Inspired by the growing interest in probing many-body phases in novel two-dimensional lattice geometries we investigate the properties of cold atoms as they could be observed in an optical Lieb lattice. We begin by computing Wannier functions localised at individual sites for a realistic experimental setup, and determining coefficients for a Hubbard-like model. Based on this, we show how experiments could probe the robustness of edge states in a Lieb lattice with diagonal boundary conditions to the effects of interactions and realise strongly correlated many-body phases in this geometry. We then generalise this to interacting particles in a half-filled 1D Lieb ladder, where excitations are dominated by flat band states. We show that for strong attractive interactions, pair correlations are enhanced even when there is strong mixing with the Dirac cone. These findings in 1D raise interesting questions about the phases in the full 2D Lieb lattice which we show can be explored in current experiments.
Inspired by the growing interest in probing many-body phases in novel two-dimensional lattice geometries we investigate the properties of cold atoms as they could be observed in an optical Lieb lattice. We begin by computing Wannier functions localised at individual sites for a realistic experimental setup, and determining coefficients for a Hubbard-like model. Based on this, we show how experiments could probe the robustness of edge states in a Lieb lattice with diagonal boundary conditions to the effects of interactions and realise strongly correlated many-body phases in this geometry. We then generalise this to interacting particles in a half-filled 1D Lieb ladder, where excitations are dominated by flat band states. We show that for strong attractive interactions, pair correlations are enhanced even when there is strong mixing with the Dirac cone. These findings in 1D raise interesting questions about the phases in the full 2D Lieb lattice which we show can be explored in current experiments.
Abstract Inspired by the growing interest in probing many-body phases in novel two-dimensional lattice geometries we investigate the properties of cold atoms as they could be observed in an optical Lieb lattice. We begin by computing Wannier functions localised at individual sites for a realistic experimental setup, and determining coefficients for a Hubbard-like model. Based on this, we show how experiments could probe the robustness of edge states in a Lieb lattice with diagonal boundary conditions to the effects of interactions and realise strongly correlated many-body phases in this geometry. We then generalise this to interacting particles in a half-filled 1D Lieb ladder, where excitations are dominated by flat band states. We show that for strong attractive interactions, pair correlations are enhanced even when there is strong mixing with the Dirac cone. These findings in 1D raise interesting questions about the phases in the full 2D Lieb lattice which we show can be explored in current experiments.
Optical lattices provide ideal experimental tools for simulating a wide variety of physical systems. They exhibit excellent coherence properties, their microscopic dynamics are well-understood, and they can be dynamically controlled with great precision. Recent advances in cooling and trapping gases of dipolar molecules and Rydberg atoms have further increased the versatility of optical lattices as quantum simulators, particularly for solid state systems possessing long-range interactions. In this thesis we further examine the microscopic properties of cold atoms trapped in an optical lattice, going beyond the case where the lattice axes are orthogonal. We present a transformation of the single-particle energy eigenstates to a complete, orthonormal basis of so-called 'generalised Wannier functions' that are localised to each lattice site. Generalised Wannier functions, as opposed to ordinary Wannier functions, are formed by mixing states between bands and are required when the bands are degenerate. We modify an algorithm devised by Marzari and Vanderbilt (Phys. Rev. B 56(20) 12847) for calculating maximally localised generalised Wannier functions, specifically to determine these functions for optical lattice systems of arbitrary geometry, including those with degenerate lower bands. Moreover, we overcome an issue with the initialisation of the original algorithm to find the maximally localised set without fail. We then present results for a variety of optical lattice systems in one- and two-dimensions, including hexagonal and Kagomé geometries. In each case we use the generalised Wannier functions to calculate the hopping and interaction parameters, and thus determine a reduced form of the system's Hamiltonian that contains only local interactions. We also study two spin-1/2 systems with long-range interactions as paradigm examples of the models that can be simulated using optical lattices loaded with dipolar molecules. We first examine the critical properties of a 1D spin-1/2 Ising chain possessing exponentially decaying interactions as the range of the interactions is increased. Using the infinite density matrix renormalisation group method, we calculate the location of the critical point and observe a smooth approach to the classical mean-field result in the limit of infinite-ranged interactions. We go on to consider the quantum dynamics of a single spin-1/2 coupled to a bath of interacting spins as a model for decoherence in solid state quantum memories. The large degree of symmetry of the bath, described by the infinite-ranged Lipkin-Meshkov-Glick model, allows us to find parameter regimes where the initial qubit state is revived at well-defined times after the qubit preparation. We observe that these times may become independent of the bath size for large baths and thus enable faithful qubit storage even in the presence of strong coupling to the bath.
There is a growing interest in using cold-atom systems to explore the effects of strong interactions in topological band structures. Here we investigate interacting bosons in a Cruetz ladder, which is characterised by topological flat energy bands where it has been proposed that interactions can lead to the formation of bound atomic pairs giving rise to pair superfluidity. By investigating realistic experimental implementations, we understand how the lattice topology enhances the properties of bound pairs giving rise to relatively large effective pair-tunnelling in these systems which can lead to robust pair superfluidity, and we find lattice supersolid phases involving only pairs. We identify schemes for preparation of these phases via time-dependent parameter variation and look at ways to detect and characterise these systems in a lattice. This work provides a starting point for investigating the interplay between the effects of topology, interactions and pairing in more general systems, with potential future connections to quantum simulation of topological materials.
Abstract There is a growing interest in using cold-atom systems to explore the effects of strong interactions in topological band structures. Here we investigate interacting bosons in a Cruetz ladder, which is characterised by topological flat energy bands where it has been proposed that interactions can lead to the formation of bound atomic pairs giving rise to pair superfluidity. By investigating realistic experimental implementations, we understand how the lattice topology enhances the properties of bound pairs giving rise to relatively large effective pair-tunnelling in these systems which can lead to robust pair superfluidity, and we find lattice supersolid phases involving only pairs. We identify schemes for preparation of these phases via time-dependent parameter variation and look at ways to detect and characterise these systems in a lattice. This work provides a starting point for investigating the interplay between the effects of topology, interactions and pairing in more general systems, with potential future connections to quantum simulation of topological materials.
The Lieb lattice possesses three bands and with intrinsic spin orbit coupling λ, supports topologically non-trivial band insulating phases. At half filling the lower band is fully filled, while the upper band is empty. The chemical potential lies in the flat band (FB) located at the middle of the spectrum, thereby stabilizing a flat band insulator. At this filling, we introduce on-site Hubbard interaction U on all sites. Within a slave rotor mean field theory we show that, in spite of the singular effect of interaction on the FB, the three bands remain stable up to a fairly large critical correlation strength (Ucrit), creating a correlated flat band insulator. Beyond Ucrit, there is a sudden transition to a Mott insulating state, where the FB is destroyed due to complete transfer of spectral weight from the FB to the upper and lower bands. We show that all the correlation driven insulating phases host edge modes with linearly dispersing bands along with a FB passing through the Dirac point, exhibiting that the topological nature of the bulk band structure remains intact in presence of strong correlation. Furthermore, in the limiting case of U introduced only on one sublattice where λ=0, we show that the Lieb lattice can support mixed edge modes containing contributions from both spinons and electrons, in contrast to purely spinon edge modes arising in the topological Mott insulator.
In this thesis, we discuss quantum phase transitions in low-dimensional optical lattices, namely one- and two-dimensional lattices. The dimensional confinement is realized in experiments by suppressing the hopping in the extra dimensions through a deep potential barrier that prevents the atoms to tunnel across dimensions. The dimensionality considerably affects the physical behavior of the system. For instance, in one dimension, the celebrated Fermi-liquid theory does not describe the many-body behavior of a fermionic system. There are no single particle excitations, but instead collective behavior, and spin and charge degrees of freedom are decoupled . In the first part of this thesis, we consider one-dimensional fermionic gases under the effect of time-dependent driving forces. First, we describe a proposal to realize non-standard Hubbard models that display correlated-hopping terms by modulating interactions through a time-dependent magnetic field. The extra terms describe hopping processes that depend on the particle-density difference in neighboring sites. The effective model is obtained in arbitrary spatial dimensions, but only in one-dimension it is known to be exactly solvable for specific values of the parameters. Moreover, an SU(2) symmetry in the charge sector allows to construct non-trivial superconducting eigenstates (eta-states), discovered by Yang as excited states of the Hubbard model. Then, we study a one-dimensional bipartite optical lattice with fermionic atoms at half-filling. The lattice potential barriers have heights alternating in magnitude. As a consequence, the hopping amplitudes also alternate in magnitude and the elementary unit cell has two sites. This system is known to display Peierls dimerization at half-filling: the fermions form dimers in neighboring sites that inhibit transport and open a gap in the spectrum. After applying shaking on this system we discover that one can dynamically induce a phase transition to an unconventional metallic state with four Fermi-points. The second part of this thesis describes bosonic systems in two-dimensional optical lattices. First, we consider a square optical lattice with alternating deep-shallow wells arranged in a checkerboard pattern. This lattice is realized in the experimental laboratory of Prof. Hemmerich in Hamburg. In the experiment, the visibility is extracted from the time-of-flight imaging to quantify the phase coherence of the bosonic gas. We develop a mean-field theory and determine the phase diagram of this model. We interpret the experimental data by assuming that at large detuning, only the deeper wells are populated and form an imbalanced Mott insulator with empty shallow wells and integer density on the deeper wells. We support this hypothesis by large-scale Gutzwiller calculations. Then, we investigate a system of bosons in the Lieb lattice, which is the lattice geometry of the Cu-O plane in cuprates . We consider configurations of the optical potential where s and p orbitals sitting on neighboring wells become resonant and hybridize. The interest in this problem comes from the possibility to realize time-reversal broken phases, usually known as Varma phases for electronic systems, which have been proposed to explain the pseudo-gap regime of cuprate superconductors. We show the conditions under which the symmetry breaking in a four-band model can lead to states with loop currents. Moreover, the lowest branch of the excitation spectrum has a non-vanishing Chern number and therefore supports topological excitations.