In the last decade there has been an intense activity aimed at the quantum simulation of interactingudmany-body systems using cold atoms [1, 2]. The idea of quantum simulations traces back to Feynmanud[3], who argued that the ideal setting to study quantum systems would be a quantum experimentaludsetup rather then a classical one - the latter one being fundamentally limited due to its hardware classicaludstructure. This is a particularly important problem given the intrinsic complexity of interactingudmany-body problems, and the difficulties that arise when tackling them with numerical simulations -udtwo paradigmatic examples being the sign(s) problem affecting Monte Carlo simulations of fermionicudsystems, and the real time dynamics in more than one spatial dimension.udUltracold atoms offer a very powerful setting for quantum simulations. Atoms can be trappedudin tailored optical and magnetic potentials, also controlling their dimensionality. The inter-atomicudinteractions can be tuned by external knobs, such as Feshbach resonances. This gives a large freedomudon model building and, with suitable mappings, they allow the implementation of desired target models.udThis allowed an impressive exploitation of quantum simulators on the context of condensed matterudphysics.udThe simulation of high-energy physics is an important line of research in this field and it is lessuddirect. In particular it requires the implementation of symmetries like Lorentz and gauge invarianceudwhich are not immediately available in a cold atomic setting. Gauge fields are ubiquitous inudphysics ranging from condensed matter [4–6] and quantum computation [7, 8] to particle physics [9],udan archetypical example being Quantum Chromodynamics (QCD) [10,11], the theory of strong nuclearudforces. Currently open problems in QCD, providing a long-term goal of cold atomic simulations, includeudconfinement/deconfinement and the structure of color superconducting phases at finite chemicaludpotential [12]. Even though QCD is a very complicated theory (to simulate or study), it is possibleudto envision a path through implementation of simpler models. Furthermore, it is also expected thatudinteresting physics is found on such “intermediate models” which may deserve attention irrespectivelyudof the QCD study. A very relevant model in this regard is the Schwinger model (Quantum Electrodynamicsudin 1+1 dimensions) [13]. This theory exhibits features of QCD, such as confinement [14],udand is at the very same time amenable to both theoretical studies and simpler experimental schemes.udThis model was the target of the first experimental realization of a gauge theory with a quantumudsimulator [15].udThe work on this Thesis is, in part, motivated by the study of toy models which put in evidenceudcertain aspects that can be found in QCD. Such toy models provide also intermediate steps in theudpath towards more complex simulations. The two main aspects of QCD which are addressed hereudare symmetry-locking and confinement. The other main motivation for this study is to develop audsystematic framework, through dimensional mismatch, for theoretical understanding and quantumudsimulations of long-range theories using gauge theories.udThe model used to study symmetry-locking consists of a four-fermion mixture [16]. It has the basicudingredients to exhibit a non-Abelian symmetry-locked phase: the full Hamiltonian has an SU (2) ×udSU (2) (global) symmetry which can break to a smaller SU (2) group. Such phase is found in a extensiveudregion of the phase diagram by using a mean-field approach and a strong coupling expansion. A possibleudrealization of such system is provided by an Ytterbium mixture. Even without tuning interactions,udit is shown that such mixture falls inside the the locked-symmetry phase pointing towards a possibleudrealization in current day experiments.udThe models with dimensional mismatch investigated here have fermions in a lower dimensionalityudd + 1 and gauge fields in higher dimensionality D + 1. They serve two purposes: establish mappingsudto non-local theories by integration of fields [17] and the study of confinement [18].udIn the particular case of d = 1 and D = 2 it is found that some general non-local terms can beudobtained on the Lagrangian [17]. This is found in the form of power-law expansions of the Laplacianudmediating either kinetic terms (for bosons) or interactions (for fermions). The fact that such expansionsudare not completely general is not surprising since constraints do exist, preventing unphysical featuresudlike breaking of unitarity. The non-local terms obtained are physically acceptable, in this regard, sinceudthey are derived from unitary theories. The above mapping is done exactly. In certain cases it isudshown that it is possible to construct an effective long-range Hamiltonian in a perturbative expansion.udIn particular it is shown how this is done for non-relativistic fermions (in d = 1) and 3 + 1 gaugeudfields. These results are relevant in the context of state of the art experiments which implementudmodels with long-range interactions and where theoretical results are less abundant than for the caseudof local theories. The above mappings establish a direct relation with local theories which allowudtheoretical insight onto these systems. Examples of this would consist on the application of Mermin-udWagner-Hohenberg theorem [19, 20] and Lieb-Robinson bounds [21], on the propagation of quantumudcorrelations, to non-local models. In addition they can also provide a path towards implementation ofudtunable long-range interactions with cold atoms. Furthermore, in terms of quantum simulations, theyudare in between the full higher dimensional system and the full lower dimensional one. Such property isudattractive from the point of view of a gradual increase of complexity for quantum simulations of gaugeudtheories.udAnother interesting property of these models is that they allow the study of confinement beyond theudsimpler case of the Schwinger model. The extra dimensions are enough to attribute dynamics to theudgauge field, which are no longer completely fixed by the Gauss law. The phases of the Schwinger modeludare shown to be robust under variation of the dimension of the gauge fields [18]. Both the screenedudphase, of the massless case, and the confined phase, of the massive one, are found for gauge fields inud2+1 and 3+1 dimensions. Such results are also obtained in the Schwinger- Thirring model. This showsudthat these phases are very robust and raises interesting questions about the nature of confinement.udRobustness under Thirring interactions are relevant because it shows that errors on the experimentaludimplementation will not spoil the phase. Even more interesting is the case of gauge fields in higheruddimensions since confinement in the Schwinger model is intuitively atributed to the dimensionality ofudthe gauge fields (creating linear potentials between particles).udThis Thesis is organized as follows. In Chapter 1, some essential background regarding quantumudsimulations of gauge theories is provided. It gives both a brief introduction to cold atomic physics andudlattice gauge theories. In Chapter 2, it is presented an overview over proposals of quantum simulatorsudof gauge potentials and gauge fields. At the end of this Chapter, in Section 2.3 it is briefly presentedudongoing work on a realization of the Schwinger model that we term Half Link Schwinger model. There,udit is argued, some of the generators of the gauge symmetry on the lattice can be neglected withoutudcomprimising gauge invariance. In Chapter 3, the results regarding the phase diagram of the fourfermionudmixture, exhibiting symmetry-locking, is presented. In Chapter 4, the path towards controllingudnon-local kinetic terms and interactions is provided, after a general introduction to the formalism ofuddimensional mismatch. At the end the construction of effective Hamiltonians is described. Finally,udChapter 5 concerns the study of confinement and the robustness of it for 1+1 fermions. The first partudregards the Schwinger-Thirring with the presence of a -term while, in the second part, models withuddimensional mismatch are considered. The thesis ends with conclusions and perspectives of futureudwork based on the results presented here.
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