We systematically explore the ground state properties of one dimensional fermions with long-range interactions decaying in a power law ∼[Formula: see text] through the density matrix renormalization group algorithm. By comparing values of Luttinger liquid parameters precisely measured in two different ways, we show convincing evidence that Luttinger liquid theory is valid if [Formula: see text] is larger than some threshold, otherwise the theory breaks down. Combining analysis on structure factor, charge gap and charge stiffness, we determine how the metal-insulator transition point develops as the interaction range is continuously tuned. A region in the range of [Formula: see text] has small interactions and finite charge gaps, but, interestingly, it shows metallic nature at the same time. We obtain approximate phase diagrams for the entire parameter space and for band fillings equal to 1/2 and 1/3. Finally, we compare certain bosonization and field theory formulas with our quasi-exact numerical results, from which disagreements are found.
We systematically explore the ground state properties of one dimensional fermions with long-range interactions decaying in a power law ∼[Formula: see text] through the density matrix renormalization group algorithm. By comparing values of Luttinger liquid parameters precisely measured in two different ways, we show convincing evidence that Luttinger liquid theory is valid if [Formula: see text] is larger than some threshold, otherwise the theory breaks down. Combining analysis on structure factor, charge gap and charge stiffness, we determine how the metal-insulator transition point develops as the interaction range is continuously tuned. A region in the range of [Formula: see text] has small interactions and finite charge gaps, but, interestingly, it shows metallic nature at the same time. We obtain approximate phase diagrams for the entire parameter space and for band fillings equal to 1/2 and 1/3. Finally, we compare certain bosonization and field theory formulas with our quasi-exact numerical results, from which disagreements are found.
We solve a very general two-channel fermion-boson model describing charge transport within some background medium by means of a refined pseudosite density-matrix renormalization group technique. Performing a careful finite-size scaling analysis, we determine the ground-state phase diagram and convincingly prove that the model exhibits a metal-insulator quantum phase transition for the half-filled band case. In order to characterize the metallic and insulating regimes we calculate, besides the local particle densities and fermion-boson correlation functions, the kinetic energy, the charge-structure factor, the Luttinger liquid charge exponent, and the single-particle excitation gap for a one-dimensional infinite system.
We examine the ground-state phase diagram of the t-J model in one dimension by means of the density matrix renormalization group. This model is characterized by a rich phase diagram as a function of the exchange interaction J and the density n, displaying Luttinger-liquid (LL) behavior both of repulsive and attractive (i.e., superconducting) natures, a spin-gap phase, and phase separation. The phase boundaries separating the repulsive from the attractive LL phase as J is increased, and also the boundaries of the spin-gap region at low densities, and phase separation at even larger J, are determined on the basis of correlation functions and energy gaps. In particular, we shed light on a contradiction between variational and renormalization-group (RG) results about the extent of the spin-gap phase, which results larger than the variational but smaller than the RG one. Furthermore, we show that the spin gap can reach a sizable value (0.1t) at low enough filling, such that preformed pairs should be observable at temperatures below these energy scales. No evidence for a phase with clustering of more than two particles is found on approaching phase separation.
Experiments on quasi-one-dimensional systems such as quantum wires and metallic chains on surfaces suggest the existence of electron-electron interactions of substantial range and hence physics beyond the Hubbard model. We therefore investigate one-dimensional, quarter-filled chains with a Coulomb potential with variable screening length by quantum Monte Carlo methods and exact diagonalization. The Luttinger liquid interaction parameter ${K}_{\ensuremath{\rho}}$ decreases with increasing interaction strength and range. Experimentally observed values close to 1/4 require strong interactions and/or large screening lengths. As predicted by bosonization, we find a metal-insulator transition at ${K}_{\ensuremath{\rho}}=1/4$. Upon increasing the screening length, the charge and spin correlation functions reveal the crossover from dominant 2kF spin correlations to dominant 4kF charge correlations, and a strong enhancement of the charge velocity. In the metallic phase, the signatures of spin-charge separation in the single-particle spectrum, spinon and holon bands, remain robust even for rather long-ranged interactions. The charge-density-wave state exhibits backfolded shadow bands.
We study systematically the effect of long-range dipolar interactions on the ground-state phase diagram of the one-dimensional model by means of the density matrix renormalization group. While the basic phases described by the Luttinger parameter , namely, the repulsive Luttinger liquid (metallic phase, ), attractive Luttinger liquid (superconducting phase, ), and the phase separation , are similar to those of the conventional model, the presence of the long-range dipolar interactions leads to significant differences. i) At high-density regime, the phase boundaries of these three phases are even pushed to the large- J region; ii) at low-density regime, these phase boundaries shift toward the small- J region and, most importantly, iii) the spin-gap region spreads across the boundary of , suggesting an exotic metallic phase with spin-gap, which is absent in the conventional model. The result indicates that the long-range dipolar interactions have a significant influence on the ground-state properties of the one-dimensional model. Its implication on the pseudogap phenomenon of the hole-doped cuprates is briefly discussed.
We provide solid evidence for the long-standing presumption that model Hamiltonians with short-range interactions faithfully reproduce the physics of the long-range Coulomb interaction in real materials. For this aim, we address a generic Hubbard model that captures the quantum phase transitions between metal, Mott insulator, and charge-density-wave (CDW) insulator, in the absence of Fermi-surface nesting. By comparing the quantum phase diagrams for the 1 / r -Hubbard model on a half-filled chain with nearest-neighbor and 1 / r -long-range interactions, we argue that the inclusion of long-range interactions is not crucial for a proper description of interacting many-electron systems. To this end, we employ the density matrix renormalization group method on finite lattices and antiperiodic boundary conditions to determine the quantum phase transitions between the metallic Luttinger liquid for weak interactions, the Mott-Hubbard insulator for dominant on-site interactions, and the CDW insulator for dominant intersite interactions. The two phase diagrams qualitatively agree inasmuch as the quantum phase transitions are continuous in both cases. Moreover, simple Hartree-Fock theory and the atomic limit provide renormalization factors that allow us to quantitatively map the two phase diagrams onto each other. As a practical advantage, our findings imply that computational efforts can be reduced tremendously by using models with short-range interactions only.
Since the advent of high-Tc cuprate superconductors in 1986, strongly correlated electron systems have attracted much attention. Since the cuprates are essentially two-dimensional, low-dimensional systems have moved into the focus of condensed-matter theory. From a theoretical point of view, one-dimensional systems are of particular interest because there are exact numerical and analytical methods which permit detailed studies and deep insights into the many-body problem. In the first part of this thesis, using the numerically exact methods Exact Diagonalization and the Density-Matrix Renormalization Group (DMRG), we investigate the properties of the Tomonaga–Luttinger liquid which is the generic metallic state of matter in one dimension. In particular, we concentrate on the investigation of the so-called Tomonaga–Luttinger liquid parameter which determines the critical exponent α for the density of states near the Fermi energy. Experimental results for some quasi one-dimensional materials report α≳1, which would imply Kρ≲0.17, a value which cannot be reconciled with the bare Hubbard model where KρH≥0.5, i.e., αH≲1/8. We develop new accurate numerical methods to obtain Kρ and investigate how to obtain such small values for Kρ for slightly doped charge-density-wave insulators. In the second part of this thesis, using the Thermodynamic Bethe Ansatz (TBA) as exact analytical method, we investigate the one-dimensional Hubbard model in the spindisordered regime, which is characterized by the temperature being much larger than the magnetic energy scale but small compared to the Mott–Hubbard gap. Our study is motivated by the controversy about the Mott–Hubbard insulator in infinite dimensions whose ground state is also spin-disordered. In this system the determination of the precise value of the critical interaction strength Uc where the Mott–Hubbard gap closes is still unsolved. Therefore, we provide an example of a Hubbard-type model with a disordered spin background which can be solved exactly. The thermodynamics of our model can be understood in terms of gapped charged excitations with an effective dispersion which are decoupled form the spin degrees of freedom; the latter contribute only entropically. An interpretation of this regime in terms of a putative interacting-electron system at zero temperature leads to a metal-insulator transition at a finite interaction strength above which the gap opens linearly. Our exact results indicate that the strong-coupling expansion of the ground-state energy cannot be used to locate Uc. However, the strong-coupling expansion of the gap permits a reliable extrapolation of the critical interaction strength.