The half-filled one-dimensional Holstein-Hubbard model presents, at zero temperature, a charge-density-wave (CDW) phase and a Mott insulator phase. Recent results have shown that the transition from one phase to the other might proceed through an intermediate metallic phase. In this work, we determine the CDW phase boundary using the variational cluster approximation. Using exact diagonalization and cluster perturbation theory, we study both the pair susceptibility and the spectral gaps in the non-CDW part of the phase diagram. We cannot rule out the existence of an intermediate metallic phase.
The variational cluster approach (VCA) is applied to study spontaneous ferromagnetism in the Hubbard model at zero temperature. We discuss several technical improvements of the numerical implementation of the VCA which become necessary for studies of a ferromagnetically ordered phase, e.g., more accurate techniques to evaluate the variational ground-state energy, improved local as well as global algorithms to find stationary points, and different methods to locate the magnetic phase transition. Using the single-site VCA, i.e., the dynamical impurity approximation (DIA), the ferromagnetic phase diagram of the model in infinite dimensions is worked out. The results are compared with previous dynamical mean-field studies for benchmarking purposes. The DIA results provide a unified picture of ferromagnetism in the infinite-dimensional model by interlinking different parameter regimes that are governed by different mechanisms for ferromagnetic order. Using the DIA and the VCA, we then study ferromagnetism in one-dimensional Hubbard chains with nearest- and next-nearest-neighbor hopping
t2. In comparison with previous results from the density-matrix renormalization group, the phase diagram is mapped out as a function of the Hubbard-
U, the electron filling, and
t2. The stability of the ferromagnetic ground state against local and short-range nonlocal quantum fluctuations is discussed.
The variational cluster approximation is used to study the spontaneous symmetry breaking of the excitonic insulator (EI) state in the two-orbital Hubbard model defined on the two-dimensional square lattice. Evaluating the order parameter of the EI state, staggered magnetization, and number of particles on each orbital, we obtain the ground-state phase diagram of the model in a wide parameter space of the intra and interorbital Coulomb interactions. We also calculate the single-particle and anomalous Green's functions. We show that the normal metallic phase is unstable to the formation of the EI state and the EI phase appears in a wide parameter region between the band-insulator and Mott-insulator phases.