During this Ph.D, we studied one-dimensional systems with long-range couplings. In the first part, we demonstrate that power-law couplings lead to an algebraic decay of correlations at long distances in disordered quantum wires. In the second chapter, we analyzed an extended Hubbard model where particles interact via a finite-range potential that induces frustration and new exotic phases. In the third chapter, we demonstrated that restoring energy extensivity has an influence on the low-energy properties of quantum model in the thermodynamic limit. Finally, we provide preliminary results on the modification of Anderson localization due to the coupling to a cavity mode.
During this Ph.D, we studied one-dimensional systems with long-range couplings. In the first part, we demonstrate that power-law couplings lead to an algebraic decay of correlations at long distances in disordered quantum wires. In the second chapter, we analyzed an extended Hubbard model where particles interact via a finite-range potential that induces frustration and new exotic phases. In the third chapter, we demonstrated that restoring energy extensivity has an influence on the low-energy properties of quantum model in the thermodynamic limit. Finally, we provide preliminary results on the modification of Anderson localization due to the coupling to a cavity mode.
We analyze the effects of disorder on the correlation functions of one-dimensional quantum models of fermions and spins with long-range interactions that decay with distance $\ensuremath{\ell}$ as a power law $1/{\ensuremath{\ell}}^{\ensuremath{\alpha}}$. Using a combination of analytical and numerical results, we demonstrate that power-law interactions imply a long-distance algebraic decay of correlations within disordered-localized phases, for all exponents $\ensuremath{\alpha}$. The exponent of algebraic decay depends only on $\ensuremath{\alpha}$, and not, e.g., on the strength of disorder. We find a similar algebraic localization for wave functions. These results are in contrast to expectations from short-range models and are of direct relevance for a variety of quantum mechanical systems in atomic, molecular, and solid-state physics.
The central aim of this thesis is the study of the low-energy and low-temperature properties of strongly correlated systems of bosonic particles interacting via finite- and long-range potentials, and relevant to experimental realization with cold atomic gases. This study is carried out with a combination of state-of-the-art numerical techniques such as Path Integral Monte Carlo and analytical techniques. The main result of my work is the demonstration of the existence of a stripe supersolid phase and of a rare transition between isotropic and anisotropic supersolids in a finite-range interacting model of hard-bosons on a square lattice. I also investigate the out-of-equilibrium scenarios of such models via simulated temperature quenches. Finally, I investigate how restoring energy extensivity in long-range interacting systems can have a profound incidence on the low-energy properties in the thermodynamic limit.
This dissertation examines the impact of long-range interactions in lattice models of low-dimensional strongly correlated systems with a focus on their relevance to anomalous electronic transport. Traditionally, the Hubbard model forms the basis of theoretical and numerical investigations of strongly correlated quantum matter. This effective model constitutes the simplest model of electronic correlation and relies on an approximation that truncates electron-electron interactions to a local, on-site potential. While this approximation has proved valid in several instances, certain unexplained phenomena in correlated materials challenge us to develop new perspectives and approaches.In the first part of this thesis, we discuss the historical motivation for our current work from both a theoretical and an experimental viewpoint. In particular, we cover the discovery of bad metallic transport and the associated breakdown of Boltzmann theory of electron transport. We further review examples of strongly correlated systems that exhibit bad metallic transport and other exotic phenomena that suggest the possibility of poorly screened electronic interactions acting as a common microscopic mechanism. This general feature justifies the development of a theoretical framework that is capable of examining long-range interactions in a controlled, systematic manner.Motivated by this, we analyze a long-range interacting lattice model for spinless electrons in two dimensions via exact diagonalization on a finite cluster in real space. As this method is susceptible to finite-size errors, we discuss at length its implementation and technical details designed specifically to reduce these errors, namely the use of twisted boundary conditions and the Ewald summation. We determine the phase diagram at zero temperature and demonstrate that the long-range interactions act as a source of self-generated disorder that drives the development of a strongly correlated pseudogap phase, in a manner reminiscent of the classical Efros-Shklovskii Coulomb gap. Furthermore, we establish that this phenomenon is independent of lattice geometry, a fact that represents the broad potential of long-range interactions in giving rise to a novel form of strongly correlated behavior.Following our zero-temperature study, we address the question of whether this self-generated disorder can survive in the presence of thermal fluctuations and local Coulomb repulsion (Hubbard terms) relevant in Mott systems, a situation closer to those in experimental systems. By means of classical Monte Carlo calculations, we show that not only does the self-generated disorder survive, but that it also acts as a microscopic mechanism that generates bad metallic transport. Upon further investigation, we determine that the long-range interactions, in combination with quantum fluctuations, cause a transient localization of the charge carriers, thereby suppressing the conductivity values below those predicted in conventional Boltzmann theory, as observed experimentally.In conclusion, we have established that long-range interactions in electronic models act as a microscopic mechanism that is capable of causing exotic phenomena in numerous materials. We conclude that a proper treatment of both quantum fluctuations and long-range interactions should next be studied with the finite temperature Lanczos method, which will act as a bridge between our exact diagonalization and Monte Carlo studies. Overall, this dissertation emphasizes the capability of long-range interactions to produce novel phenomena and elicits further thought-provoking questions concerning the interplay with Mott-Hubbard physics and external disorder, crucial in metallic systems.
Topic of the thesis is a theoretical description of the ultracold atomic gases in one- and two-dimensional optical lattices in the presence of the disorder leading to the Anderson localization. The disorder is created by interaction of the main fraction of atoms with the second immobilized fraction distributed randomly over the lattice. In low-dimensional systems there is no transition from the Anderson localized to the conducting phase, although in the presence of correlations a discrete set of extended states can exist. The first part of the thesis is devoted to properties of such states. In the finite size lattices, the presence of those states results in the appearance of `windows of transport' -- energy ranges, in which the localization length is longer than the system size.The analytical method of determining the extended states energies for correlations of generalized N-mers type is presented, along with a proof that indeed those states are extended in the infinite system. Subsequently, the way of experimental creation of this type of correlations is proposed, as well as the technique of generation of the disorder in the tunneling amplitudes, which significantly enhances tunability of the proposed energy filters. The second part of the thesis describes the method which allows to simulate a specific type of the disorder: random magnetic field. Systems of such a class, created in a two dimensional lattice using simultaneous fast periodic modulation of the lattice height and interactions with immobilized species, may allow in future to investigate the range of topics from the condensed matter physics, for example fractional quantum Hall effect at half-filling. In the thesis, the most interesting features observed upon investigation of such systems are presented. Especially, the anomalously low localization length for correlated disorder is explained.
We investigate the zero-temperature phase diagram of the one-dimensional Bose-Hubbard model with power-law hopping decaying with distance as 1/r^{α} using exact large scale quantum Monte Carlo simulations. For all 1<α≤3 the quantum phase transition from a superfluid and a Mott insulator at unit filling is found to be continuous and scale invariant, in marked contrast with the Berezinskii-Kosterlitz-Thouless (BKT) scenario that is recovered only for α>3. By performing finite-size scaling collapses of the superfluid stiffness and extracting dynamical and correlation-length exponents from the low-energy spectrum, we establish that these transitions define a distinct universality class throughout the long-range regime 1<α≤3. Analysis of the single-particle correlation functions and grand canonical phase diagram further reveals a sequence of ordering regimes within the superfluid phase: true long-range order for α≤2, anomalous quasi-long-range order for 2<α≤3, and conventional algebraic decay for α>3. Our exact numerical results provide a benchmark to compare theories of long-range quantum models and are relevant for experiments with cold neutral atom, molecules and ion chains.
We analyze the quantum phases, correlation functions and edge modes for a class of spin-1/2 and fermionic models related to the one-dimensional Ising chain in the presence of a transverse field. These models are the Ising chain with anti-ferromagnetic long-range interactions that decay with distance r as 1 / r α , as well as a related class of fermionic Hamiltonians that generalize the Kitaev chain, where both the hopping and pairing terms are long-range and their relative strength can be varied. For these models, we provide the phase diagram for all exponents α , based on an analysis of the entanglement entropy, the decay of correlation functions, and the edge modes in the case of open chains. We demonstrate that violations of the area law can occur for α ≲ 1 , while connected correlation functions can decay with a hybrid exponential and power-law behavior, with a power that is α -dependent. Interestingly, for the fermionic models we provide an exact analytical derivation for the decay of the correlation functions at every α . Along the critical lines, for all models breaking of conformal symmetry is argued at low enough α . For the fermionic models we show that the edge modes, massless for α ≳ 1 , can acquire a mass for α < 1 . The mass of these modes can be tuned by varying the relative strength of the kinetic and pairing terms in the Hamiltonian. Interestingly, for the Ising chain a similar edge localization appears for the first and second excited states on the paramagnetic side of the phase diagram, where edge modes are not expected. We argue that, at least for the fermionic chains, these massive states correspond to the appearance of new phases, notably approached via quantum phase transitions without mass gap closure. Finally, we discuss the possibility to detect some of these effects in experiments with cold trapped ions.