Cold atoms in optical lattices offer unprecedented control over strongly correlatedmany-body states. For this reason they represent an excellent tool for the implementation ofa “quantum simulator”, which can be used to realize experimentally several Hamiltonians ofsystems of physical interest. In particular, they enable the engineering of artificial gaugefields, which gives access to the physics of frustrated magnetism. In this work, we study thethermodynamics of cold atoms both from a theoretical and a numerical point of view. Atpresent days, the most effective method used in this field is the quantum Monte Carlo. Butbecause of the so-called “sign problem” it can only be applied to a limited class of systems,which for example do not include frustrated systems. The interest of this thesis is to developof a new approximated method based on a Monte Carlo approach. The first part of this workis dedicated to theoretical considerations concerning the spatial structure of quantum andclassical correlations. These results permit to develop, in the second part, an approximationcalled quantum mean-field. This latter allows to propose, in the third part, a numericalmethod that we call “auxiliary-field Monte Carlo” and that we apply to some systems ofphysical interest, among which the frustrated triangular lattice.
Cold atoms in optical lattices offer unprecedented control over strongly correlatedmany-body states. For this reason they represent an excellent tool for the implementation ofa “quantum simulator”, which can be used to realize experimentally several Hamiltonians ofsystems of physical interest. In particular, they enable the engineering of artificial gaugefields, which gives access to the physics of frustrated magnetism. In this work, we study thethermodynamics of cold atoms both from a theoretical and a numerical point of view. Atpresent days, the most effective method used in this field is the quantum Monte Carlo. Butbecause of the so-called “sign problem” it can only be applied to a limited class of systems,which for example do not include frustrated systems. The interest of this thesis is to developof a new approximated method based on a Monte Carlo approach. The first part of this workis dedicated to theoretical considerations concerning the spatial structure of quantum andclassical correlations. These results permit to develop, in the second part, an approximationcalled quantum mean-field. This latter allows to propose, in the third part, a numericalmethod that we call “auxiliary-field Monte Carlo” and that we apply to some systems ofphysical interest, among which the frustrated triangular lattice.
Many strongly correlated quantum systems are difficult to study numerically because of the exponential growth of the Hilbert space combined with the failure of numerical methods in important parameter regimes. This includes the long-standing problems of high-temperature superconductivity and frustrated magnetism. A promising solution to this dilemma are quantum simulators, well-controlled (synthetic) quantum systems, emulating model Hamiltonians not only of condensed matter systems, where parameters can be tuned at will and model complexity can be added step by step. This dissertation presents numerical simulation studies that are motivated by quantum simulators. Using large-scale quantum Monte Carlo computer simulations and mean-field theory the semiquantitative ground state phase diagram of a Penning ion trap quantum simulator, realizing a two-dimensional model of quantum magnetism with long-range interactions, is obtained. We study also thermal and quantum phase transitions in one-dimensional transverse-field Ising models with long-range interactions, which exhibit a Kosterlitz-Thouless transition and universality classes with continuously varying critical exponents. The second part of this thesis is concerned with the Fermi-Hubbard model, which has been realized recently with ultracold atoms in optical lattices, both for repulsive and attractive interactions. Equipped with a quantum gas microscope, which allows single-site single-atom detection, these experimental setups can collect histograms of the particle number from repeated projective measurements. This dissertation presents a method to compute such full counting statistics for quadratic operators such as particle number or magnetization in the framework of determinantal quantum Monte Carlo. From an analysis of the full counting statistics of the particle number for different subsystem sizes the size of preformed pairs or Cooper pairs in the attractive Hubbard model is inferred. Furthermore, excellent agreement of the numerically computed probability distribution of the staggered magnetization with a recent quantum gas experiment demonstrates that current experiments are capable of resolving the differences between the Hubbard model and its limiting Heisenberg model. An extensive method section reviews the stochastic series expansion quantum Monte Carlo method for Ising models in a transverse field, which is free of the sign problem for arbitrary interactions, with special emphasis on the efficient simulation of long-range interactions. Also a self-contained introduction to the determinantal quantum Monte Carlo method is provided.
We introduce a new numerical technique, the bosonic auxiliary-field Monte Carlo method, which allows us to calculate the thermal properties of large lattice-boson systems within a systematically improvable semiclassical approach, and which is virtually applicable to any bosonic model. Our method amounts to a decomposition of the lattice into clusters, and to an ansatz for the density matrix of the system in the form of a cluster-separable state-with nonentangled, yet classically correlated clusters. This approximation eliminates any sign problem, and can be systematically improved upon by using clusters of growing size. Extrapolation in the cluster size allows us to reproduce numerically exact results for the superfluid transition of hard-core bosons on the square lattice, and to provide a solid quantitative prediction for the superfluid and chiral transition of hardcore bosons on the frustrated triangular lattice.
A quantum simulator is a well-controlled quantum system which can be used to simulate another quantum system. Quantum degenerate gases confined by optical standing waves(optical lattices) have been considered to simulate condensed matter systems. We focus on two issues in this problem. The first one is the realization of interesting quantum phases by engineering Hamiltonians with the cold atoms toolbox. Secondly, the systems we studied are often intractable to analytical methods, which makes it a crucial issue to solve them with numerical methods. For the first issue we find several interesting phases can be studied with cold atoms in optical lattices. We show that a p-wave superfluid is stablized through a dissipation blockade mechanism. This mechanism also induces a new insulator formed by p-wave Feshbach molecules. We find that the anisotropic nature of interaction between polar molecules or dipolar atoms can be used to realize two charge density waves with different patterns and a supersolid phase which has not been found conclusively in the condensed matter system. Finally, we propose to realize a long sought spin liquid phase with hard-core bosons subject to a spin-dependent lattice and Raman induced hoppings. For the second issue we choose to numerically study these systems with the so-called tensor network algorithm which is a variational method based on the matrix(tensor) product states product states which are designed to approximate generic wave functions from their entanglement properties. In general the tensor network algorithm has advantages that it is applicable beyond one dimension, and can be used to study physical properties in the thermodynamic limit directly. We apply this method to map out the phase diagrams of the above systems. Furthermore, we show that it is capable of capturing the phase diagram of a frustrated magnetism model, the Heisenberg model on a checkerboard lattice. The great controllability and isolation from the environment of a cold atom system offer a playground to study interesting physics in strongly correlated systems. Several proposed control schemes in this thesis have been demonstrated in experiments. By providing detailed evidence from numerical simulations, this work addresses practical routes to novel quantum phases.
A major focus in condensed matter physics is to study the origin of exotic quantum phases such as coexistent and inhomogeneous phases, quantum criticality, and secondary ordered phases close to quantum critical points. Exotic phenomena in strongly correlated systems occur due to competing complex interactions of spin, charge, lattice, and orbital degrees of freedom. Complex quantum phases in strongly correlated systems are challenging, but they might be very useful due to their possible functionality to make advance devices. In order to understand, utilize, and optimize such behaviors, we need to improve our understanding of these systems. Studies in cold atom systems are of interest since cold atom experiments provide a control on model parameters. In this thesis, we use novel analytical and computational techniques to treat strongly interacting bosonic systems. Taking advantage of the very versatile quantum Monte Carlo Stochastic Green Function algorithm, we studied several interesting problems. First, we explore a recently developed new confining method for cold atoms on optical lattices. Atoms are confined via a hopping integral that decreases as a function of the distance from the center of the lattice. This method might lead to lower temperatures than existing diagonal confinement methods. Next, we study the ground state phase diagram of interacting bosons on a ring-shape lattice with a region of weak hopping integrals. The model, an extension of the well known Bose-Hubbard model, develops a novel local Mott phase in addition to the usual Mott and superfluid phases in the homogeneous system. This might provide a new insight to the description of atomtronics applications. Finally, we study the two species Bose Hubbard model in a two-dimensional lattice. This model presents novel phases due to the complexity associated with multiple species. Its phase diagram shows ordered and coexistence phases including a ferromagnetic phase separated phase with high entropy. This phase might be accessible experimentally. The novel phases found from our studies are linked to experiments on ultra-cold atoms trapped by laser beams.
Beautiful experiments have shown that bosons in optical lattices provide a physical system with an unprecedented degree of control; that they can generate strongly correlated states; and that, as long-term goals, they may be useful for simulating other many-body systems, or realising a quantum computer. Recently a practical scheme for measuring temperature in an optical lattice has been demonstrated, which marks an important milestone necessary for the detailed study of the thermodynamics of this system. In this thesis, we consider both the translationally-invariant lattice, and the combined harmonic trap and optical lattice. We use an extended Bose-Hubbard Hamiltonian which goes beyond the usual Bose-Hubbard approach, and is valid for shallower lattices and higher temperatures, by allowing for beyond nearest-neighbour hopping and excited bands, and we have developed an approximation scheme for off-site interactions. We derive the equations of the Popov approximation to the Hartree-Fock-Bogoliubov method for our Hamiltonian, and diagonalise these equations in the local density approximation (LDA). We examine the density of states of the optical lattice in detail and in various limits. We derive new results on the structure of the density of states, and, in the ideal case, we compare the density of states from the full diagonalisation with our LDA calculation. We make an efficient numerical implementation of our theory and compare the results with the full diagonalisation (for the non-interacting case) and with the limited experimental results currently available. We consider the significance of beyond nearest-neighbour hopping and excited bands and illustrate the properties of our model. In contrast to the trapped gas with no lattice, few thermodynamic results for cold bosons in an optical lattice have been calculated. We analytically derive the first practical formula for the critical temperature in an optical lattice by using simplified shapes for the density of states. We derive corrections for the influence of excited bands, the finite-size effect and mean-field interactions. In all of these cases, we compare our results to full numerical calculations and show that the validity range of our method is complementary to that of the effective-mass approximation, so that the simple descriptions extend over a wide range.
The investigation of quantum many-particle systems is a central goal of modern condensed-matter physics. Complex physical phenomena such as high-temperature superconductivity, heavy fermion behavior, and correlation-driven metal-insulator transitions are central aspects of this particular field of science. The Hubbard model is a paradigm of correlated-electron physics. It is one of the simplest quantum-mechanical lattice models capable of capturing relevant physical aspects of strongly correlated electron systems. The Hubbard model also plays an important role in the explanation of many-body phenomena observed in ultra-cold atoms trapped in optical lattices. In this thesis, we study the three-dimensional Hubbard model using quantum Monte Carlo simulations (QMC). Our aim is a precise numerical study of its properties in the thermodynamic limit. However, a direct investigation of lattice models in the thermodynamic limit is generally impossible because QMC simulations cannot treat infinite lattice sizes. Therefore, approximative schemes that allow calculations directly in the thermodynamic limit are advantageous. A widely applied approximation of this kind is the dynamical mean-field theory (DMFT) which maps the infinite-lattice problem onto a single-site impurity model embedded in a mean field. We apply cluster mean-field theories which represent a systematic extension of the DMFT by expanding the single impurity to a finite cluster. The second focus of this thesis is the calculation of spectral properties of the Hubbard model. QMC methods map the quantum-mechanical lattice model on a classical one at the expense of an additional imaginary time dimension. For this reason, QMC algorithms can only provide dynamical data on the imaginary time or frequency axis. The necessary analytic continuation to the physically relevant real axis has proven to be a difficult problem that has to be approached by advanced data analysis tools. The maximum-entropy method (MEM) is the standard technique to handle this problem. We apply the MEM to the problem of extracting momentum-resolved single particle spectra from QMC simulations of the 3D Hubbard model. Furthermore, we discuss and expand algorithms that were recently proposed as alternatives to the standard MEM approach.