Ultra-cold fermionic atoms trapped in optical lattices may be a candidate for the discovery of new novel phenomena in condensed matter systems.Experiments afford the creation of virtually any lattice geometry, and physicalparameters of tight binding type lattice models can be acurately and easily tuned. Although some theoretical work has been conducted, few have used the power of the functional renormalization group method to unearth rigorous methods for determining collective many-body phases in this regime. Motivated byrecent theoretical achievements, we investigate novel condensed matter systemsinvolving interacting fermions which are engineered to be confined in differentdimensions. In this sense, we seek low energy effective theories for low-dimensional fermionic lattice systems embedded into higher dimensional lattice systems, and show how tuning physical quantities, such as the fillingor density, can have dramatic effects on the behavior of the lower dimensionalsystem.
Ultra-cold fermionic atoms trapped in optical lattices may be a candidate for the discovery of new novel phenomena in condensed matter systems.Experiments afford the creation of virtually any lattice geometry, and physicalparameters of tight binding type lattice models can be acurately and easily tuned. Although some theoretical work has been conducted, few have used the power of the functional renormalization group method to unearth rigorous methods for determining collective many-body phases in this regime. Motivated byrecent theoretical achievements, we investigate novel condensed matter systemsinvolving interacting fermions which are engineered to be confined in differentdimensions. In this sense, we seek low energy effective theories for low-dimensional fermionic lattice systems embedded into higher dimensional lattice systems, and show how tuning physical quantities, such as the fillingor density, can have dramatic effects on the behavior of the lower dimensionalsystem.
Using two-dimensional optical lattices, it has recently become possible to split up clouds of ultra cold atoms into collections of parallel one-dimensional tubes in which the dynamics is restricted to the direction along their axis. Since the Feshbach-resonance interactions among atoms are short ranged, they can be approximated as contact interactions. It is thus possible to realize optical-trap versions of some classic integrable systems first studied by Lieb, Gaudin, Yang and others. Such systems can display exotic physics like spin-charge separation and unusual paired states. We focus on the study of systems of fermions in one-dimension. At finite temperatures, regimes such as the spin-coherent and the spin-incoherent Luttinger liquids can be realized by tuning the inter-atomic interaction strength and trap parameters. We identify the noise correlation of density fluctuations as a robust observable to discriminate between these two regimes. We also address the effects of spin imbalance for the case of attractive interactions, and study the one-dimensional analog of spatially modulated (FFLO) superconducting states. Our focus is on how the temperature affects the density profiles that are being measured in the experiments. The theoretical study of temperature effects in these systems is of experimental relevance for the problem of thermometry.
Recent experiments demonstrate the ability to construct cold-atom mixtures with species-selective optical lattices. This allows for the possibility of a mixed-dimension system, where one fermionic atomic species is confined to a two-dimensional lattice, while another species is confined to a three-dimensional lattice that contains the two-dimensional one. We show that by tuning the density of an arbitrary number of three-dimensional atomic species, we can engineer an arbitrary, rotationally symmetric, density-density, effective interaction for the two-dimensional particles. This possibility allows for an effective interaction that favors triplet pairing for two-dimensional, SU(2) symmetric particles. Using a functional renormalization-group analysis for the two-dimensional particles, we derive and numerically confirm that the critical temperature for triplet pairing depends exponentially on the effective interaction strength. We then analyze how the stability of this phase is affected by the particle densities and the fine tuning of interaction parameters. We conclude by briefly discussing experimental considerations and the potential to study triplet-pairing physics, including Majorana fermions and spin textures, with cold atoms on optical lattices.
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.
In the past fifteen years ultracold atoms confined to optical lattices, which are artificial crystals of light, have proven to be powerful tools in the study of strongly correlated many-body systems widely known from a condensed matter context. The detailed knowledge, high tunability and control of microscopic parameters has enabled the creation and control of highly non-trivial quantum systems for which a direct comparison of model calculations to experimental measurements is now possible. In order to gain precise information on the properties of ultracold gases in optical lattices, many powerful methods have been developed. In particular spectroscopic methods have enabled to probe different characteristic excitations. However, many fascinating properties of strongly interacting quantum many-body systems are not yet fully characterized. Therefore, this thesis explores further possibilities of lattice modulation spectroscopy which is a time-periodic modulation of the optical lattice amplitude. We theoretically investigate the response of different (mostly one-dimensional) cold atom systems to a lattice modulation by means of the time-dependent density matrix renormalization group method and different analytical approximations. In addition to the normal lattice modulation, we consider a superlattice modulation scheme where the lattice amplitude is modulated in a dimerized fashion, coupling to a certain class of excitations in the system. The lattice modulation spectroscopy is a versatile probe which we use to study different situations relevant for experimental applications. We show that a detection of signatures of the exotic bond order wave phase is feasible in a system of interacting fermions on a one-dimensional optical lattice with staggered energy offset. Moreover, we demonstrate that a temperature measurement of fermions confined to an optical lattice is possible as well as a determination of the interaction strength of bosonic atoms confined to a one-dimensional optical lattice. In addition, various excitations in all these systems are investigated in detail.
We study quantum phases and transitions in ultra-cold quantum gases, beyond the scope of conventional condensed matter systems. The study focuses on three major thrusts. The first thrust is the study of cold ensembles of bosonic or fermionic atoms in the higher orbital bands of optical lattices. Orbital and topological quantum physics is systematically explored for various lattice geometries, to be outlined below. The second is the low dimensional spin-imbalanced fermions in the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) phase. We analyze its thermal instability towards Berezinsky-Kosterlitz-Thouless (BKT) phase transitions driven by different topological defects at finite temperature. The third is the theoretical search for a mechanism to achieve a Bose-Einstein crystal phase for bosons in the ground state. A new class of two-body interactions characterized by strong momentum-dependence is found to give rise to such a quantum state. It is different than but related to the long-debated concept of supersolidity. The first thrust forms the main part of the dissertation. Meta-stable phases of high orbital bosons and fermions have been explored for different lattice geometries such as square and bipartite lattices. Exotic Mott and superfluid phases are studied in both one and two dimensions. Quantum and thermal phase transitions, associated with time reversal symmetry breaking and their experimental signatures are discussed. In particular a quantum ``disordered" superfluid phase, beyond the mean field description, is found in one dimension. The challenging problem of probing time reversal symmetry breaking is solved. Besides, a one-dimensional fermionic optical ladder with coupled s- and p-orbitals is shown to mimic spin-orbit interactions and exhibit topological insulator and topological superconducting phases. Charge fractionalization is shown to be realizable in such an sp-orbital ladder.
Engineering long-range interactions in cold-atom quantum simulators can lead to exotic quantum many-body behavior. Fermionic atoms in ultracold atomic mixtures can act as mediators, giving rise to long-range Ruderman-Kittel-Kasuya-Yosida-type interactions characterized by the dimensionality and density of the fermionic gas. Here, we propose several tuning knobs, accessible in current experimental platforms, that allow one to further control the range and shape of the mediated interactions, extending the existing quantum simulation toolbox. In particular, we include an additional optical lattice for the fermionic mediator, as well as anisotropic traps to change its dimensionality in a continuous manner. This allows us to interpolate between power-law and exponential decays, introducing an effective cutoff for the interaction range, as well as to tune the relative interaction strengths at different distances. Finally, we show how our approach allows one to investigate frustrated regimes that were not previously accessible, where symmetry-protected topological phases as well as chiral spin liquids emerge.