is important to study behaviors of the pair-correlation function $\ensuremath{\langle}{a}_{i}{b}_{i}^{\ifmmode\dagger\else\textdagger\fi{}}{a}_{j}^{\ifmmode\dagger\else\textdagger\fi{}}{b}_{j}\ensuremath{\rangle}$. For this purpose, we propose a choice of the worm head for calculating the pair-correlation function. From this pair correlation, we confirm the Kosterlitz-Thouless character of the SCF phase. The simulation efficiency is also improved in the SCF phase.
Several microscopic theories point out that Bose-Einstein condensation (BEC), i.e., a macroscopic occupation of the lowest energy single particle state in many-boson systems, may appear also in quantum fluids and solids and that it is at the origin of the phenomenon of superfluidity. Nevertheless, the connection between BEC and superfluidity is still matter of debate, since the experimental evidences indicating a non zero condensate fraction in superfluid helium are only indirect. In the theoretical study of BEC in quantum fluids and solids, perturbative approaches are useless because of the strong correlations between the atoms, arising both from the interatomic potential and from the quantum nature of the system. Microscopic Quantum Monte Carlo simulations provide a reliable description of these systems. In particular, the Path Integral Monte Carlo (PIMC) method is very suitable for this purpose. This method is able to provide exact results for the properties of the quantum system, both at zero and finite temperature, only with the definition of the Hamiltonian and of the symmetry properties of the system, giving an easy picture for superfluidity and BEC in many-boson systems. In this thesis, we apply PIMC methods to the study of several quantum fluids and solids. We describe in detail all the features of PIMC, from the sampling methods to the estimators of the physical properties. We present also the most recent techniques, such as the high-order approximations for the thermal density matrix and the worm algorithm, used in PIMC to provide reliable simulations. We study the liquid phase of condensed 4He, providing unbiased estimations of the one-body density matrix g1(r). We analyze the model for g1(r) used to fit the experimental data, highlighting its merits and its faults. In particular we see that, even if it presents some difficulties in the description of the overall behavior of g1(r), it can provide an accurate estimation of the kinetic energy K and of the condensate fraction n0 of the system. Furthermore, we show that our results for n0 as a function of the pressure are in a good agreement with the most recent experimental results. The study of the solid phase of 4He is the most significant part of this thesis. The recent observation of non classical rotational inertia (NCRI) effects in solid helium has generated big interest in the study of an eventual supersolid phase, characterized at the same time by crystalline order and superfluidity. Nevertheless, until now it has been impossible to give a theoretical model able to describe all the experimental evidences. In this work, we perform PIMC simulations of 4He at high densities, according to different microscopic configurations of the atoms. In commensurate crystals we see that BEC does not appear, our model being able to reproduce the momentum distribution obtained form neutron scattering experiments. In a crystal with vacancies, we have been able to see a transition to a superfluid phase at temperatures in agreement with experimental results if the vacancy concentration is low enough. In amorphous solids, superfluid effects are enhanced but appear at temperatures higher than the experimental estimation for the transition temperature. Finally, we study also metastable disordered configurations in molecular para-hydrogen at low temperature. The aim of this study is to investigate if a Bose liquid other than helium can display superfluidity. Choosing accurately a ¿quantum liquid¿ initial configuration and the dimensions of the simulation box, we have been able to frustrate the formation of the crystal and to calculate the temperature dependence of the superfluid density, showing a transition to a superfluid phase at temperatures close to 1 K.