This thesis is divided into five parts. The first part, comprising Chapters 1 and 2, is devoted to the theoretical background of the Bose-Hubbard model. The second part, containing Chapters 3 and 4, concerns the theoretical methods that we have applied to study the system. The third part is constituted by Chapters 5 and 6, which presents the main contributions from this work. In the fourth part, corresponding to Chapters 7 and 8, we present some preliminary calculations that could be used to extend the results of the present thesis as well as to produce new and insightful results for the problem examined here. Finally, in the fifth part, we summarize the contribution of this work, presenting the concluding remarks.
In Chapter 1, we start by introducing the coupling between atoms and the radia- tion field via the optical dipole force. We then demonstrate how one can use counter- propagating laser beams to generate the periodic structure of optical lattices. We also discuss how a random external potential can be created using laser speckles. Next, we introduce the Bose-Hubbard model, first considering the single-particle picture and then deriving the Hamiltonian within the formalism of second quantization. To connect the lattice modulation with the competitions between the parameters of the Bose-Hubbard Hamilto- nian, we show how such parameters scale with the lattice potential depth up to the first approximation. We then describe the ground state of the system, as well as the techniques used to experimentally probe them.
Chapter 2 comprises a review of the important aspects of phase transitions between the ground states of the Bose-Hubbard model. Firstly, we present the definition of quantum phase transitions and important aspects concerning the scaling behavior near criticality.
Then, we discuss the Landau theory as a method for investigating the symmetry-breaking
mechanism that occurs in continuous phase transitions. These critical properties of quan- tum phase transitions are discussed in the context of the Mott-superfluid phase transition in the clean case. Next, we work how the presence of disorder affects the phase diagram, specifically how the Bose-glass state intervenes.
We present the principal characteristics of the single-particle Green’s functions in Chapter 3. First, we discuss how these quantities emerge from linear response theory.
Then, we present the general form of the Källén-Lemann representation of Green’s func- tions and derive the spectral function, which incorporates the singularities of the Green’s functions. To set the stage for our perturbation treatment, we introduce the Matsubara formalism. Next, we discuss some general properties of Green’s functions in disordered systems. We then consider the physical interpretation of Green’s functions in the context of elementary excitations and apply these ideas to the disordered Bose-Hubbard model in the limit of strong interactions. To close this chapter, we briefly report the techniques of Bragg spectroscopy and the radio-frequency transfer method, which can be applied in experiments to measure Green’s and spectral functions.
In Chapter 4, we develop the perturbation method used to obtain the main results of this thesis. The method relies on a hopping expansion constructed in the functional integral formalism. We first demonstrate how to calculate corrections to the Matsubara Green’s function in terms of the tunneling energy at a tree-level approximation. After that, we combine an infinite subset of terms in the hopping expansion of the Green’s function by applying the technique of partial summation. This allows us to obtain a result that takes into account the corrections to the path of an excitation which hops through the lattice and undergoes infinitely many scattering events against the disordered potential at each site. By considering the first relevant correction to the local Green’s function in the hopping expansion, we then use the Poincaré-Lindstedt method to obtain a renormalized expression for the local density of states.
Chapter 5 is devoted to the analysis of the effect of disorder on the excitation spec- trum of the Bose-Hubbard model in the limit of strong interactions. To this end, we derive the spectral function that follows from the partial summation result of the Green’s function.
First, we present the details of the spectral function in the clean case. Then, we discuss the changes that occur in the disordered case by presenting the results for a bounded uni- form disorder distribution. We then analyze the propagation of excitations by observing the spatiotemporal profile of the retarded Green’s function.
The results for the phase diagram in the disordered case at finite temperatures and for small values of the tunneling energy are presented in Chapter 6. Firstly, we derive the superfluid phase boundary from the partial summation result of the Green’s function.
Secondly, by analyzing the renormalized local density of states obtained with the Poincaré- Lindstedt method, we obtain an analytical expression for the phase boundary of the Mott- Bose glass phase transition. Then, we present the phase diagram for a uniform disorder
distribution. Finally, we compare our results with numerical calculations from the literature.
In Chapter 7, we present some preliminary calculations that serve as the starting point for investigating the effects of disorder on the excitation spectra of the superfluid phase. These calculations follow from the application of the effective-action approach. We first show how to derive the effective-action functional to recover the results for the exci- tation spectrum obtained in Chapter 5. By considering first-order fluctuations around its equilibrium value, we discuss how one could obtain the energy spectra for the superfluid excitations within this approach. We then present a possible order parameter that can be used to investigate the excitations in the Bose-glass phase.
Chapter 8 contains the development of a disorder expansion perturbation theory. In this chapter, we consider free lattice bosons in the presence of weak disorder and construct a diagrammatic expansion for the Green’s function within the functional integral formalism.
By considering a single-site approximation, it is possible to obtain a result for the singulari- ties of the Green’s function, which is similar to the one obtained with the partial summation method. We then discuss how to investigate the interacting system by considering the hop- ping and disorder expansion together, where the fluctuations of each perturbation method could be treated separately.
Finally, Chapter 9 comprises a summary and concluding remarks that contextualize the contributions of the present work. There, we also discuss some prospects and insights that could be topics for future research.
Appendices A, B, and C contain the derivations of the functional integral formula- tion of the Bose-Hubbard model, the sum rule of the spectral function, and the asymptotic behavior of the Green’s function, respectively.