This thesis discusses mainly two Fermionic lattice systems, first a Kondo lattice with additional Hubbard interaction and second a Hubbard Hamiltonian augmented with additional spin and charge interactions. We first introduce the Quantum Monte Carlo technique, which is then employed to study the two respective systems. We present an innovation that allows to calculate time displaced Greens functions more efficiently. Compared with previously used numerically stable algorithms the new method gains an order of magnitude in speed, but is just as precise, and very simple to implement. In the second chapter we consider the Kondo lattice model in two dimensions at half filling. In addition to the Fermionic hopping integral t and the superexchange coupling J the role of a Coulomb repulsion U in the conduction band is investigated. We find the model to display a magnetic order-disorder transition in the U-J plane with a critical value of Jc which is decreasing as a function of U. The single-particle spectral function A(k,ω) is computed across this transition. We conclude that (i) the local screening of impurity spins determines the low-energy behavior of the spectral function and (ii) one cannot deform continuously the spectral function of the half-filled Hubbard model at J=0 to that of the Kondo insulator at J>Jc. In the third chapter we investigate the phase diagram of a new model that exhibits a first order transition between s-wave superconducting and antiferromagnetic phases. The model, a generalized Hubbard model augmented with competing spin-spin and pair-pair interactions, was investigated using the projector quantum Monte Carlo method. Upon varying the Hubbard U from attractive to repulsive, we find a first order phase transition between superconducting and antiferromagnetic states.
Ground-state properties of the one-dimensional Kondo lattice model are investigated using the quantum Monte Carlo (QMC) simulation method. To reduce the difficulty of the negative sign problem, we employ the variationally optimized singlet wave function. Since the improvement of Monte Carlo sampling efficiency is achieved in the half-filling case, we can perform the simulations on large systems (up to 26-site system). The QMC data on the local spin correlations are closely compared with those of the variational Monte Carlo (VMC) calculations as well as with the exact diagonalization data of the Ruderman-Kittel-Kasuya-Yoshida (RKKY) lattice model. We find that the local spin correlation decays exponentially with the correlation length ${\ensuremath{\xi}}_{\mathit{f}\mathit{f}}$\ensuremath{\sim}2.54 in the QMC data, while ${\ensuremath{\xi}}_{\mathit{f}\mathit{f}}$\ensuremath{\sim}1.08 in the VMC results (J/t=1). This implies the absence of the long-range order (the existence of the spin gap) and the importance of intersite correlation effects which are not taken into account in the VMC results. On the other hand, the system size dependence is almost absent in the Kondo lattice model results, which contrasts the behavior in the RKKY data.
The interplay between the Kondo effect, indirect magnetic interaction, and geometrical frustration is studied in the Kondo lattice on the one-dimensional zigzag ladder. Using the density-matrix renormalization group, the ground-state and various short- and long-range spin- and density-correlation functions are calculated for the model at half filling as a function of the antiferromagnetic Kondo interaction down to
J=0.3t, where
t is the nearest-neighbor hopping on the zigzag ladder. Geometrical frustration is shown to lead to at least two critical points: Starting from the strong-
J limit, where almost local Kondo screening dominates and where the system is a nonmagnetic Kondo insulator, antiferromagnetic correlations between nearest-neighbor and next-nearest-neighbor local spins become stronger and stronger, until at ${J}_{\mathrm{c}}^{\mathrm{dim}}\ensuremath{\approx}0.89t$ frustration is alleviated by a spontaneous breaking of translational symmetry and a corresponding transition to a dimerized state. This is characterized by antiferromagnetic correlations along the legs and by alternating antiferro- and ferromagnetic correlations on the rungs of the ladder. A mechanism of partial Kondo screening that has been suggested for the Kondo lattice on the two-dimensional triangular lattice is not realized in the one-dimensional case. Furthermore, within the symmetry-broken dimerized state, there is a magnetic transition to a ${90}^{\ensuremath{\circ}}$ quantum spin spiral with quasi-long-range order at ${J}_{\mathrm{c}}^{\mathrm{mag}}\ensuremath{\approx}0.84t$. The quantum-critical point is characterized by a closure of the spin gap (with decreasing
J) and a divergence of the spin-correlation length and of the spin-structure factor
S(q) at wave vector $q=\ensuremath{\pi}/2$. This is opposed to the model on the one-dimensional bipartite chain, which is known to have a finite spin gap for all
J>0 at half filling.
The ground state properties of the one-dimensional Kondo lattice with an f^2 configuration at each site are studied by the density matrix renormalization group method. At half-filling, competition between the Kondo exchange J and the antiferromagnetic intra f-shell exchange I leads to reduction of energy gaps for spin, quasi-particle and charge excitations. The attractive force among conduction electrons is induced by the competition and the bound state is formed. As J/I increases the f^2 singlet gives way to the Kondo singlet as the dominant local correlation. The remarkable change of the quasi-particle gap is driven by the change of the spin-1/2 excitation character from the itinerant one to the localized one. Possible metal-insulator transition is discussed which may occur as the ratio J/I is varied by reference to mean-field results in the f^2 lattice system and the two impurity Kondo system.
We investigate the phase diagram of the half-filled
SU(N) Hubbard-Heisenberg model with hopping
t, exchange
J, and Hubbard
U, on a two-dimensional square lattice. In the large-
N limit, and as a function of decreasing values of
t∕J, the model shows a transition from a
d-density wave state to a spin dimerized insulator. A similar behavior is observed at
N=6 whereas at
N=2 a spin density wave insulating ground state is stabilized. The
N=4 model, has a
d-density wave ground state at large values of
t∕J which as a function of decreasing values of
t∕J becomes unstable to an insulating state with no apparent lattice and spin broken symmetries. In this state, the staggered spin-spin correlations decay as a power law, resulting in gapless spin excitations at $\stackrel{P\vec}{q}=(\ensuremath{\pi},\ensuremath{\pi})$. Furthermore, low lying spin modes with small spectral weight are apparent around the wave vectors $\stackrel{P\vec}{q}=(0,\ensuremath{\pi})$ and $\stackrel{P\vec}{q}=(\ensuremath{\pi},0)$. This gapless spin liquid state is equally found in the SU(4) Heisenberg $(U∕t\ensuremath{\rightarrow}\ensuremath{\infty})$ model in the self-adjoint antisymmetric representation. An interpretation of this state in terms of a $\ensuremath{\pi}$-flux phase is offered. Our results stem from projective
(T=0) quantum Monte Carlo simulations on lattice sizes ranging up to $24\ifmmode\times\else\texttimes\fi{}24$.
Motivated by the recent proposal that the Monte Carlo sign problem might be used as an indicator of quantum phase transitions, we simulate the periodic Anderson model at half-filling with a staggered ionic potential using the determinant quantum Monte Carlo (DQMC) approach. We observe a phase transition from a band insulator to a Kondo insulator as the hybridization increases. At finite temperature, an intermediate metallic region emerges between two insulating phases, which is characterized by a pseudogap in the spectral functions near the boundary of the intermediate region and a quasiparticle peak in the middle. The latter coincides with a crossing point of the
f-electron occupation on different sublattices as well as the maximum antiferromagnetic (AFM) correlations. The metallic region extrapolates to a single quantum critical point at zero temperature, which may result in exotic quasiparticle excitations associated with the emergent metallic state. The Monte Carlo sign problem is found to be most pronounced in the quantum critical region, implying a close connection between the two phenomena. Our results are qualitatively supported by the Hartree-Fock (HF) mean-field calculations. Our work reveals interesting physics in inhomogeneous Kondo lattice systems that may be worthwhile of more investigations in the future.