We theoretically investigate the light absorption spectrum of two-dimensional (2D) Mott insulators. We employ the numerical diagonalization method to calculate the light absorption spectrum with using the effective Hamiltonian for the extended Hubbard model, which is valid when the on-site Coulomb interaction energy
U is much larger than the nearest-neighbor transfer integral
t and the nearest-neighbor Coulomb interaction energy
V. For
V=0, the absorption spectrum consists of a broad band with a width of about
16t and a sharp central peak, and the energy eigenstates contributing to the absorption spectrum do not have the antiferromagnetic (AF) spin order, when
U is sufficiently larger than
t. These features result from the spin-charge interplay inherent in the very strong correlation region where charge transfer term is dominant. With decreasing
U∕t, the peak structure in absorption spectrum becomes unclear and some low energy eigenstates have the AF spin order, as a result of the spin-spin interaction term. For large
V∕t, the dominant peak arises in the lower energy region of the spectrum. A large number of holon-doublon bound states, which have nearly the same charge but different spin structures, are responsible for this peak, in contrast to the conventional exciton state. This is also in contrast to the charge bound states in one-dimensional (1D) Mott insulators, where a single energy eigenstate dominates the optical transition moment as a result of the spin-charge separation. The essentially different absorption spectra between the 1D and 2D Mott insulators originate from the difference in the coupling between spin and charge degrees of freedom.
We study the third-order nonlinear optical susceptibility ${\ensuremath{\chi}}^{(3)}$ and photoexcited states of two-dimensional (2D) Mott insulators by using an effective model in the strong-coupling limit of a half-filled Hubbard model. In the numerically exact diagonalization calculations on finite-size clusters, we find that the coupling of charge and spin degrees of freedom plays a crucial role in the distribution of the dipole-allowed states with odd parity and the dipole-forbidden states with even parity in the photoexcited states. This is in contrast with the photoexcited states in one dimension, where the charge and spin degrees of freedom are decoupled. In the third-harmonic generation (THG) spectrum, the main contribution is found to come from the process of three-photon resonance associated with the odd-parity states. As a result, the two-photon resonance process is less pronounced in the THG spectrum. The calculated THG spectrum is compared with recent experimental data. We also find that ${\ensuremath{\chi}}^{(3)}$ with a cross-polarized configuration of pump and probe photons shows spectral distributions similar to ${\ensuremath{\chi}}^{(3)}$ with a copolarized configuration, although the weight is small. These findings will help analyses of the experimental data of ${\ensuremath{\chi}}^{(3)}$ in 2D Mott insulators.
One of the most interesting and challenging problems in physics is understanding strongly correlated many-body systems, where strong interactions can yield many remarkable phenomena such as superfluidity in 4He, high-temperature superconductivity, etc. In order to attack these problems, we often need to reduce the complexity of the systems to simple models in hopes of getting better insights into the properties of the systems. The Hubbard model, the focus of this dissertation, is one of the most famous examples of such model, which describes a tunneling of electrons between nearest neighbor sites of a lattice with on-site interactions. This simple model is an important concept in condensed matter physics and provides rich understandings of electronic and magnetic properties of materials. Despite its simplicity, there is no general analytical solution to the Hubbard model beyond 1D.;The discovery of ultracold atoms and optical lattices opens up the possibility of emulating the Hubbard model in experiments. Optical lattices provide an ideal realization of the Hubbard model where relevant parameters can be tuned systematically. It makes theoretical studies of the Hubbard model increasingly attractive since a direct comparison between theoretical calculations and experimental results becomes more and more possible.;In this dissertation, the ground-state properties of the repulsive Hubbard model for weak to intermediate interaction strengths in two, three dimensions and their dimensional crossover are studied within the mean field theory. We show that the system exhibits unidirectional spin-density wave (SDW) order with antiferromagnetic correlations and a long wavelength modulation. The modulating wave is along the [0011-direction at low interaction strength U/t and along the [1111-direction at higher U/t. The evolution of the wavelength of the SDW is determined as a function of U/t, the density, and t⊥/t. With an analysis of the pairing of spins based on nesting and deformation of the Fermi surface, we discuss how these results can be rationalized and how a simple, predictive model can be constructed for the properties of the SDW states.
We examine the third-order non-linear optical response of a one-dimensional Mott insulator coupled with phonons. The Mott insulator is described by an extended Hubbard-Holstein model. The third harmonic generation (THG) of the model is calculated by dynamical density-matrix renormalization group. We find that the electron-phonon interaction enhances the intensity of THG if the on-site Coulomb interaction is comparable to the band width, but the enhancement is small for a realistic parameter value of a typical Mott insulator Sr
2CuO
3. The low-energy spin excitation does not contribute to the optical response if the electron-phonon coupling is neglected. We find that the THG spectrum can detect the spin excitations even for no electron-phonon coupling. The introduction of the electron-phonon coupling leads to a slight increase of low-energy weight in THG without peak structure through phonon-assisted spin excitation. To fully understand the phonon-assisted spin excitation, we additionally calculate the linear susceptibility of a doped Hubbard-Holstein model, showing a spectral distribution different from that at half filling.
Systems of fermions described by the three-dimensional (3D) repulsive Hubbard model on a cubic lattice have recently attracted considerable attention due to their possible experimental realization via cold atoms in an optical lattice. Analytical and numerical results are limited away from half-filling. We study the ground state of the doped system from weak to intermediate interaction strengths within the generalized Hartree-Fock approximation. The exact solution to the self-consistent-field equations in the thermodynamic limit is obtained and the ground state is shown to exhibit antiferromagnetic order and incommensurate spin-density waves (SDW). At low interaction strengths, the SDW has unidirectional character with a leading wavevector along the [100]-direction, and the system is metallic. As the interaction increases, the system undergoes a simultaneous structural and metal-to-insulator transition to a unidirectional SDW state along the [111]-direction, with a different wavelength. We systematically determine the real and momentum space properties of these states. The crossover from 3D to two dimensions (2D) is then studied by varying the inter-plane hopping amplitude, which can be experimentally realized by tuning the distance between a stack of square-lattice layers. Detailed comparisons are made between the exact numerical results and predictions from the pairing model, a variational ansatz based on the pairing of spins in the vicinity of the Fermi surface. Most of the numerical results can be understood quantitatively from the ansatz, which provides a simple picture for the nature of the SDW states.