The one-band two-dimensional Hubbard Hamiltonian is analyzed using a real-space renormalization-group block method. The renormalization-group method, previously applied to the one-dimensional case has been extended to two dimensions. It is also shown how to avoid the proliferation of new terms during the process if the shape of the blocks and the symmetries of the kept states are chosen conveniently. We characterize the ground state by its energy per site, the gap of charge excitations, the double occupancy, and the effective hopping. The system shows an insulating behavior in the case of half filling for all nonzero interaction parameter, U, if the hopping is isotropic (
ty=
tx); that is to say, the gap opens for all positive U, as in the one-dimensional case. If the hopping anisotropy, \ensuremath{\alpha}=
ty/
tx, is different from the unity and zero the ground state is a conductor up to a critical value (U/t
)c1, which depends on \ensuremath{\alpha}. We have analyzed in the same way another filling, n=7/9. The system behaves as a conductor, paramagnetic up to a value (U/t
)c2\ensuremath{\sim}2 and weakly ferromagnetic for higher values of U.
The ground state of the half-filled Hubbard model is studied using a real-space renormalization-group technique. We first study in detail the one-dimensional case. We find the ground state to be insulating for all finite coupling, in agreement with the exact solution. We compute the ground-state energy, localization length, energy gap, magnitude of the local moment, and spin-density autocorrelation function. For those quantities that are exactly known we find good agreement with the exact results. Using a simple extension of our one-dimensional calculation, we are able to study approximately two- and three-dimensional lattices. We find a Mott transition at finite interaction for these cases. The critical exponents for these transitions are found to satisfy an approximate interdimensional scaling law.
By renormalization-group methods we obtain nonperturbative results about a
d=1 system of interacting spinless fermions in a periodic potential when the conduction band is filled. Both the strength of the interaction and the amplitude of the periodic potential are assumed to be small. We determine that the large-distance asymptotic behavior of the two-point Schwinger function is anomalous and described by two critical indices, explicitly computed by convergent series, related to the renormalization of the spectral gap and of the discontinuity at the Fermi surface.
The real-space renormalization-group technique is applied to the Hubbard model on a linear chain with a half-filled band; a recently proposed modified scheme is used for the renormalization of the parameters. The result for the ground-state energy is quite improved with respect to the earlier results of the existing scheme. Both the energy and its derivative with respect to the on-site repulsion term agree reasonably well with the exact solution. However, it is indicated that the calculation of the correlation function and energy gap requires further refinement of this scheme.
A zero-temperature real-space renormalization-group method for quantum systems is applied to the
S=21XY model with a
Z field with the Hamiltonian $H=\ensuremath{-}\frac{1}{2}J\ensuremath{\Sigma}{〈i, j〉}^{}[(1+\ensuremath{\gamma}){S}_{i}^{x}{S}_{j}^{x}+(1\ensuremath{-}\ensuremath{\gamma}){S}_{i}^{y}{S}_{j}^{y}]\ensuremath{-}h\ensuremath{\Sigma}{i}^{}{S}_{i}^{z}$ on quadratic, triangular, and hexagonal lattices in two dimensions (2D) and on a cubic lattice in 3D. In all the isotropic ($\ensuremath{\gamma}=0$) 2D cases the method gives no order in the ground state
∣0〉 but changes in the correlation functions at ${(\frac{h}{J})}_{c}=\frac{1}{2}{z}^{\ensuremath{'}}$ [${z}^{\ensuremath{'}}$ is the number of nearest neighbors (NN)] are observed. An argument is presented that at
(Jh)c the method becomes exact in the limit of infinite-size spin blocks. It is suggested that the correlation functions $〈0|{{S}_{0}}^{\ensuremath{\alpha}}{{S}_{R}}^{\ensuremath{\alpha}}|0〉$ ($\ensuremath{\alpha}=x, y$) for large
R have a power-law behavior $\ensuremath{\sim}{R}^{\ensuremath{-}\ensuremath{\eta}}$ for
0< ∣Jh∣<∣(Jh)c∣ with $\ensuremath{\eta}\ensuremath{\approx}1.5$ for
∣Jh∣<(Jh)c∣ and $\ensuremath{\eta}=2$ at
(Jh)c. The dynamical exponent
z is obtained to be \ensuremath{\approx} 0.3 for $|\frac{h}{J}|<\frac{1}{2}{z}^{\ensuremath{'}}$ and at
(Jh)c close to 2. An indirect method of calculating the ground-state energy is developed, and the results are compared with the known estimates. For $\ensuremath{\gamma}\ensuremath{\ne}0$ and
h=0 the $\ensuremath{\gamma}$ dependences of the order in the ground state ${\ensuremath{\mu}}^{x}(\ensuremath{\gamma})$ is calculated to be ${\ensuremath{\mu}}^{x}\ensuremath{\sim}{\ensuremath{\gamma}}^{0.21}$ for $\ensuremath{\gamma}\ensuremath{\rightarrow}0$ and
h=0. In a 3D cubic lattice ($\ensuremath{\gamma}=0$) the analysis of recursion relations indicates the existence of order, in accord with recent exact theorems.