Continuous-time quantum Monte Carlo solver for dynamical mean field theory in the compact Legendre representation

We investigate the phenomenon of high-temperature superconductivity within a strong coupling perspective.The occurence is traced to a quantum critical point that is in the phase diagram of the plaquette's t, t -Hubbard model.We develop a bottom-up approach combining several methods, i.e. exact diagonalization of an isolated plaquette, the Lanczos-method for a plaquette within a bath and cluster dynamical Mean-Field theory with continuous time quantum Monte-Carlo solver to embedd the plaquette in a lattice environment.The quantum critical point is located where the N = 2, 3, 4-sectors of the plaquette cross.This point is also found to show optimal doping.The d x 2 -y 2 -wave order turns out to be largest at the localized-itinerant transition of the electrons.Furthermore, we present an explenation for the pseudo-gap phenomenon, that is explained by a soft mode related to local singlets of the plaquette.The theory presented here is similar to the resonating valence bond theory, but stresses the importance of local singlets.

Lanczos-based Low-Rank Correction Method for Solving the Dyson Equation in Inhomogenous Dynamical Mean-Field Theory

An acceleration of continuous time quantum Monte Carlo (CTQMC) methods is a potentially interesting branch of work as they are matchless as impurity solvers of a density functional theory in combination with a dynamical mean field theory approach for the description of electronic structures of strongly correlated materials. The inversion of the $k \times k$ matrix given by the diagram expansion order $k$ in the CTQMC update and the multiplication of the $k \times k$ matrix and the non-interacting Green's function to measure the impurity Green's function are computationally time-consuming. Here, we propose the CTQMC method in combination with a machine learning technique, which would eliminate the need for multiplication of the matrix with the non-interacting Green's function. This method predicts the accurate impurity Green's function and double occupancy at low temperature, and also considers the physical properties of high Matsubara frequency in a much shorter computational time than the conventional CTQMC method.

We studied several aspects of the Mott metal-insulator transition in the disordered case. The model on which we based our analysis is the disordered Hubbard model, which is the simplest model capable of capturing the Mott metal-insulator transition. We investigated this model through statistical dynamical mean-field theory (statDMFT). This theory is a natural extension of dynamical mean-field theory (DMFT), which has been used with relative success in the past several years with the purpose of describing the Mott transition in the clean case. As is the case for the latter theory, statDMFT incorporates the electronic correlation effects only in their local manifestations. Disorder, on the other hand, is treated in such a way as to incorporate Anderson localization effects. With this technique, we analyzed the disordered two-dimensional Mott transition, using the quantum Monte Carlo algorithm to solve the associated single-impurity problems. We found spinodal lines at which the metal and insulator ceased to be metastable. We also studied spatial fluctuations of local quantities, such as self-energy and local Green's function, and showed the appearance of metallic regions within the insulator and vice versa. We carried out an analysis of finite-size effects and showed that, in agreement with the theorems of Imry and Ma [Y. Imry and S. K. Ma, Phys. Rev. Lett. 35, 1399 (1975).], the first-order transition is smeared in the thermodynamic limit. We analyzed transport properties by means of a mapping to a random classical resistor network and calculated both the average current and its distribution across the metal-insulator transition.

Several recent works have introduced highly compact representations of single-particle Green's functions in the imaginary time and Matsubara frequency domains, as well as efficient interpolation grids used to recover the representations. In particular, the intermediate representation with sparse sampling and the discrete Lehmann representation (DLR) make use of low rank compression techniques to obtain optimal approximations with controllable accuracy. We consider the use of the DLR in dynamical mean-field theory (DMFT) calculations, and in particular show that the standard full Matsubara frequency grid can be replaced by the compact grid of DLR Matsubara frequency nodes. We test the performance of the method for a DMFT calculation of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ at temperature $50\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ using a continuous-time quantum Monte Carlo impurity solver, and demonstrate that Matsubara frequency quantities can be represented on a grid of only 36 nodes with no reduction in accuracy, or increase in the number of self-consistent iterations, despite the presence of significant Monte Carlo noise.

A new coarse-mesh computational method for the numerical solution of heat conduction and fluid flow problems is formally developed and applied to sample problems. The method is based upon formal use of Green's functions, which are defined locally over subdomains of the original system under consideration. The formal development of the local Green's function method for the solution of heat conduction problems is presented and discussed. Numerical solutions of sample problems for one-dimensional heat conduction with constant thermal conductivity, one-dimensional heat conduction with temperature-dependent thermal conductivity, and two-dimensional heat conduction with constant thermal conductivity are given, and these results are compared with results obtained using the finite difference and finite element methods. The formal development of the local Green's function method for the solution of fluid flow problems is then also presented and discussed; the numerical solution of a sample problem for simple one-dimensional incompressible fluid flow with viscous heating is also given.

formula omitted]QIST v0.7: An open source continuous-time quantum Monte Carlo impurity solver toolkit

We present a new charge self-consistent scheme combining Density Functional\nand Dynamical Mean Field Theory, which uses Green's function of multiple\nscattering-type. In this implementation the many-body effects are incorporated\ninto the Kohn-Sham iterative scheme without the need for the numerically\nill-posed analytic continuation of the Green's function and of the self-energy.\nThis is achieved by producing the Kohn-Sham Hamiltonian in the sub-space of\ncorrelated partial waves and allows to formulate the Green's function directly\non the Matsubara axis. The spectral moments of the Matsubara Green's function\nenable us to put together the real space charge density, therefore the charge\nself-consistency can be achieved. Our results for the spectral functions\n(density of states) and equation of state curves for transition metal elements,\nFe, Ni and FeAl compound agree very well with those of Hamiltonian based\nLDA+DMFT implementations. The current implementation improves on numerical\naccuracy, requires a minimal effort besides the multiple scattering formulation\nand can be generalized in several ways that are interesting for applications to\nreal materials.\n

In condensed matter physics, the idea of localized magnetic moments and the scattering of conduction electrons by those local moments are of central importance not only for explaining the transport of electrons at low temperature region but also for developing the related models and theories to extend our understanding of novel physics like non-conventional superconductor, quantum phase transition and non-Fermi liquid behavior. In the first chapter of this work, I will look back at the discovery of Kondo scattering, the introduction of Anderson impurity scenario, the extension to lattice version of the Anderson impurities, the periodic Anderson model, and how these ideas are used to the heavy fermion materials. In order to attack those problems and models numerically, in the second chapter, I will discuss the algorithm of dynamical mean-field theory, where interaction expansion continuous time quantum Monte Carlo is used as impurity solver. The introduction of dynamical mean-field theory and its related methods are one of the largest victory of computational condensed matter physics in the past three decades. By mapping the lattice problem to an impurity problem and solving it self-consistently, we can get the static and dynamic outputs in the thermal dynamical limit from dynamical mean-field theory simulation by solving a single site problem. In the infinite coordinate number limit, the dynamical mean-field theory results converge to the exact results. Quantum Monte Carlo is widely used in computational physics. Compared with the conventional quantum Monte Carlo, the recently introduced interaction expansion continuous time quantum Monte Carlo is free of decomposition error in imaginary time. In the third chapter, the effect of phonon coupling to the conduction band of the periodic Anderson model is discussed. In the periodic Anderson model at low temperature region, the local moments on f -band are screened by the conduction electrons and form the coherent Kondo singlet states. When the lattice oscillation degree of freedom is introduced to the conduction band, the conduction electrons bond with each other because of the retarded attractive interaction incurred by the phonons. As the electron-phonon interaction is large, the competition of electronic bonding on the c -band will compete with the Kondo singlet bonding between the conduction electrons and the local moments and finally lead to a Kondo collapse. In the fourth chapter, the pressure induced volume collapse problem of Cerium is explored. It has been discovered that Cerium will experience a 15% volume collapse from a

A Singular value decomposition (SVD)-enhanced Least-square fitting technique is discussed. By automatic identifying, ordering, and selecting dominant SVD modes of the derivative matrix that responds to the variations of the variables, the converging process of the Least Square fitting is significantly enhanced. Thus the fitting speed can be fast enough for a fairly large system. This technique has been successfully applied to precision PEP-II optics measurement in which we determine all quadrupole strengths (both normal and skew components) and sextupole feed-downs as well as all BPM gains and BPM cross-plane couplings through Least-Square fitting of the phase advances and the Local Green's functions as well as the coupling ellipses among BPMs. The local Green's functions are specified by 4 local transfer matrix components R12, R34, R32, R14. These measurable quantities (the Green's functions, the phase advances and the coupling ellipse tilt angles and axis ratios) are obtained by analyzing turn-by-turn Beam Position Monitor (BPM) data with a high-resolution model-independent analysis (MIA). Once all of the quadrupoles and sextupole feed-downs are determined, we obtain a computer virtual accelerator which matches the real accelerator in linear optics. Thus, beta functions, linear coupling parameters, and interaction point (IP) optics characteristics can be measured and displayed.

The dynamical mean field theory (DMFT), which is successful in the study of strongly correlated fermions, was recently extended to boson systems [Phys. Rev. B {\textbf 77}, 235106 (2008)]. In this paper, we employ the bosonic DMFT to study the Bose-Hubbard model which describes on-site interacting bosons in a lattice. Using exact diagonalization as the impurity solver, we get the DMFT solutions for the Green's function, the occupation density, as well as the condensate fraction on a Bethe lattice. Various phases are identified: the Mott insulator, the Bose-Einstein condensed (BEC) phase, and the normal phase. At finite temperatures, we obtain the crossover between the Mott-like regime and the normal phase, as well as the BEC-to-normal phase transition. Phase diagrams on the $\mu/U-\tilde{t}/U$ plane and on the $T/U-\tilde{t}/U$ plane are produced ($\tilde{t}$ is the scaled hopping amplitude). We compare our results with the previous ones, and discuss the implication of these results to experiments.

An exact real-space renormalization-group approach is developed to calculate the local Green's function and the local density of states at any site in an infinite Fibonacci chain, in which three basic renormalization transformations ${\mathit{T}}_{\mathrm{\ensuremath{\alpha}}}$, ${\mathit{T}}_{\mathrm{\ensuremath{\beta}}}$, and ${\mathit{T}}_{\ensuremath{\gamma}}$ are introduced. Transformations ${\mathit{T}}_{\mathrm{\ensuremath{\beta}}}$ and ${\mathit{T}}_{\ensuremath{\gamma}}$ divide the Fibonacci chains of different generations into two distinct subclasses with key sites ${\mathit{S}}_{\mathrm{\ensuremath{\beta}}}$ and ${\mathit{S}}_{\ensuremath{\gamma}}$, respectively. The local Green's function and the local density of states at the key site ${\mathit{S}}_{\mathrm{\ensuremath{\beta}}}$ or ${\mathit{S}}_{\ensuremath{\gamma}}$ can be obtained by successive iterations of the transformation ${\mathit{T}}_{\mathrm{\ensuremath{\beta}}}$ or ${\mathit{T}}_{\ensuremath{\gamma}}$. Any other site can be transferred to the key site of a renormalized Fibonacci chain by suitable combinations of the transformations ${\mathit{T}}_{\mathrm{\ensuremath{\alpha}}}$, ${\mathit{T}}_{\mathrm{\ensuremath{\beta}}}$, and ${\mathit{T}}_{\ensuremath{\gamma}}$. In this approach, the relevant scale concerning the key site is ${\mathrm{\ensuremath{\tau}}}^{2}$, where \ensuremath{\tau} is the golden mean.

We present a strategy to alleviate the sign problem in continuous-time quantum Monte Carlo (CTQMC) simulations of the dynamical-mean-field-theory (DMFT) equations for the spin-orbit-coupled multiorbital Hubbard model. We first identify the combinations of rotationally invariant Hund coupling terms present in the relativistic basis which lead to a severe sign problem. Exploiting the fact that the average sign in CTQMC depends on the choice of single-particle basis, we propose a bonding-antibonding basis $V_{j3/2\mathrm{BA}}$ which shows an improved average sign compared to the widely used relativistic basis for most parameter sets investigated. We then generalize this procedure by introducing a stochastic optimization algorithm that exploits the space of single-particle bases and show that $V_{j3/2\mathrm{BA}}$ is very close to optimal within the parameter space investigated. Our findings enable more efficient DMFT simulations of materials with strong spin-orbit coupling.

We develop a reliable parameter-free analytic continuation method for quantum\nmany-body calculations. Our method is based on a kernel grid, a causal spline,\na regularization using the second-derivative roughness penalty, and the L-curve\ncriterion. We also develop the L-curve averaged deviation to estimate the\nprecision of our analytic continuation. To deal with statistically obtained\ndata more efficiently, we further develop a bootstrap-averaged analytic\ncontinuation method. In the test using the exact imaginary-frequency Green's\nfunction with added statistical error, our method produces the spectral\nfunction that converges systematically to the exact one as the statistical\nerror decreases. As an application, we simulate the two-orbital Hubbard model\nfor various electron numbers with the dynamical-mean field theory in the\nimaginary time and obtain the real-frequency self-energy with our analytic\ncontinuation method, clearly identifying a non-Fermi liquid behavior as the\nelectron number approaches the half filling from the quarter filling. Our\nanalytic continuation can be used widely and it will facilitate drawing clear\nconclusions from imaginary-time quantum many-body calculations.\n

Efficient continuous time quantum Monte Carlo (CT-QMC) algorithms that do not suffer from time discretization errors have become the state-of-the-art for most discrete quantum models. They have not been widely used yet for fermionic quantum lattice models, such as the Hubbard model, due to a suboptimal scaling of $O(\beta^3)$ with inverse temperature $\beta$, compared to the linear scaling of discrete time algorithms. Here we present a CT-QMC algorithms for fermionic lattice models that matches the scaling of discrete-time methods but is more efficient and free of time discretization errors. This provides an efficient simulation scheme that is free from the systematic errors opening an avenue to more precise studies of large systems at low temperatures.