Abstract In this work for an interacting electrons system that obeys the Hubbard model, a new quantum Monte Carlo algorithm introduces for calculation of average Green function. This algorithm is applied to investigate the effects of electrons onsite Coulomb repulsion on the band structure of a square lattice in both single-site approximations such as dynamical mean field theory (DMFT) and multi-site approximations such as effective medium supercell approximation (EMSCA). The advantages of our algorithm in comparison to the Hirsch-Fye algorithm and also the Blankenbecler, Scalapino, and Sugar (BSS) algorithm are the elimination of instabilities resulting from the Metropolis algorithm in the accepting and rejecting configurations, stability at low temperatures, the elimination of systematic errors resulting from the update of the Green's function in the quantum Monte Carlo process, and considering different probabilities for each possible configuration. Finally, by using our algorithm, it is possible to calculate the interacting three-dimensional system's band structure and the density of states that obey the Hubbard model. We have applied our algorithm to an interactive two-dimensional square lattice. As a result, phase transition boundaries can be easily recognized through calculated bands for different repulsions. Our results show that critical Coulomb repulsion values for Mott transition are uc=9.05t and uc=2.4t for DMFT and BEMSCA respectively. This means that DMFT significantly overestimates band splitting due to electrons' Coulomb repulsion. We found by starting at low repulsions and then increasing electrons' Coulomb repulsion, a partially flatted valance band around the center of the first Brillouin zone appears, but disappears at high repulsions.
Abstract In this work for an interacting electrons system that obeys the Hubbard model, a new quantum Monte Carlo algorithm introduces for calculation of average Green function. This algorithm is applied to investigate the effects of electrons onsite Coulomb repulsion on the band structure of a square lattice in both single-site approximations such as dynamical mean field theory (DMFT) and multi-site approximations such as effective medium supercell approximation (EMSCA). The advantages of our algorithm in comparison to the Hirsch-Fye algorithm and also the Blankenbecler, Scalapino, and Sugar (BSS) algorithm are the elimination of instabilities resulting from the Metropolis algorithm in the accepting and rejecting configurations, stability at low temperatures, the elimination of systematic errors resulting from the update of the Green's function in the quantum Monte Carlo process, and considering different probabilities for each possible configuration. Finally, by using our algorithm, it is possible to calculate the interacting three-dimensional system's band structure and the density of states that obey the Hubbard model. We have applied our algorithm to an interactive two-dimensional square lattice. As a result, phase transition boundaries can be easily recognized through calculated bands for different repulsions. Our results show that critical Coulomb repulsion values for Mott transition are uc=9.05t and uc=2.4t for DMFT and BEMSCA respectively. This means that DMFT significantly overestimates band splitting due to electrons' Coulomb repulsion. We found by starting at low repulsions and then increasing electrons' Coulomb repulsion, a partially flatted valance band around the center of the first Brillouin zone appears, but disappears at high repulsions.
The Quantum Monte Carlo simulations of the ionic Hubbard model on a two-dimensional square lattice at half filling were performed. The method based on the direct-space, proposed by Suzuki and al., Hirsch and al., was used. Cycles of increasing and decreasing values of the Coulomb interaction $ U $ were performed for fixed temperature ($ kT=0.01 $). Results indicate that, at low temperature, the two insulator phases are separated by a metallic phase for weak to intermediate values of the staggered potential $ Δ$. For large Coulomb repulsion the system is in a Mott insulator with an antiferromagnetism order. On increasing and decreasing the Coulomb interaction $ U $ the metal-Mott insulator transition shows an hysteresis phenomenon while the metal-band insulator transition is continue. For large $ Δ$ it seems that the metallic region shrinks to a single metallic point. However, the band insulator to the Mott insulator transition is not direct for the studied model. A phase diagram is drawn for the temperature $ kT=0.01 $. For $ Δ=0.5 $ cycles of increasing and decreasing temperature were programmed for different values of the Coulomb interaction $ U $ . A behaviour change appears for $ U\simeq 1.75 $. This suggests that a crossover line divides the metallic region of the phase diagram.
We study the Hubbard model using the cellular dynamical mean-field theory (CDMFT) with quantum Monte Carlo (QMC) simulations. We present the algorithmic details of CDMFT with the Hirsch-Fye QMC method for the solution of the self-consistently embedded quantum cluster problem. We use the one- and two-dimensional half filled Hubbard model to gauge the performance of CDMFT+QMC particularly for small clusters by comparing with the exact results and also with other quantum cluster methods. We calculate single-particle Green's functions and self-energies on small clusters to study their size dependence in one and two dimensions. It is shown that in one dimension, CDMFT with two sites in the cluster is already able to describe with high accuracy the evolution of the density as a function of the chemical potential and the compressibility divergence at the Mott transition, in good agreement with the exact Bethe ansatz result. With increasing U the result on small clusters rapidly approaches that of the infinite size cluster. Large scattering rate and a positive slope in the real part of the self-energy in one dimension suggest that the system is a non-Fermi liquid for all the parameters studied here. In two dimensions, at intermediate to strong coupling, even the smallest cluster $({N}_{c}=2\ifmmode\times\else\texttimes\fi{}2)$ accounts for more than 95% of the correlation effect of the infinite-size cluster in the single particle spectrum, suggesting that some of the important problems in strongly correlated electron systems may be studied highly accurately with a reasonable computational effort. Finally, as an application that is sensitive to details of correlations, we show that CDMFT+QMC can describe spin-charge separated Luttinger liquid physics in one dimension. The spinon and holon branches appear only for sufficiently large system sizes.
In this thesis we consider three different models for strongly correlated electrons, namely a multi-band Hubbard model as well as the spinless Falicov-Kimball model, both with a semi-elliptical density of states in the limit of infinite dimensions d, and the attractive Hubbard model on a square lattice in d=2.\n\nIn the first part, we study a two-band Hubbard model with unequal bandwidths and anisotropic Hund's rule coupling (J_z-model) in the limit of infinite dimensions within the dynamical mean-field theory (DMFT). Here, the DMFT impurity problem is solved with the use of quantum Monte Carlo (QMC) simulations. Our main result is that the J_z-model describes the occurrence of an orbital-selective Mott transition (OSMT), in contrast to earlier findings. We investigate the model with a high-precision DMFT \nalgorithm, which was developed as part of this thesis and which supplements QMC with a high-frequency expansion of the self-energy.\nThe main advantage of this scheme is the extraordinary accuracy of the numerical solutions, which can be obtained already with moderate computational effort, so that studies of multi-orbital systems within the DMFT+QMC are strongly improved. We also found that a suitably defined\nFalicov-Kimball (FK) model exhibits an OSMT, revealing the close connection of the Falicov-Kimball physics to the J_z-model in the OSM phase.\n\nIn the second part of this thesis we study the attractive Hubbard model in two spatial dimensions within second-order self-consistent perturbation theory.\nThis model is considered on a square lattice at finite doping and at low temperatures. Our main result is that the predictions of first-order perturbation theory (Hartree-Fock approximation) are renormalized by a factor of the order of unity even at arbitrarily weak interaction (U->0). The \nrenormalization factor q can be evaluated as a function of the filling n for 0<n<1. In the limit n->0, the q-factor vanishes, signaling the divergence of self-consistent perturbation theory in this limit. Thus we present the first asymptotically exact results at weak-coupling for the negative-U Hubbard model in d=2 at finite doping.
This thesis will describe efforts to enhance our ability to simulate the 2D Hubbard model. Chapter 2 provides a background on the main computational techniques used throughout the work, including quantum Monte Carlo (QMC), dynamical mean field theory (DMFT), the dynamical cluster approximation (DCA), the continuous time auxiliary field algorithm (CTAUX), and the Maximum Entropy Method (MEM) for numeric analytic continuation. Chapter 3 presents new work on applying Twisted Boundary Conditions to the DCA framework. This method is applied in a effort to access thermodynamic (i.e. large system) information about the Hubbard model without the immense computational expense required to simulate large lattices directly. Chapter 4 describes efforts to study the extended 2D Hubbard model, which re-introduces non-local interactions between electrons that drive the formation of charge ordered phases. The work includes a thorough analysis of the phase diagram of the model away from half-filling, as well as analysis of the effect of non-local interactions on anti-ferromagnetic fluctuations and competition between the charge order and AFM states. Chapter 5 includes a derivation of the dual fermions diagrammatic expansion and the dual fermions ladder approximation used to compute corrections to single site DMFT calculations. The method is applied to the 2D Hubbard model in order to study the evolution of its spectral function, Fermi surface, and momentum dependent mass renormalization as the temperature, interaction strength, and doping are changed. The great advantage of this method is that we are able to access arbitrary momenta throughout the Brillouin zone, in contrast to the limitation to a few cluster momenta that characterize cluster methods like DCA.
Strong electronic correlations are the root of many fascinating phenomena of condensed matter physics. However, they also make calculations notoriously difficult. This thesis is concerned with the development and improvement of computational methods for strongly correlated electron systems. After a brief recapitulation of the Green’s function formalism,which is the theoretical basis of most such methods, we discuss the analytical continuation from imaginary to real frequencies. Especially the maximum entropy method is explained in detail and an implementation there of is presented.The remainder of this thesis deals with methods to compute Green’s functions on thei maginary frequency axis. First, the Anderson impurity model (AIM) is considered. It can be solved numerically – a task for which several methods exist. We focus on continuous-time quantum Monte Carlo with worm sampling, through which various kinds of correlation functions can be calculated. On this basis we derive and implement symmetric improved estimators that allow for the computation of the self-energy and vertex functions where the level of stochastic noise is largely reduced with respect to conventional estimators.As a next step the dynamical mean-field theory (DMFT) is introduced as a method to capture local correlations in the Hubbard model (HM) by self-consistently mapping it onto the AIM. We study the convergence behavior of this iterative procedure and the improvement of convergence due to mixing. Then, we compare electron scattering rates from DMFT in the weak-coupling regime of the square-lattice HM to results from Boltzmann scattering theory.While local correlations account for many interesting phenomena, such as the Mott-Hubbard metal-insulator transition, they are not sufficient to correctly describe, e.g., the physics of phase transitions in two- or three-dimensional systems. It is thus the final step in this thesis to also include nonlocal correlations by means of the dynamical vertex approximation (DΓA). For this we develop a new self-consistency scheme, which is then applied to several systems: We study the square-lattice HM, where we compare to benchmark methods in the weak-coupling regime, but also present results for intermediate and strong coupling. Furthermore, a two-band model with doping and a three-band Wannier model for strontium vanadate are studied. As an application of our new self-consistent DΓA, we study the effects of geometrical frustration in the kagome lattice, where we also compare our results to determinant quantum Monte Carlo.
We study an ionic Hubbard model in one dimension by using a quantum Monte Carlo method. The ionic Hubbard model we study is a strongly correlated model for electrons in alternating potentials with a lattice period of 2. By modifying the Hirsch-Fye algorithm, we incorporate the local variation of the potential in the weight of the partition function at each imaginary-time segment. For a given alternating potential of strength Δ = 1.0, as we change the local Coulomb repulsion, U , between spin-up and spin-down electrons, we calculate various physical quantities, including kinetic energies, correlation energies, and band energies. We try to find the signature of the phase transitions from the behavior of the kinetic energy and that of double occupancies at finite temperatures.