Unconventional and intertwined orders of the low-dimensional Hubbard model

The understanding of superconductivity exhibited at high critical temperature by certain transition metal oxides remains a central issue in theoretical condensed matter physics. In this context, and since the historical proposal by P. W. Anderson, the repulsive Hubbard model in two dimensions became a paradigm in an attempt to capture the essential properties of non-conventional superconducting materials. However, the determination of the exact ground state encounters the exponential complexity of the quantum many-body problem. The main purpose of this thesis is to develop a variational scheme free of any hypothesis concerning magnetic, charge or superconducting orders likely to emerge from the Hamiltonian at low energy. The originality of the approach is found in the introduction of correlations by restoring, before variation, symmetries deliberately broken in a trial state given by a superposition of versatile wavefunctions of Hartree-Fock and Bogoliubov-de Gennes types. For small clusters of two and four sites, we show analytically that this symmetry entangled mean field method allows to find the exact ground state regardless of the strength of the on-site interaction. For larger hole-doped clusters and in the strongly correlated regime, we highlight an arrangement of magnetic moments in a spiral or in a spin density wave that is then accompanied by inhomogeneities in the form of regularly distributed stripes. Moreover, such orders are intertwined with long range d-wave pairing correlations, which, in the thermodynamic limit, sign superconductivity. These results are obtained through systematic simulations in a four-leg tube geometry that can be realized experimentally using cold atoms trapped in optical lattices. (author)