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.
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.
In this work we build the foundations of a quantum Monte Carlo as a stochastic numerical method to solve lattice many-body quantum systems with nearest-neighbor interactions at most. As motivation, we briefly describe the bilinear-biquadratic Heisenberg model with an external field, for spin-1 particles, as an effective Hamiltonian of the Bose-Hubbard model with an external quadratic Zeeman field in the Mott insulator phase at unit filling. Then, we discuss how to implement the world line Monte Carlo with local updates to circumvent the difficulties that arise on these type of systems by mapping the quantum partition function into the one of an effective classical model, in one additional dimension, given by the imaginary time evolution of the system. Such a mapping is performed by means of the Suzuki-Trotter decomposition, which transforms the original partition function into a summation of weights given by the classical configurations. Later, we present a set of observables that can be measured through this method and show how to use a Metropolis update scheme to accomplish the measurements. At last, we present the maximization of the configuration weights for three parameter sets as the first and relevant step to perform future measurements.
The precision achieved by experimental techniques with ultracold atoms has enabled the creation of quantum simulators, facilitating the exploration of one-dimensional systems in the laboratory. In these systems, fluctuations are inherently collective and propagate over large distances due to frequent collisions. As a result, theoretical frameworks have been developed that leverage these fluctuations to describe such systems. Ultracold gases confined in optical lattices not only serve as quantum simulators for traditional condensed matter systems but also act as platforms to create new many-body quantum systems. In this work, two numerical techniques are proposed to explore a spin-1 bosonic gas contained in a one-dimensional optical lattice, in the strongly interacting phase and in the Mott insulator regime. The first technique is a mean-field approach, known as the Gutzwiller ansatz, which approximates the many-body wave function as the product of individual contributions at each site. Additionally, a more advanced ansatz has been developed that considers pairs of coupled sites, allowing for the exploration of regions in the phase space that are not accessible with the traditional ansatz. The second technique is based on tensor networks, specifically representing the state as a Matrix Product State (MPS), combined with the Time-Evolving Block Decimation (TEBD) algorithm. This technique is used to analyze the ground state properties and magnetic behavior of the system under the influence of an external magnetic field. Both techniques were evaluated within the context of a Bilinear-Biquadratic Heisenberg Hamiltonian, incorporating perturbative corrections to account for the quadratic Zeeman effect. The results of this work demonstrate that tensor networks, particularly MPS, provide a robust numerical technique for capturing the quantum phenomena exhibited by spin-1 bosons. The calculations of ground state energy and magnetization profiles confirm the effectiveness of tensor networks in representing the essential physics of the system. Furthermore, the investigation of entanglement properties reveals that the system’s entanglement entropy obeys an area law, characteristic of gapped one-dimensional systems. Chirality and spin correlation observables were also explored, revealing relevant information about the system’s behavior under various interaction parameters. The results are consistent with previous theoretical and numerical studies, highlighting the potential of tensor networks to address challenges in many-body quantum physics. This work introduces a novel numerical perspective within the context of Solid State research at the University of Valle, paving the way for future studies utilizing advanced tensor network techniques.
Since the experimental realization of Bose-Einstein condensation in 1995, the field of ultracold atoms has expanded rapidly, and has become an important arena for exploring diverse quantum many-body physics, as highlighted by the experimental observation of quantum phase transitions in the system of bosons in a periodic potential, which is described by the Bose-Hubbard model.Recently, ultracold atoms with spin degrees of freedom are actively studied because the coupling between the internal spin degrees of freedom and the spatial degrees of freedom can give rise to novel and interesting physics.A minimal model for studying such physics is the spin-1 Bose-Hubbard model, which describes spin-1 bosons in a periodic potential.The model has a spin-dependent interaction, which can be ferromagnetic or antiferromagnetic, and it is known that although the ferromagnetic model resembles the spinless counterpart, the antiferromagnetic model exhibits totally different features, which make the model much more intensively studied compared with the ferromagnetic one.However, the presence of an external magnetic field can change the situation, because the positive quadratic Zeeman energy competes with the ferromagnetic interaction, which can give rise to rich phases.In this thesis, we explore the ground-state and nonzero-temperature phase diagrams of the spin-1 Bose-Hubbard model under an external magnetic field.To analyze the model, we use an approximation method called the decoupling approximation, or equivalently, the Gutzwiller variational ansatz.It is shown that, in addition to the Mott insulator phase, the model possesses two superfluid phases.Moreover, for large values of a spin-dependent interaction, the phase transitions between the two superfluid phases can be discontinuous.Such discontinuous phase transitions are shown to survive at nonzero but sufficiently low temperature.The discontinuous phase transitions are in clear contrast to the mean-field theory of weakly interacting spin-1 bosons without a lattice, which predicts that the corresponding phase transition is continuous, signifying the importance of the effect of strong interactions on the interplay between the spatial and spinor degrees of freedom. 4.1.2The grand Hamiltonian . . . . . . . . .
The Bose-Hubbard model describes the physics of a system of bosonic ultracold atoms in an optical lattice, in which a phase transition is present between a superfluid phase and a Mott insulator one. \nThe exact solution of this Hamiltonian is only feasible to find the ground-state of small systems, while other techniques (as mean-field schemes or quantum Monte Carlo) are necessary to study systems of larger size. \n \nAs a first application, we study the trapped model - relevant for the comparison with current experiments - through an inhomogeneous mean-field scheme. We describe some signatures of the phase crossover between superfluid and Mott insulator. In particular, the visibility of the quasimomentum distribution shows some kinks as a function of the lattice depth; we describe these features and we link them with the ones observed in other works in the literature. \n \nAs a second application, we use quantum Monte Carlo techniques to study the one-dimensional Bose-Hubbard model with long-range interactions and we focus on the appearance of the Haldane insulating phase, distinguishable from the Mott one through the presence of non-local hidden order. \nNon-local correlation functions are also used to describe the difference between the superfluid phase and the Mott insulator one.
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.
Boson lattices are theoretically well described by the Hubbard model. The basic model and its variants can be effectively simulated using Monte Carlo techniques. We describe two newly developed approaches, the Stochastic Series Expansion (SSE) with directed loop updates and continuous–time Diffusion Monte Carlo (CTDMC). SSE is a formulation of the finite temperature partition function as a stochastic sampling over product terms. Directed loops is a general framework to implement this stochastic sampling in a non–local fashion while maintaining detailed balance. CTDMC is well suited to finding exact ground–state properties, applicable to any lattice model not suffering from the sign problem; for a lattice model the evolution of the wave function can be performed in continuous time without any time discretization error. Both the directed loop algorithm and the CTDMC are important recent advances in development of computational methods. Here we present results for a Hubbard model for anti–ferromagnetic spin–1 bosons in one dimensions, and show evidence for a dimerized ground state in the lowest Mott lobe.