We study antiferromagnetism and single-particle properties in the two-dimensional half-filled Hubbard model at low temperature. Collective spin fluctuations are governed by a non-linear sigma model that we derive from the Hubbard model for any value of the Coulomb repulsion. As the Coulomb repulsion increases, the ground state progressively evolves from a Slater to a Mott-Heisenberg antiferromagnet. At finite temperature, we find a metal-insulator transition between a pseudogap phase at weak coupling and a Mott-Hubbard insulator at strong coupling.
Using the Hubbard chain at quarter filling as a model system, we study the ground state properties of highly doped antiferromagnets. In particular, the Hubbard chain at quarter filling is unstable against
2kF- and
4kF-periodic potentials, leading to a large variety of charge and spin ordered ground states. Employing the density matrix renormalization group method, we compare the energy gain of the ground state induced by different periodic potentials. For interacting systems the lowest energy is found for a
2kF-periodic magnetic field, resulting in a band insulator with spin gap. For strong interaction, the
4kF-periodic potential leads to a half-filled Heisenberg chain and thus to a Mott insulating state without spin gap. This ground state is more stable than the band insulating state caused by any non-magnetic
2kF-periodic potential. Adding more electrons, a cluster-like ordering is preferred.
We investigate the ground-state phase diagram of the half-filled one-dimensional Hubbard model with next-nearest-neighbor hopping using the density-matrix renormalization-group technique as well as an unrestricted Hartree-Fock approximation. We find commensurate and incommensurate disordered magnetic insulating phases and a spin-gapped metallic phase in addition to the one-dimensional Heisenberg phase. At large on-site Coulomb repulsion U, we make contact with the phase diagram of the frustrated Heisenberg chain, which has spin-gapped phases for sufficiently large frustration. For weak U, sufficiently large next-nearest-neighbor hopping
t2 leads to a band structure with four Fermi points rather than two, producing a spin-gapped metallic phase. As U is increased in this regime, the system undergoes a Mott-Hubbard transition to a frustrated antiferromagnetic insulator.
The half-filled Hubbard model on the Bethe lattice with coordination number
z=3 is studied using the density-matrix renormalization group (DMRG) method. Ground-state properties such as the energy
E, average local magnetization $ $, its fluctuations $ - ^2$ and various spin correlation functions $ - $ are determined as a function of the Coulomb interaction strength
U/t. The calculated local magnetic moments $ $ increase monotonically with increasing Coulomb repulsion
U/t forming an antiferromagnetic spin-density-wave state which matches the two sublattices of the bipartite Bethe lattice. At large
U/t, $ $ is strongly reduced with respect to the saturation value 1/2 due to exchange fluctuations between nearest neighbors (NN) spins (
∣∣≃0.35 for
U/t→+∞). $ - ^2$ shows a maximum for
U/t=2.4--2.9 which results from the interplay between the usual increase of $ $ with increasing
U/t and the formation of important permanent moments $ $ at large
U/t. NN sites show antiferromagnetic spin correlations which increase with increasing Coulomb repulsion. In contrast next NN sites are very weakly correlated over the whole range of
U/t. The accuracy of the DMRG results is discussed by comparison with tight-binding exact results, independent DMRG calculations for the Heisenberg model and simple first-order perturbation estimates.
Low energy behavior of the S = 1/2 antiferromagnetic XXZ chains with Fibonacci exchange/field modulation is studied by the density matrix renormalization group method. For both types of modulation, the energy gap of the chain with length N scales as exp(-cNω ) with nonuniversal exponent ω if the z-component of the exchange coupling is antiferromagnetic. For the field modulation case, the possibility of fully localized ground state is suggested. This is in contrast to the XY chains for which the gap scales as N−z .
The low energy behavior of the S=1/2 antiferromagnetic XY-like XXZ chains with precious mean quasiperiodic exchange modulation is studied by the density matrix renormalization group method. It is found that the energy gap of the chain with length N scales as
exp(−cNω) with nonuniversal exponent
ω if the Ising component of the exhange coupling is antiferromagnetic. This behavior is expected to be the characteristic feature of the quantum spin chains with relevant aperiodicity. This is in contrast to the XY chain for which the precious mean exchange modulation is marginal and the gap scales as
N−z. On the contrary, it is also verified that the energy gap scales as
N−1 if the Ising component of the exhange coupling is ferromagnetic. Our results are not only consistent with the recent bosonization analysis of Vidal, Mouhanna and Giamarchi but also clarify the nature of the strong coupling regime which is inaccesssible by the bosonization approach.