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arXiv:1402.7087 · Physical Review B

Ergodicity of the hybridization-expansion Monte Carlo algorithm for broken-symmetry states

P. Sémon, G. Sordi, A.–M. S. Tremblay

Published Apr 11, 2014• 22 citations
arXivHTMLPDF

Abstract

With the success of dynamical mean-field theories, solvers for quantum-impurity problems have become an important tool for the numerical study of strongly correlated systems. Continuous-time quantum Monte Carlo sampling of the expansion in powers of the hybridization between the ``impurity'' and the bath provides a powerful solver when interactions are strong. Here we show that the usual updates that add or remove a pair of creation-annihilation operators are rigorously not ergodic for several classes of broken symmetries that involve spatial components. We show that updates with larger numbers of simultaneous updates of pairs of creation-annihilation operators remedy this problem. As an example, we apply the four-operator updates that are necessary for ergodicity to the case of $d$-wave superconductivity in plaquette dynamical mean-field theory for the one-band Hubbard model. While the results are qualitatively similar to those previously published, they are quantitatively better than previous ones, being closer to those obtained by other approaches.

DOI
10.1103/physrevb.89.165113

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With the success of dynamical mean-field theories, solvers for quantum-impurity problems have become an important tool for the numerical study of strongly correlated systems. Continuous-time quantum Monte Carlo sampling of the expansion in powers of the hybridization between the ``impurity'' and the bath provides a powerful solver when interactions are strong. Here we show that the usual updates that add or remove a pair of creation-annihilation operators are rigorously not ergodic for several classes of broken symmetries that involve spatial components. We show that updates with larger numbers of simultaneous updates of pairs of creation-annihilation operators remedy this problem. As an example, we apply the four-operator updates that are necessary for ergodicity to the case of ddd-wave superconductivity in plaquette dynamical mean-field theory for the one-band Hubbard model. While the results are qualitatively similar to those previously published, they are quantitatively better than previous ones, being closer to those obtained by other approaches.
We present a new continuous-time solver for quantum impurity models such as those relevant to dynamical mean field theory. It is based on a stochastic sampling of a perturbation expansion in the impurity-bath hybridization parameter. Comparisons with Monte Carlo and exact diagonalization calculations confirm the accuracy of the new approach, which allows very efficient simulations even at low temperatures and for strong interactions. As examples of the power of the method we present results for the temperature dependence of the kinetic energy and the free energy, enabling an accurate location of the temperature-driven metal-insulator transition.
Abstract: Originally designed for the study of strong electronic correlations in model Hamiltonians, dynamical mean field theory (DMFT) has become, in combination with density functional theory (DFT), a powerful tool for ab initio simulations of real materials. At the heart of DMFT lies the solution of a quantum impurity problem. While only the continuous-time quantum Monte Carlo (CT-QMC) impurity solvers yield (statistically) exact solutions of a general impurity problem, they are quite complex and computationally expensive. Hence, in this thesis we are interested in improving the CT-QMC impurity solvers. After a short introduction to DMFT and its cluster extensions, we begin by reviewing two of the CT-QMC impurity solvers, the interaction expansion or “Rubtsov” solver (CT- INT) and the hybridization expansion solver (CT-HYB). Focussing on the latter, which is the algorithm of choice within real material simulations, we then show how to reduce a sign problem, allowing us to address the unusual criticality found in layered organic superconductors. With high-T c superconductivity as example, we further discuss how to ensure ergodicity of the CT-HYB solver in the context of broken symmetries. Finally, algorithmic optimizations of CT-HYB are presented and combined, leading to speedups of up to 500 within the context of real material simulations. // Résumé: Initialement conçue pour traiter les fortes corrélations électroniques dans des hamiltoniens modèles, la théorie du champ moyen dynamique (DMFT) est devenue, en combinaison avec la théorie de la densité fonctionnelle (DFT), un outil puissant pour la simulation de matériaux réels. Au cœur de la DMFT se trouve la solution d'un modèle d'impureté quantique. Seulement les solutionneurs d'impureté Monte Carlo en temps continu (CT-QMC) donnent des solutions exactes. En même temps, ces solutionneurs sont plutôt complexes et gourmands en temps de calcul. Le but de cette thèse est donc d'améliorer les solutionneurs d'impureté CT-QMC. Après une courte introduction à la DMFT et à ses extensions pour les amas, on commence par une revue de deux des solutionneurs CT-QMC, celui en développement d'interaction ou de "Rubtsov" (CT-INT) et celui en développement d'hybridation (CT-HYB). Mettant l'accent sur le dernier, qui est l'algorithme de choix dans le cadre des matériaux réels, on montre alors comment réduire un problème de signe, nous permettant ainsi de traiter la criticalité inhabituelle des organiques en couche. Avec la supraconductivité à haute température critique comme exemple, on discute ensuite comment assurer l'ergodicité du solutionneur CT-HYB dans le cadre des symétries brisées. Finalement, des optimisations algorithmiques sont présentées et combinées, amenant à des accélérations allant jusqu'à un facteur de 500 dans le contexte des matériaux réels.
We develop a fast impurity solver which is based on the combination of Hubbard-I approximation and hybridization expansion continuous-time quantum Monte Carlo algorithm. This solver inherits the advantages of both algorithms. In order to demonstrate the power and usefulness of this solver, we use it to study the magnetic phase transitions of single-band and two-band Hubbard models in the framework of single-site dynamical mean-field theory. The calculated results agree well with those obtained by hybridization expansion quantum impurity solver. It is suggested that this solver is very suitable to solve the multi-orbital quantum impurity models efficiently and can be used to study more realistic systems with magnetic long-range order in the future.
Due to the intrinsic complexity of the quantum many-body problem, quantum Monte Carlo algorithms and their corresponding Monte Carlo configurations can be defined in various ways. Configurations corresponding to few Feynman diagrams often lead to severe sign problems. On the other hand, computing the configuration weight becomes numerically expensive in the opposite limit in which many diagrams are grouped together. Here we show that for continuous-time quantum Monte Carlo in the hybridization expansion the efficiency can be substantially improved by dividing the local impurity trace into fragments, which are then sampled individually. For this technique, which also turns out to preserve the fermionic sign, a modified update strategy is introduced in order to ensure ergodicity. Our (super)state sampling is particularly beneficial to calculations with many ddd orbitals and general local interactions, such as full Coulomb interaction. For illustration, we reconsider the simple albeit well-known case of a degenerate three-orbital model at low temperatures. This allows us to quantify the coherence properties of the ``spin-freezing'' crossover, even close to the Mott transition.
We investigate the sampling efficiency in a segment-based continuous-time hybridization-expansion quantum Monte Carlo solver within dynamical mean-field theory for the doped SU(3) Fermi-Hubbard model. Focusing on the low-temperature and strong-coupling regime, where a ferromagnetic state appears upon hole doping away from one-third filling, we compare several update schemes. Adding the double-flip update to the basic local update yields the smallest integrated autocorrelation time of the majority-flavor occupation in the ferromagnetic regime, whereas adding the flavor-permutation update is the most effective in the paramagnetic regime. We further clarify the origin of this difference by analyzing the characteristic occupation changes caused by accepted updates.
The development of polynomial cost solvers for correlated quantum impurity models, with controllable errors, is a central challenge in quantum many-body physics, where these models find applications ranging from nanoscience to the dynamical mean-field theory (DMFT). Here, we describe how configuration interaction (CI) approximations to exact diagonalization (ED) may be used as solvers in DMFT. CI approximations retain the main advantages of ED, such as the ability to treat general interactions and off-diagonal hybridizations and to obtain real spectral information, but are of polynomial cost. Furthermore, their errors can be controlled by monitoring the convergence of physical quantities as a function of the CI hierarchy. Using benchmark DMFT applications, such as single-site DMFT of the one-dimensional Hubbard model and $2\ifmmode\times\else\texttimes\fi{}2$ cluster DMFT of the two-dimensional Hubbard model, we show that CI approximations allow us to obtain near-exact ED results for a tiny fraction of the cost. This is true over the entire range of interaction strengths including ``difficult'' regimes, such as in the pseudogap phase of the two-dimensional Hubbard model. We use the ability of CI approximations to treat large numbers of orbitals to demonstrate convergence of the bath representation in the $2\ifmmode\times\else\texttimes\fi{}2$ cluster DMFT using a 24-bath orbital representation. CI approximations thus form a promising route to extend ED to problems that are currently difficult to study using other solvers such as continuous-time quantum Monte Carlo, including impurity models with large numbers of orbitals and general interactions.