We show how it is possible to include the constraint of single site-occupancy in the large U limit of the Hubbard model at half-filling by means of a local chemical potential. The formalism which is in principle exact allows for finite temperature mean field results satisfying the constraint within 10%. The method is here applied to the dimerized phase and the commensurate flux phase which are still found as local saddle points of the action. The symmetry properties of these actions have been studied. One loop static corrections have been added to obtain an effective free energy and an estimate of the specific heat. In the limit of zero temperature our results reproduce those of the conventional mean field theory of these phases, because quantum fluctuations have not been included. We find that the temperature destabilizes the flux phase more than it does with the dimer phase, especially due to the constraint, which does not suppress phase fluctuating modes, provided the flux is conserved.
We implement the constraint of no-double-site occupancy exactly, in the spin-1/2 Hubbard-Heisenberg model at half filling. The staggered Peierls and the flux phase are again found to be saddle points for the effective action, also at finite temperature. Their mean-field critical temperatures are higher when the accuracy restriction is taken into account. Gaussian fluctuations of the auxiliary field are reasonably small for a large temperature range, but have to be necessarily included, when extending the free energy to finite temperatures.
We have calculated the ground-state energy and the fluctuation contribution to the free-energy for the Hubbard model at half filling. We have used a gauge that leads to the Hartree-Fock solution in the saddle-point approximation. The mean-field ground-state energy calculated here agrees with the results obtained by Dagotto et al. [Phys. Rev. B 45, 10 741 (1992)] with use of the exact-diagonalization techniques on a 4\ifmmode\times\else\texttimes\fi{}4 site cluster. In the large-U limit, the ground-state energy per site is obtained as
Eg/N=-U/4-J+O(
J2/U). The fluctuation contribution to the free-energy was found to have a stronger temperature and U dependence than the mean-field contribution, and to play a dominant role for strongly correlated systems.
We report on a nonperturbative approach to the one-dimensional (1D) and two-dimensional (2D) Hubbard models that is capable of recovering both strong- $(U\ensuremath{\gg}t)$ and weak-coupling $(U\ensuremath{\ll}t)$ limits, with U the on-site Coulomb repulsion and t the kinetic energy. Dynamical corrections to the electron self-energy in the single particle Green function are explicitly included by expanding in terms of the 16 eigenstates that characterize two nearest neighbor sites. We first show that even when U is much smaller than the bandwidth, the Mott-Hubbard gap never closes at half filling in both 1D and 2D. Consequently, the Hubbard model at half filling is always in the strong-coupling nonperturbative regime. For both large and small U, we find that the population of nearest-neighbor singlet states approaches a value of order unity as $\stackrel{\ensuremath{\rightarrow}}{T}0$ as would be expected for antiferromagnetic order. We also find that the double occupancy is a smooth monotonic function of U and approaches the anticipated noninteracting limit of
1/4 as $\stackrel{\ensuremath{\rightarrow}}{U}0$ and vanishes as $\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty}.$ Finally, we compute the heat capacity
[C(T,U)] for both 1D and 2D. Our results for 1D at moderate to high temperatures are in quantitative agreement with those of the exact Bethe ansatz solution, differing by no more than
1 In addition, we find that in 2D, the
C(T,U) curves vs T for different values of U exhibit a universal crossing point at two characteristic temperatures $T\ensuremath{\approx}1.7t\ifmmode\pm\else\textpm\fi{}0.1t$ and $T\ensuremath{\approx}0.4\ifmmode\pm\else\textpm\fi{}0.1t$ as is seen universally in Hubbard models and experimentally in a wide range of strongly correlated systems such as
3He, UBe3, and ${\mathrm{CeCu}}_{6\ensuremath{-}x}{\mathrm{Al}}_{x}.$ The success of this method in recovering well-established results that stem fundamentally from the Coulomb interaction suggests that local dynamics are at the heart of the physics of strongly correlated systems.