In this thesis, we investigate zero temperature and low temperature properties of lattice models as realized in and applicable to condensed matter systems and optical lattices. Optical lattices created using counterpropagating laser beams can mimic a clean lattice model of a material, thereby opening up the doorway towards accurate simulation of the latter. We treat lattice models of these systems where the interactions between the constituent particles range from very weak to very strong. In the first part of the thesis, we separately consider the effects of the kinetic energies and potential energies of the constituent particles in the lattice model. In the non-interacting limit, we use the formalism of lattice Green’s functions to compute the density of states of three related lattices: kagome, diced and hyperkagome. Such tight-binding descriptions are realized in materials like graphene and are useful as starting points to analyse the effects of turning on the inter-particle interactions. In the opposite limit of immobile and interacting particles, we consider Ising spins on quasi-one dimensional ladders and the kagome lattice at their respective saturation fields, the latter physics being realized at the saturation field of the spin ice compound Dy2Ti2O4. We accurately evaluate the zero temperature entropy at this phase point using transfer matrix methods and Binder’s algorithm, and compare with the available experimental value. Our method of constructing higher dimensional lattices from lower dimensional ones can be similarly used, in conjunction with Binder’s algorithm, to accurately evaluate the partition functions of large spin systems in complex lattice geometries. In the next part of the thesis, the effects of both kinetic and potential energy terms are included in the lattice model. The soft-core and hard-core Bose-Hubbard models, which are realized in optical lattices and bosonic systems like Helium-4, are investigated using linked cluster expansions and spinwave theory. We study the effect of introducing three and four body interactions in the soft-core BoseHubbard model in one and two dimensional systems using linked cluster expansions. We formulate a novel scheme to fully renormalize the two body interactions from the addition of these higher body interactions, and show that the latter preserve the universality of the transition between two of the zero temperature phases in the model the Mott insulator and superfluid. These results will aid in a better understanding of the physics in optical lattice experiments where higher body interactions in bosons can be controllably manipulated. In the hard-core Bose-Hubbard model with nearest neighbour interactions, we formulate a spin wave theory for a d-dimensional hypercubic lattice and compare with quantum Monte Carlo simulations for the square and simple cubic lattice models; we find good agreement between the two approaches in the weak to moderate coupling limits. Our comparison of the phase diagrams in one to three dimensions using various methods provides a good gauge for the effect of fluctuations in the model upon changing dimensionality. On the kagome lattice, the same model is investigated using numerical linked cluster expansions at finite temperatures. We compute thermodynamic quantities like system compressibility, internal energy, specific heat and Gruneisen parameter for various coupling strengths and fillings. We provide evidence for a weak first order transition between the valence bond solid and the superfluid phases at zero temperature using Gruneisen parameter scaling. In the immobile or Ising limit, the transition to the polarized state from the 1/3-filled phase is shown to be a first order transition. Low temperature scaling of the specific heat in the Ising limit at small fillings suggests the presence of low energy excitations in this compressible regime. Our results are relevant for extensitons to the recent realizations of hard-core bosons on optical lattices, and the investigation of the transitions and phases therein.
In this thesis, we investigate zero temperature and low temperature properties of lattice models as realized in and applicable to condensed matter systems and optical lattices. Optical lattices created using counterpropagating laser beams can mimic a clean lattice model of a material, thereby opening up the doorway towards accurate simulation of the latter. We treat lattice models of these systems where the interactions between the constituent particles range from very weak to very strong. In the first part of the thesis, we separately consider the effects of the kinetic energies and potential energies of the constituent particles in the lattice model. In the non-interacting limit, we use the formalism of lattice Green’s functions to compute the density of states of three related lattices: kagome, diced and hyperkagome. Such tight-binding descriptions are realized in materials like graphene and are useful as starting points to analyse the effects of turning on the inter-particle interactions. In the opposite limit of immobile and interacting particles, we consider Ising spins on quasi-one dimensional ladders and the kagome lattice at their respective saturation fields, the latter physics being realized at the saturation field of the spin ice compound Dy2Ti2O4. We accurately evaluate the zero temperature entropy at this phase point using transfer matrix methods and Binder’s algorithm, and compare with the available experimental value. Our method of constructing higher dimensional lattices from lower dimensional ones can be similarly used, in conjunction with Binder’s algorithm, to accurately evaluate the partition functions of large spin systems in complex lattice geometries. In the next part of the thesis, the effects of both kinetic and potential energy terms are included in the lattice model. The soft-core and hard-core Bose-Hubbard models, which are realized in optical lattices and bosonic systems like Helium-4, are investigated using linked cluster expansions and spinwave theory. We study the effect of introducing three and four body interactions in the soft-core BoseHubbard model in one and two dimensional systems using linked cluster expansions. We formulate a novel scheme to fully renormalize the two body interactions from the addition of these higher body interactions, and show that the latter preserve the universality of the transition between two of the zero temperature phases in the model the Mott insulator and superfluid. These results will aid in a better understanding of the physics in optical lattice experiments where higher body interactions in bosons can be controllably manipulated. In the hard-core Bose-Hubbard model with nearest neighbour interactions, we formulate a spin wave theory for a d-dimensional hypercubic lattice and compare with quantum Monte Carlo simulations for the square and simple cubic lattice models; we find good agreement between the two approaches in the weak to moderate coupling limits. Our comparison of the phase diagrams in one to three dimensions using various methods provides a good gauge for the effect of fluctuations in the model upon changing dimensionality. On the kagome lattice, the same model is investigated using numerical linked cluster expansions at finite temperatures. We compute thermodynamic quantities like system compressibility, internal energy, specific heat and Gruneisen parameter for various coupling strengths and fillings. We provide evidence for a weak first order transition between the valence bond solid and the superfluid phases at zero temperature using Gruneisen parameter scaling. In the immobile or Ising limit, the transition to the polarized state from the 1/3-filled phase is shown to be a first order transition. Low temperature scaling of the specific heat in the Ising limit at small fillings suggests the presence of low energy excitations in this compressible regime. Our results are relevant for extensitons to the recent realizations of hard-core bosons on optical lattices, and the investigation of the transitions and phases therein.
This dissertation serves as a summary of my Ph.D. work numerically studying equilibrium and non-equilibrium properties of strongly-interacting one-dimensional (1D) boson systems. This work is motivated by the fact that 1D systems are realizable and highly controllable with ultracold atoms in optical lattice and atom chip experiments. We apply a recent worm algorithmic Monte Carlo approach developed for 1D continuous systems to study their equilibrium properties, both with and without an underlying lattice. We also apply an exact lattice approach based on the Bose-Fermi mapping to check our Monte Carlo results in the Tonks-Girardeau limit, and more importantly, to study far-from-equilibrium expansion dynamics of the systems.
We first study the scaling of one-particle correlations of the harmonically trapped Lieb-Liniger gas with changing temperature and interaction strength. Based on the universal behaviors of the density and momentum profiles, we are able to determine the effective parameters needed to fully characterize the system. We also find that the Tonks-Girardeau limit at low temperatures is the ideal regime for the experimental observation of the 1/k4 momentum tail.
An extra periodic lattice can drive the transition from superfluid to Mott insulator states. Exact and complete phase diagrams for such transitions are available only in the weak interacting and deep lattice limit, in which the system can be described using one-band Bose-Hubbard model. Beyond this limit, we use the worm algorithm in continuous space to map out the phase diagrams at various interaction strengths. We compare our phase diagrams with one-band Bose-Hubbard predictions and identify the regime where the one-band description breaks down. We introduce an inverse confined scattering solution to obtain effective Hubbard parameters, with which the Bose-Hubbard model provides correct results for strong interactions and deep lattices at unit filling.
In addition to the equilibrium properties, we also study the expansion dynamics of ultracold atoms in the hard-core limit. Experimentally, this is usually achieved by turning off confining potentials and letting atoms expand in optical lattices. Theoretical studies from initial ground states predicted the occurrence of fermionization of the momentum distribution after long expansion times. In addition, quasicondensation at finite momenta emerges when expanding from Mott insulating domains. Here, we develop a finite-temperature extension of the lattice approach for dynamics. We find the dynamical ferminoization of the momentum distributions at all temperatures. For expansion from initial Mott domains, we observe enhanced correlations reminiscent of dynamical quasicondensation. Surprisingly, we find the systems appear to cool down during the melting of the Mott domains. We use an emergent local Hamiltonian to understand these emergent phenomena.
In this dissertation I present results for lattice boson systems based on quantum Monte Carlo simulations by the Worm algorithm. Lattice bosons can be realized by ultracold gases in optical lattices and, under certain conditions, they are described by the Bose-Hubbard model. I will discuss the ground state, i.e. the phase diagram and properties of elementary excitations, as well as finite temperature thermodynamic properties of the two- and three-dimensional systems. These results provide bench-marks and guidance for experimentalists. I will also discuss the phase diagram of the one-dimensional hard-core Bose-Hubbard model with three-body interactions. The model is realized by cold polar molecules in optical lattices. The ground state presents a novel solid phase at unconventional filling, characterized by coexisting charge density wave and bond orders. Such phase has not been predicted by the preexisting mean field theory.
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.
We study properties of ultracold quantum gases trapped in 1D and under the influence of an underlying optical lattice. We analyze both fermionic and bosonic systems following different numerically exact approaches, like quantum Monte Carlo for soft-core bosons and two-species fermions, and a recently developed exact approach for HCB. The exposition is organized as follows. In Chap. 2, we study single species degenerated fermions on 1D lattices. We show that a splitting of the system takes place in a region of the spectrum with eigenstates that have a non-vanishing weight only in a fraction of the system. We also show that if on top of the lattice an alternating potential is introduced, doubling the original periodicity, an additional slicing' of the system can be achieved. The width and number of such regions can be controlled in a given energy range by the amplitude of the new modulation. By filling these systems the ground state may exhibit insulating regions due to the presence of local gaps, which can be observed in the local density of states. As a complement to these 1D results we present in Appendix A results obtained in 2D lattices. In Chap. 3 we review in some detail the two quantum Monte Carlo (QMC) techniques that are relevant to our study of confined soft-core bosons and two-species fermions on optical lattices, the worldline QMC and the zero temperature projector QMC, respectively. The basis of the classical Monte Carlo method are also presented, included a proof of the Central Limit theorem (Appendix B) that establishes its foundations. Properties of soft-core bosons confined on optical lattices are presented in Chap. 4. QMC simulations show that due to the presence of the slowly varying confining potential the concept of commensurability, typical for periodic systems, loses its meaning. Mott domains appears for a continuous range of fillings and always coexist with superfluid phases, as it was mentioned before. The latter feature is reflected by the behavior of the global compressibility, which never vanishes. A local compressibility, proportional to the variance of the local density, is defined in order to characterize the local Mott-insulating phases. Finally, a phase diagram for trapped systems is also presented. In Chap. 5 we study the ground state properties of two-components fermions confined on optical lattices. Like for the bosonic case, Mott domains also appear for a continuous range of fillings and always coexist with compressible phases. We define a local order parameter (that we denominate local compressibility), which characterizes the Mott insulating phases in an unambiguous way. By means of this local compressibility, we study in detail the interphase between the metallic and insulating region finding that critical behavior sets in revealing a new critical exponent. Furthermore, the behavior of the local compressibility and the variance of the density are found to be universal in this case, independently of the confining potential and the strength of the interaction, as usual in critical phenomena. We present in Chap. 6 a recently developed exact numerical approach that allows to study ground state properties of HCB confined on 1-D lattices. This exact treatment is applied to study the off-diagonal behavior of the one-particle density matrix (OPDM) and related quantities in the equilibrium case. We find that the OPDM decays as a power-law x^-1/2 for large-x, irrespective of the confining potential chosen. The power-law above is shown to determine the scaling of the occupation of the lowest natural orbital in the thermodynamic limit. This scaling and its finite size corrections are also studied for arbitrary powers of the confining potential. In addition, we find a power-law decay of the NO occupations at low densities, and show that its exponent is also universal. The low density limit in the lattice, equivalent to continuous systems, is also analyzed in detail. The approach above is generalized in Chap. 7 to study the non-equilibrium dynamics of HCB in 1-D configurations with an underlying lattice. The presence of the lattice enables the creation of pure Fock states of HCB (a HCB per lattice site) where there is no coherence in the system. We show that quasi-long range correlations develop in the equal-time-one-particle density matrix when such states are allowed to evolve freely, and that they lead to the formation of quasi-condensates of HCB at finite momentum. In addition,we obtain an universal power-law describing the population of the quasi-condensate as a function of time. Finally, we discuss how such systems can be used to create atom lasers with a wave-length that can be controlled through the lattice parameter.
Ultracold atomic gases may be the ultimate quantum simulator.These isolated systems have the lowest temperatures in the observable universe, and their properties and interactions can be precisely and accurately tuned across a full spectrum of behaviors, from few-body physics to highly-correlated many-body effects.The ability to impose potentials on and tune interactions within ultracold gases to mimic complex systems mean they could become a theorist's playground.One of their great strengths, however, is also one of the largest obstacles to this dream: isolation.This thesis touches on both of these themes.First, methods to characterize phases and quantum critical points, and to construct finite temperature phase diagrams using experimentally accessible observables in the Bose Hubbard model are discussed.Then, the transition from a weakly to a strongly interacting Bose-Fermi mixture in the continuum is analyzed using zero temperature numerical techniques.Real materials can be emulated by ultracold atomic gases loaded into optical lattice potentials.We discuss the characteristics of a single boson species trapped in an optical lattice (described by the Bose Hubbard model) and the hallmarks of the quantum critical region that separates the superfluid and the Mott insulator ground states.We propose a method to map the quantum critical region using the single, experimentally accessible, local quantity R, the ratio of compressibility to local number fluctuations.The procedure to map a phase diagram with R is easily generalized to inhomogeneous systems and generic many-body Hamiltonians.We illustrate it here using quantum Monte Carlo simulations of the 2D Bose Hubbard model.Secondly, we investigate the transition from a degenerate Fermi gas weakly coupled List of Tables Table Page 2.1 Techniques to Probe Atoms in Optical Lattices . . . . . .