Iqist an Open Source Continuous time Quantum Monte Carlo Impurity Solver Toolkit

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$i\mathrm{QIST}$ : An open source continuous-time quantum Monte Carlo impurity solver toolkit☆

Abstract

Quantum impurity solvers have a broad range of applications in theoretical studies of strongly correlated electron systems. Especially, they play a key role in dynamical mean-field theory calculations of correlated lattice models and realistic materials. Therefore, the development and implementation of efficient quantum impurity solvers is an important task. In this paper, we present an open source interacting quantum impurity solver toolkit (dubbed $i\mathrm{QIST}$ ). This package contains several highly optimized quantum impurity solvers which are based on the hybridization expansion continuous-time quantum Monte Carlo algorithm, as well as some essential pre- and post-processing tools. We first introduce the basic principle of continuous-time quantum Monte Carlo algorithm and then discuss the implementation details and optimization strategies. The software framework, major features, and installation procedure for $i\mathrm{QIST}$ are also explained. Finally, several simple tutorials are presented in order to demonstrate the usage and power of $i\mathrm{QIST}$ .

Program summary

Program title: $i\mathrm{QIST}$

Catalogue identifier: AEWQ_v1_0

Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEWQ_v1_0.html

Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland

Licensing provisions: GNU General Public License, version 3

No. of lines in distributed program, including test data, etc.: 226270

No. of bytes in distributed program, including test data, etc.: 5263144

Distribution format: tar.gz

Programming language: Fortran 2008 and Python.

Computer: Desktop PC, laptop, high performance computing cluster.

Operating system: Unix, Linux, Mac OS X, Windows.

Has the code been vectorized or parallelized?: Yes, it is parallelized by MPI and OpenMP

RAM: Depends on the complexity of the problem

Classification: 7.3.

External routines: BLAS, LAPACK, Latex is required to build the user manual.

Nature of problem:

Quantum impurity models were originally proposed to describe magnetic impurities in metallic hosts. In these models, the Coulomb interaction acts between electrons occupying the orbitals of the impurity atom. Electrons can hop between the impurity and the host, and in an action formulation, this hopping is described by a time-dependent hybridization function. Nowadays quantum impurity models have a broad range of applications, from the description of heavy fermion systems, and Kondo insulators, to quantum dots in nano-science. They also play an important role as auxiliary problems in dynamical mean-field theory and its diagrammatic extensions [1–3], where an interacting lattice model is mapped onto a quantum impurity model in a self-consistent manner. Thus, the accurate and efficient solution of quantum impurity models becomes an essential task.

Solution method:

The quantum impurity model can be solved by the numerically exact continuous-time quantum Monte Carlo method, which is the most efficient and powerful impurity solver for finite temperature simulations. In the $i\mathrm{QIST}$ software package, we implemented the hybridization expansion version of continuous-time quantum Monte Carlo algorithm. Both the segment representation and general matrix formalism are supported. The key idea of this algorithm is to expand the partition function diagrammatically in powers of the impurity-bath hybridization, and to stochastically sample these diagrams to all relevant orders using the Metropolis Monte Carlo algorithm. For a detailed review of the continuous-time quantum Monte Carlo algorithms, please refer to [4].

Running time:

Depends on the complexity of the problem. The sample run supplied in the distribution takes about 1.5 minutes.

References:

[1] A. Georges, G. Kotliar, W. Krauth, and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996)

[2] G. Kotliar, S. Y. Savrasov, K. Haule, V. S. Oudovenko, O. Parcollet, and C. A. Marianetti, Rev. Mod. Phys. 78, 865 (2006)

[3] T. Maier, M. Jarrell, T. Pruschke, and M. H. Hettler, Rev. Mod. Phys. 77, 1027 (2005)

[4] E. Gull, A. J. Millis, A. I. Lichtenstein, A. N. Rubtsov, M. Troyer, and Philipp Werner, Rev. Mod. Phys. 83, 349 (2011)

Introduction

In this paper we present $i\mathrm{QIST}$ (abbreviation for 'interacting quantum impurity solver toolkit'), an open source project for recently developed hybridization expansion continuous-time quantum Monte Carlo impurity solvers  [1] and corresponding pre- and post-processing tools.

Dynamical mean-field theory (DMFT)  [2], [3] and its cluster extensions  [4] play an important role in contemporary studies of correlated electron systems. The broad applications of this technique range from the study of Mott–Hubbard metal–insulator transitions  [5], unconventional superconductivity in Cu- and Fe-based superconductors  [6], [7], [8], [9], and non-Fermi liquid behaviors in multi-orbital systems  [10], [11], [12], [13], to the investigation of anomalous transport properties of transition metal oxides  [14]. For many of these applications, DMFT is the currently most powerful and reliable (sometimes the only) technique available and has in many cases produced new physical insights. Furthermore, the combination of ab initio calculation methods (such as density function theory) with DMFT  [3] allows to capture the subtle electronic properties of realistic correlated materials, including those of partially filled $3d$ - and $4d$ -electron transition metal oxides, where lattice, spin and orbital degrees of freedom are coupled  [14].

The key idea of DMFT is to map the original correlated lattice model onto a quantum impurity model whose mean-field bath is determined self-consistently  [2], [3], [4]. Thus, the central task of a DMFT simulation becomes the numerical solution of a quantum impurity problem. During the past several decades, many methods have been developed and tested as impurity solvers, including the exact diagonalization (ED)  [15], equation of motion (EOM)  [16], Hubbard-I approximation (HIA)  [17], iterative perturbation theory (IPT)  [18], non-crossing approximation (NCA)  [19], fluctuation-exchange approximation (FLEX)  [20], and quantum Monte Carlo (QMC)  [21], [22], etc. Among the methods listed above, the QMC method has several very important advantages, which makes it so far the most flexible and widely used impurity solver. First, it is based on the imaginary time action, in which the infinite bath has been integrated out. Second, it can treat arbitrary couplings, and can thus be applied to all kinds of phases including the metallic phase, insulating state, and phases with spontaneous symmetry breaking. Third, the QMC method is numerically exact with a "controlled" numerical error. In other words, by increasing the computational effort the numerical error of the QMC simulation can be systematically reduced. For these reasons, the QMC algorithm is considered as the method of choice for many applications.

Several QMC impurity solvers have been developed in the past three decades. An important innovation was the Hirsch–Fye QMC (HF-QMC) impurity solver  [21], [22], in which the time axis is divided into small time steps and the interaction term in the Hamiltonian is decoupled on each time step by means of a discrete Hubbard–Stratonovich auxiliary field. HF-QMC has been widely used in the DMFT context  [2], [3], [4], but is limited by the discretization on the time axis and also by the form of the electronic interactions (usually only density–density interactions can be efficiently treated). Recently, a new class of more powerful and versatile QMC impurity solvers, continuous-time quantum Monte Carlo (CT-QMC) algorithms, have been invented  [1], [23], [24], [25], [26], [27]. In the CT-QMC impurity solvers, the partition function of the quantum impurity problem is diagrammatically expanded, and then the diagrammatic expansion series is evaluated by stochastic Monte Carlo sampling. The continuous-time nature of the algorithm means that operators can be placed at any arbitrary position on the imaginary time interval, so that time discretization errors can be completely avoided. Depending on how the diagrammatic expansion is performed, the CT-QMC approach can be further divided into interaction expansion (or weak coupling) CT-QMC (CT-INT)  [23], auxiliary field CT-QMC (CT-AUX)  [24], and hybridization expansion (or strong coupling) CT-QMC (CT-HYB)  [25], [26], [27].

At present, CT-HYB is the most popular and powerful impurity solver, since it can be used to solve multi-orbital impurity models with general interactions at low temperature  [1]. In single-site DMFT calculations, the computational efficiency of CT-HYB is much higher than that of CT-INT, CT-AUX, and HF-QMC, especially when the interactions are intermediate or strong. However, in order to solve more complicated quantum impurity models (for example, five-band or seven-band impurity model with general interactions and spin–orbital coupling) efficiently, further improvements of the CT-HYB impurity solvers are needed. In recent years many tricks and optimizations have been explored and implemented to increase the efficiency and accuracy of the original CT-HYB algorithm, such as the truncation approximation  [27], Krylov subspace iteration  [28], orthogonal polynomial representation  [29], [30], [31], PS quantum number  [32], lazy trace evaluation  [33], skip-list technique  [33], matrix product state implementation  [34], and sliding window sampling scheme  [34]. As the state-of-the-art CT-HYB impurity solvers become more and more sophisticated and specialized, it is not easy anymore to master all their facets and build one's implementations from scratch. Hence, we believe that it is a good time to provide a CT-HYB software package for the DMFT community such that researchers can focus more on the physics problems, instead of spending much time on (re-)implementing in-house codes. In fact, there are some valuable efforts in this direction, such as TRIQS  [35], ALPS  [36], [37], W2DYNAMICS  [32], and DMFT_W2K [27], [38]. The present implementation of the CT-HYB impurity solvers is a useful complement to the existing codes. The open source $i\mathrm{QIST}$ software package contains several well-implemented and thoroughly tested modern CT-HYB impurity solvers, and the corresponding pre- and post-processing tools. We hope the release of $i\mathrm{QIST}$ can promote the quick development of this research field.

The rest of this paper is organized as follows: In Section  2, the basic theory of quantum impurity models, CT-QMC algorithms, and its hybridization expansion version are briefly introduced. The measurements of several important physical observables are presented. In Section  3, the implementation details of $i\mathrm{QIST}$ are discussed. Most of the optimization tricks and strategies implemented in $i\mathrm{QIST}$ , including dynamical truncation, lazy trace evaluation, sparse matrix technique, PS quantum number, and subspace algorithms, etc., are reviewed. These methods ensure the high efficiency of $i\mathrm{QIST}$ . In Section  4, we first present an overview on the software architecture and component framework. Then the main features of the $i\mathrm{QIST}$ software package, including the CT-HYB impurity solvers, the atomic eigenvalue solver, and the other auxiliary tools are presented. The compiling and installation procedures, and the basic usage of $i\mathrm{QIST}$ are introduced in Section  5. Section  6 shows several simple applications of $i\mathrm{QIST}$ , ranging from self-consistent single-site DMFT calculation to one-shot post-processing calculation. These examples serve as introductory tutorials. Finally, a short summary is given in Section  7 and the future development plans for the $i\mathrm{QIST}$ project are outlined as well.

Section snippets

Basic theory and methods

In this section, we will present the basic principles of CT-QMC impurity solvers, with an emphasis on the hybridization expansion technique. For detailed derivations and explanations, please refer to Ref.  [1].

Implementations and optimizations

In this section, we will focus on the implementation details and discuss the optimization tricks adopted in the $i\mathrm{QIST}$ software package.

Features

In this section, we will introduce the software architecture and component framework of $i\mathrm{QIST}$ . The major features of its components are presented in detail.

Installation and usage

In this section, we will explain how to install and use the $i\mathrm{QIST}$ software package.

Examples

In the last few years, the $i\mathrm{QIST}$ software package has been successfully used in many projects, such as the study of the pressure-driven orbital-selective Mott metal–insulator transition in cubic CoO  [53], the metal–insulator transition in a three-band Hubbard model with or without SOC  [54], [55], the non-Fermi-liquid behavior in cubic phase BaRuO₃ [56], dynamical screening effects in the electronic structure of the strongly correlated metal SrVO₃ and local two-particle vertex functions  [57]

Future developments

In this paper, we explained and demonstrated the $i\mathrm{QIST}$ software package. $i\mathrm{QIST}$ aims to provide a complete toolkit for solving various quantum impurity systems. At first, we introduced the basic theory about quantum impurity models and the CT-QMC/CT-HYB algorithm briefly. Then various optimization tricks and algorithms implemented in $i\mathrm{QIST}$ have been discussed in detail. Following that we reviewed the software architecture and major features of $i\mathrm{QIST}$ . The compiling, setup, and workflow of $i\mathrm{QIST}$

Acknowledgments

YLW, LD and XD are supported by the National Science Foundation of China and the 973 program of China (No. 2011CBA00108). Their calculations were preformed on TianHe-1A, the National Supercomputer Center in Tianjin, China. ZYM thanks the inspiring guidance from H-Y. Kee and Y. B. Kim for bringing his attention to multi-orbital physics, he acknowledges the NSERC, CIFAR, and Centre for Quantum Materials at the University of Toronto, and the National Thousand-Young-Talents Program of China. His

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