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.
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.
The research work in this thesis is based on strongly interacting quantum lattice systems. The biggest part of research was conducted using state-of-the-art tensor network simulations. Tensor networks provide efficient and highly accurate representations of quantum states when any simply connected patch of the quantum state is slightly entangled. Matrix Product States (MPS) is a tensor network representation for quantum states which is quasi-exact for one-dimensional systems when entanglement entropy of any bipartition of the state follows ``area-law". Projected Entangled Pair States (PEPS) is the extension of MPS for higher dimensional systems, where entanglement entropy area-law is non-trivial. These formalisms and their relevant ground state optimization techniques, Density Matrix Renormalization Group (DMRG) for MPS and Simple Update (SU) with Tensor Renormalization Group (TRG) for PEPS are thoroughly analysed. Entanglement entropy area-law is usually obeyed by the ground state of local Hamiltonians, while generic highly excited states follow a volume-law (they span a finite part of the Hilbert space). Recently, a class of interacting system was shown to undergo a dynamical phase transition (Many Body Localization) where the entropy of every eigenstate follows an area-law. This transition is achieved when the system is highly disordered and the quantum many body state becomes localized similarly to the free particle Anderson localization. The area-law property makes highly excited eigenstates of Many Body Localized (MBL) systems efficiently represented by the MPS ansatz. We develop a highly optimized algorithm (eDMRG) which goes beyond ground state optimization and successfully target eigenstates in any part of the spectrum. This algorithm is used together with analytical calculations, based on local integrability of MBL systems, to identify the universal behaviour of the entanglement spectrum of highly excited eigenstates in MBL systems. In the second part we study interacting bosons in 2D optical lattices. Strongly interacting bosons are simulated using the Bose-Hubbard (BH) model when the interactions are strictly local and the Extended Bose-Hubbard (EBH) model, when additional dipolar interactions are present. The ground states of BH/EBH Hamiltonians in a hexagonal lattice are studied. The phase spectrum includes various insulating and critical phases which are studied in detail using infinite-PEPS ansatz. Additional results on entanglement entropy scaling with respect to the filling of the lattice are presented. Finally, a strongly interacting Harper-Hofstadter Hamiltonian is realized by combining synthetic fields with the BH model. The homogeneity of the system is then broken by a parabolic trapping potential, similar to the ones used in cold atom experiments. Using time-dependent Gutzwiller ansatz (GA) the expansion dynamics of the cloud in large square lattices are studied. In contrast to the expansion dynamics of the BH model, it is found that the synthetic fields generate a self-trapping effect. Using phenomenology and simulations the dynamics are studied in the hard-core and soft-core limit.
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 study of interacting quantum many-body systems is one of the central challenges in modern physics. Interestingly, by understanding the physical properties of such systems from a microscopic description, one can explain many complex quantum phenomena such as the high-temperature superconductivity, fractional quantum Hall effect, etc. However, due to high complexity of these systems, theoretical investigations such as analytical and perturbation theories are limited to studying only a small subset of problems. In the past few decades, numerical simulation has emerged as an indispensable method for studying many phenomena, and has led to immense success and progress in recent years. The main obstacle in addressing quantum many-body systems is that the number of parameters required to represent a state of the system scales exponentially with the system size. However, one hardly needs the full Hilbert space but only a small subspace to describe physical properties of the system. From the point of view in quantum information theory, quantum entanglement is one of the main factors that is responsible for the complexity of studying quantum many-body systems. In particular, the amount of quantum entanglement in such system quantifies the correlation between constituents and therefore it determines how big the subspace is required to represent the system. In this thesis, we are interested in using tensor network states, a new emerging numerical technique to study dynamical and ground state properties of the system. The key ingredient is that a tensor network state which consists of a fairly small number of parameters can represent very well the complete wave function of a system. In one spatial dimension, Matrix Product State (MPS) has become a standard tool, combined with related algorithms such as Time Evolving Block Decimation algorithm (TEBD) to study dynamical properties of a locally-perturbed quantum lattice system. In this thesis, we introduce a technique called ``infinite boundary conditions'' that allows us to use a finite-size MPS to represent the state of an infinite-size system. As a result, one can simulate dynamics of a system for a longer time with a significant computational cost reduction compared to traditional methods. In two spatial dimensions, the tensor network known as Projected Entangled Pair States (PEPS) is emerging as a useful approach. However algorithms for optimizing the matrix elements of these tensors are complex and not very efficient. In particular, the efficiency of the optimization depends critically on the quality of the so-called ``environment'', which is expensive to compute. We have introduced a new method to accelerate the convergence of calculating the PEPS ground state by recycling the environment tensors for many optimization steps, vastly reducing the number of times that the full environment needs to be calculated.
Magnetism is a subject that has fascinated mankind for countless generations. With the development of quantum mechanics around a century ago a fundamental understanding of many of the underlying causes of magnetism were obtained. However, the long range interaction combined with the complex systems magnetism appears in makes it very hard to investigate it. In our days this research area is more active than ever, mainly due to future demand of the electronic industry for components engineered down to the atomic level. Quantum effects gets more important at short scales, especially in lower dimensional materials like sheets and wires. The rapidly increasing amount of computational power available makes the numerical techniques a more important part of the research effort. Many of these are especially well suited to analyze lower dimensional quantum problems. Three of the most important techniques, (classical) Monte Carlo, Exact Diagonalization (ED) and matrix product states (MPS) based techniques, like density renormalization group (DMRG) and time evolving block decimation (TEBD) will be described in some detail and put to use later in this dissertation.With the rapid development of experimental techniques for ultracold gases in optical traps a new approach to investigate magnetic properties that are hard achieve or control in the solid state has emerged. We first study the ground-state phase diagram of a spin-1 condensate trapped in an optical trap when the magnetic dipole interaction between the atoms is taken into account along with confinement and spin precession. The boundaries between the regions of ferromagnetic and polar phases move as the dipole strength is varied and the ferromagnetic phases can be modulated. The magnetization of the ferromagnetic phase perpendicular to the field becomes modulated as a helix winding around the magnetic field direction, with a wavelength inversely proportional to the dipole strength. This modulation should be observable for current experimental parameters in 87Rb. Hence the much-sought supersolid state, with broken continuous translation invariance in one direction and broken global U(1) invariance, occurs generically as a metastable state in this system as a result of dipole interaction. The ferromagnetic state parallel to the applied magnetic field becomes striped in a finite system at strong dipolar coupling.The development of artificial gauge fields, that can mimic magnetic fields, in ultracold gases suggests that atomic realization of fractional quantum Hall physics will become experimentally practical in the near future. While it is known that bosons on lattices can support quantum Hall states, the universal edge excitations that provide the most likely experimental probe of the topological order have not been obtained. We find that the edge excitations of an interacting boson lattice model are surprisingly sensitive to interedge hybridization and edge-bulk mixing for some confining potentials. With properly chosen potentials and fluxes, the edge spectrum is surprisingly clear even for small systems with strong lattice effects such as bandwidth. Various fractional quantum Hall phases for bosons can be obtained, and the phases ν=1/2 and ν=2/3 have the edge spectra predicted by the chiral Luttinger liquid theory.Also, some of the traditional experimental techniques for detecting magnetic order and dynamics in solid state materials, like neutron scattering has had somewhat of a renaissance lately. In a recent experiment on CoNb2O6, Coldea et. al. found for the first time experimental evidence of the exceptional Lie algebra E8. The emergence of this symmetry was theoretically predicted long ago for the transverse quantum Ising chain in the presence of a weak longitudinal field. We consider an accurate microscopic model of CoNb2O6 incorporating additional couplings and calculate numerically the dynamical structure function using a recently developed matrix-product-state method. The excitation spectra show bound states characteristic of the weakly broken textrmE8 symmetry. We compare the observed bound state signatures in this model to those found in the transverse Ising chain in a longitudinal field and to experimental data.Finally, we investigate the ground state phase diagram of a related quantum spin chain, the S=2 XXZ chain with single-ion anisotropy. The interest in this system comes mainly from connecting the highly quantum mechanical spin-1 phase diagram with the classical S=∞ phase diagram. While most of these questions where believed to have been satisfactorily answered mainly with DMRG, some recent studies have questioned some of the conclusions. We use several of the recent advances within DMRG and perform a detailed analysis of the whole phase diagram. We extend the phase diagram by considering different types of single ion anisotropies which help us to answer two important questions: First we show that one can adiabatically move from the isotropic Heisenberg point to the so-called large-D phase with a continuous change of the Hamiltonian. Second, we can tune the model into a predicted intermediate phase which is equivalent to the topologically non-trivial spin-1 Haldane phase. Furthermore, we study the spin-3 XXZ chain to help explaining the development of the classical phase diagram.
This thesis treats the classical simulation of strongly-interacting many-body quantum-mechanical systems in more than one dimension using matrix product states and the more general tensor product states. Contrary to classical systems, quantum many-body systems possess an exponentially larger number of degrees of freedom, thereby significantly complicating their numerical treatment on a classical computer. For this thesis two different representations of quantum many-body states were employed. The first, the so-called matrix product states (MPS) form the basis for the extremely successful density matrix renormalization group (DMRG) algorithm. While originally conceived for one-dimensional systems, MPS are in principle capable of describing arbitrary quantum many-body states. Using concepts from quantum information theory it is possible to show that MPS provide a representation of one-dimensional quantum systems that scales polynomially in the number of particles, therefore allowing an efficient simulation of one-dimensional systems on a classical computer. One of the key results of this thesis is that MPS representations are indeed efficient enough to describe even large systems in two dimensions, thereby enabling the simulation of such systems using DMRG. As a demonstration of the power of the DMRG algorithm, it is applied to the Heisenberg antiferromagnet with spin S=1/2 on the kagome lattice. This model's ground state has long been under debate, with proposals ranging from static spin configurations to so-called quantum spin liquids, states where quantum fluctuations destroy conventional order and give rise to exotic quantum orders. Using a fully SU(2)-symmetric implementation allowed us to handle the exponential growth of entanglement and to perform a large-scale study of this system, finding the ground state for cylinders of up to 700 sites. Despite employing a one-dimensional algorithm for a two-dimensional system, we were able to compute the spin gap (i.e. the energy gap to the first spinful excitation) and study the ground state properties, such as the decay of correlation functions, the static spin structure factors, and the structure and distribution of the nearest-neighbor spin-spin correlations. Additionally, by applying a new tool from quantum information theory, the topological entanglement entropy, we could also with high confidence demonstrate the ground state of this model to be the elusive gapped Z2 quantum spin liquid with topological order. To complement this study, we also considered the extension of MPS to higher dimensions, known as tensor product states (TPS). We implemented an optimization algorithm exploiting symmetries for this class of states and applied it to the bilinear-biquadratic-bicubic Heisenberg model with spin S=3/2 on the z=3 Bethe lattice. By carefully analyzing the simulation data we were able to determine the presence of both conventional and symmetry-protected topological order in this model, thereby demonstrating the analytically predicted existence of the Haldane phase in higher dimensions within an extended region of the phase diagram. Key properties of this symmetry-protected topological order include a doubling of the levels in the entanglement spectrum and the presence of edge spins, both of which were confirmed in our simulations. This finding simultaneously validated the applicability of the novel TPS algorithms to the search for exotic order.
It appears that the tensor network ansatz, while not quite complete, is an efficient coordinate system for the tiny subset of a many-body Hilbert space which can be realized as a low energy state of a local Hamiltonian.However, we don't fully understand precisely which phases are captured by the tensor network ansatz, how to compute their physical observables (even numerically), or how to compute a tensor network representation for a ground state given a microscopic Hamiltonian. These questions are algorithmic in nature, but their resolution is intimately related to understanding the nature of quantum entanglement in many-body systems.For this reason it is useful to compute the tensor network representation of various `model' wavefunctions representative of different phases of matter; this allows us to understand how the entanglement properties of each phase are expressed in the tensor network ansatz, and can serve as test cases for algorithm development.Condensed matter physics has many illuminating model wavefunctions, such as Laughlin's celebrated wave function for the fractional quantum Hall effect, the Bardeen-Cooper-Schrieffer wave function for superconductivity, and Anderson's resonating valence bond ansatz for spin liquids.This thesis presents some results on exact tensor network representations of these model wavefunctions. In addition, a tensor network representation is given for the time evolution operator of a long-range one-dimensional Hamiltonian, which allows one to numerically simulate the time evolution of power-law interacting spin chains as well as two-dimensional strips and cylinders.