This thesis presents our recent studies on quantum simulation of the Abelian Higgs lattice gauge theory with quantum systems engineered with ultracold atoms in optical lattices and quantum equilibration and thermalization dynamics in the context of Joule expansion in cold atom systems.

In the first part, we give proposals for the analog quantum simulation of lattice gauge theories with cold atoms in optical lattices with current experimental techniques, aiming at maximal simplicity on both the theoretical and experimental side. We search for the suitable platform for quantum simulation of the $(1+1)$-dimensional Abelian Higgs model, which is the scalar quantum electrodynamics replacing the fermionic fields by complex scalar fields. We use a discrete tensor reformulation to smoothly connect the space-time isotropic version used in most numerical lattice simulations to the continuous-time limit corresponding to the Hamiltonian formulation. We propose to use the Bose Hubbard model as a quantum simulator to probe the conformal Calabrese-Cardy scaling of its O(2) limit with a chemical potential. We also propose to use a physical multileg ladder of atoms trapped in optical lattices and interacting with Rydberg-dressed interactions to quantum simulate the model and measure the Polyakov loop. Numeric results from the Monte Carlo, the tensor renormalization group and the density matrix renormalization group techniques are cross-checked to support our proposals.

In the second part of this thesis, we investigate the Joule expansion of nonintegrable quantum systems that contain bosons or spinless fermions in one-dimensional lattices. Initially a barrier confines the particles to be in half of the system in a thermal state described by the canonical ensemble and is removed at time $t = 0$. We study the properties of the time-evolved density matrix, the diagonal ensemble density matrix and the corresponding canonical ensemble density matrix with an effective temperature determined by the total energy conservation. We discuss the equilibration and thermalization for the R enyi entropy of subsystems, few-body observables and reduced density matrices in subsystems. We finally discuss the difference between bosons and fermions in finite size scaling of the effective temperature. We propose the Joule expansion as a way to dynamically create negative temperature states for fermion systems with repulsive interactions.

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 physical

parameters 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 by

recent theoretical achievements, we investigate novel condensed matter systems

involving interacting fermions which are engineered to be confined in different

dimensions. 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 filling

or density, can have dramatic effects on the behavior of the lower dimensional

system.

Trapped cold atoms at very low temperatures, now readily accessible experimentally, can be controlled and serve as a testing ground for many theoretical physical properties which are paralleled in condensed matter systems. Ultracold trapped atom experiments show promise to recreate analogous condensed matter systems in a controlled environment and to create novel many-body quantum states of matter. The tunability of trapping laser intensities and magnetically controlled Feshbach interactions are the main tools for varying lattice hopping strengths and inter-particle interactions. In this thesis, we search for the phase of a many-body system of cold fermion atoms that can convert to boson molecules and vice versa. We then look at the microscopic details of two atoms that interact to form a quasibound molecule. Our two-atom model explains the mechanism behind the magnetically tunable interparticle interaction known as the Feshbach resonance. We find a dependence of the bound pair formation on the two-particle energy. Our results show that if a magnetic field can be fine tuned to sufficient precision, of order 10 mG, a narrow resonance can be achieved where longer lived bound states can be created. This narrow resonance reconciliates the two-body physics with the effective Hamiltonian that describes the many-body system.

Graphene consists of a monolayer of carbon atoms arranged in a honeycomb lattice and has been intensively studied due to its fascinating physical properties. We investigate transport in mesoscopic graphene systems, in particular, conductance oscillations due to interference of the Dirac electrons in phase-coherent transport. We consider a measurement set-up with two STM tips. We calculate the Green's functions for pure graphene in the tight-binding model, self-energy corrections due to the presence of the leads (STM tips), and $S$-matrices for the two tips, to obtain the overall $S$-matrix and the conductance using the Landauer-B"{u}ttiker formalism. We map the oscillations in the conductance as a function of position of the tips. For fixed tip positions, we find a non-monotonic dependence of the conductance on the coupling between the tips and graphene. Regular Bloch oscillations are obtained, where the wavelength is determined by the energy. Depending on the lattice orientation, an additional fast oscillation is obtained and its origin elucidated. Richer structures arise when the presence of an impurity is considered.

Moreover, we extend our research to voltage probing with multiple STM tips, specifically, three-tips configuration (two tips act as the current source and sink, the third tip serves as a voltage probe) was considered. Voltage profile has a unique expression for different separation of the fixed two tips, along with oscillations appears when the separation distance is bigger. Voltage profile express different when impurity introduced, which may be used for sensor.

The simplest way to add interactions to a single-band model of spinful electrons hopping on a crystal lattice is the Hubbard model, and the next simplest model is the extended Hubbard model (EHM). The EHM is useful for modeling superconductive electron pairing and emergent insulating phases including a dimerized phase called the "Bond Order Wave". In this thesis, I present two projects. In the first, I applied the numerical method of Density Matrix Renormalization Group to study the entanglement entropy at critical points of the clean EHM in one dimension near the BOW phase. In the second project, I applied analytical renormalization group approaches to study the phase diagram of the one dimensional model with weak disorder at weak coupling.

Graphene consists of an atom-thick layer of carbon atoms arranged in a honeycomb lattice, and its low-energy electronic excitations are well described as massless Dirac fermions with spin half and an additional pseudospin degree of freedom. Impurities in graphene can have a significant effect on the local electronic structure of graphene when the Fermi level is near the Dirac point. We study the local electronic spectra and real-space and k-space local density of state (LDOS) maps of graphene with different impurities (diagonal and non-diagonal impurity potential) such as vacancies, substitutional impurities, and adatoms. In the presence of a perpendicular magnetic field, we use a linearization approximation for the energy dispersion and employ a $T$-matrix formalism to calculate the Green's function. We investigate the effect of an external magnetic field on the Friedel oscillations and impurity-induced resonant states.

Using a multimode description for an scanning tunneling microscope (STM) tip, we calculate STM currents for the substitutional and vacancies case and find that strong resonances in the LDOS at finite energies lead to the presence of steps in the STM current and suppression of the Fano factor. We also describe in detail the theory of scanning tunneling spectroscopy in graphene in the presence of adatoms, magnetic or not, with localized orbitals of arbitrary symmetry, corresponding to any given angular momentum state.We show that quantum interference effects

which are naturally inbuilt in the honeycomb lattice, in combination

with the orbital symmetry of the localized state, allow scanning tunneling

probes to characterize adatoms and defects in graphene.

We study a mixture of fermionic and bosonic cold atoms on a two-dimensional optical lattice, where the fermions are prepared in two isospin states and the bosons have Bose-Einstein condensed. Number density fluctuations of the condensate form delocalized bosonic modes which couple to the fermionic atoms similarly to the electron-phonon coupling in crystals. We study the phase diagram for this system at fixed fermion density of one per site. We find that tuning of the lattice parameters and interaction strengths drives the system to undergo antiferromagnetic ordering, s-wave and d-wave pairing superconductivity, or a charge density-wave phase. We use functional renormalization group analysis where retardation effects are fully taken into account. We calculate response functions and also provide estimates of the energy gap associated with the dominant order, and how it depends on different parameters of the problem.

Graphene is a carbon based material that has only one atomic layer. It has exceptional electronic and mechanical properties which makes itself an ideal system to study nanoelectromechanical behaviors. In this work I present the fabrication and electromechanical properties of single layer graphene resonator in nanoribbon geometry.

Ultra-cold atom system provides novel technology to simulate traditional solid state physics, including boson and fermion particles. Due to the flexibility of tuning pa- rameters, people can further understand basic physics behind strongly correlated effects, especially mechanism of unconventional density wave and superfluidity. Using weak cou- pling renormalization group method, we propose and study several models which can be realized in experiments. Ranging from most well-known extended Hubbard model, to spin- polarized Fermi Hubbard model and multi-flavor Fermi-Fermi mixture, we establish solid theories to explain the origin of new states of matter, and the experimental techniques to exploit them. The effects of lattice structure and particle density in fermionic system play important roles for determining phase diagram in low energy scale. Due to the presence of lattice, these systems have insulator phase at half-filled. The density imbalance, however, prevents formation of conventional density wave state, and the screening interaction dra- matically affects other species's behavior. The interplay between screening interaction and square lattice is key ingredient of unconventional density wave and superfluidity. Our study

vishows that the unconventional density wave and superfluidity can be realized and detected in ultra-cold atom experiments.