Continuous-time Quantum Monte Carlo Algorithm Research Articles

Accelerated impurity solver for DMFT and its diagrammatic extensions

We present ComCTQMC, a GPU accelerated quantum impurity solver. It uses the continuous-time quantum Monte Carlo (CTQMC) algorithm wherein the partition function is expanded in terms of the hybridisation function (CT-HYB). ComCTQMC supports both partition and worm-space measurements, and it uses improved estimators and the reduced density matrix to improve observable measurements whenever possible. ComCTQMC efficiently measures all one and two-particle Green's functions, all static observables which commute with the local Hamiltonian, and the occupation of each impurity orbital. ComCTQMC can solve complex-valued impurities with crystal fields that are hybridized to both fermionic and bosonic baths. Most importantly, ComCTQMC utilizes graphical processing units (GPUs), if available, to dramatically accelerate the CTQMC algorithm when the Hilbert space is sufficiently large. We demonstrate acceleration by a factor of over 600 (100) in a simulation of δ-Pu at 600 K with (without) crystal fields. In easier problems, the GPU offers less impressive acceleration or even decelerates the CTQMC. Here we describe the theory, algorithms, and structure used by ComCTQMC in order to achieve this set of features and level of acceleration. Program summaryProgram Title: ComCTQMCCPC Library link to program files:https://doi.org/10.17632/x2gzgm8njh.1Licensing provisions: GPLv3Programming language: C++/CUDANature of problem: In dynamical mean-field theory (DMFT), the computational bottleneck is the repeated solution of a quantum impurity problem [1]. The continuous-time quantum Monte-Carlo (CTQMC) algorithm has emerged as one of the most efficient methods for solving multiorbital impurity problems at moderate-to-high temperatures [2]. However, the low-temperature regime remains inaccessible, particularly for f-shell systems, and the measurement of two-particle correlation functions on an impurity adds a substantial computational burden. The bottleneck of the CTQMC solver is itself the computation of the local trace which includes the multiplication of many moderate-to-large sized matrices. The efficient solution of the impurity, measurement of the two-particle correlation functions, and acceleration of the trace computation are therefore critical.Solution method: ComCTQMC uses the hybridisation expansion of the impurity action to explore partition space [3]. It uses the worm algorithm [4] to explore the union of the partition space with observables spaces, e.g., the two-particle correlation functions. It uses improved estimators to more accurately measure the one- and two-particle Green's functions [5]. Identical impurities are solved across all MPI ranks (for ideal weak scaling) and the trace computations of these impurities are distributed to and accelerated by GPUs (when available). The lazy-trace algorithm [6] is used to further reduce the burden of the local trace calculation.Additional comments including restrictions and unusual features: ComCTQMC solves nearly arbitrary impurities, including those with complex valued and time-dependent interactions. However, there are two restrictions: (1) The retarded part of the interaction is described by a set of bilinears (a paired creation and annihilation operator), and these bilinears must commute with the local Hamiltonian and have real quantum numbers; (2) If a local Green's function vanishes, then the corresponding hybridisation function also vanishes.

Footprints of impurity quantum phase transitions in quantum Monte Carlo statistics

Interacting single-level quantum dot connected to BCS superconducting leads represents a well-controllable system to study the interplay between the correlation effects and the electron pairing that can result in a $0-\pi$ (singlet-doublet) quantum phase transition. The physics of this system can be well described by the impurity Anderson model. We present an analysis of the statistics of the perturbation expansion order of the continuous-time hybridization expansion quantum Monte Carlo algorithm in the vicinity of such a first-order impurity quantum phase transition. By calculating the moments of the histograms of the expansion order which deviate from the ideal Gaussian shape, we provide an insight into the thermodynamic mixing of the two phases at low but finite temperatures which is reflected in the bimodal nature of the histograms.

Mott transition of fermions in anisotropic ruby lattice

<sec>In this work, the Hubbard model is adopted to describe fermions with on-site repulsive interaction and the nearest-neighbor hopping in anisotropic ruby lattice. The combination of cluster dynamical mean field theory and continuous-time quantum Monte Carlo algorithm is used to solve the theoretical model.</sec> <sec>It is widely accepted that the density of states and the double occupancy are two important quantities for determining the phase transition of two-dimensional strongly correlated system. Therefore, based on the self-consistent calculation, using the maximum entropy method to calculate the single particle density of states and double occupancy of fermions in anisotropic ruby lattice. Here in this work, there are 6 sites in a cluster.</sec> <sec>The influences of temperature, interaction and anisotropic parameter on metal-insulator phase transition of fermions in anisotropic ruby lattice are discussed based on the calculations of single particle density of state and double occupancy. Finally, the metal-Mott insulator phase diagram which shows the competition between temperature and on-site repulsive interaction in the phase transition of fermions in anisotropic ruby lattice is presented. The results shows that the system is in metallic state for the regime of weak interaction and low temperature, and the Mott insulator appears in the regime of strong interaction and high temperature. The metallic state and Mott insulating one are separated by the second-order transition line in the phase diagram.</sec>

Generalized Quantum Monte Carlo Algorithm in Eigen Basis

New generalized continuous-time quantum Monte Carlo algorithm is developed. The algorithm is absolutely universal in the sense that it allows to calculate models specified in an arbitrary basis (in the basis of occupation numbers; in the momentum representation; in the basis of the eigenfunctions of small clusters into which the entire system is divided, or in any other basis): the algorithm requires only the specification of the matrix elements between the basis functions defined in a convenient basis. The implemented algorithm is applied to one-dimensional Hubbard model.

Mott transition in ruby lattice Hubbard model**Project supported by Inner Mongolia Natural Science Foundation, China (Grant No. 06021601).

Mott transition in a ruby lattice with fermions described by the Hubbard model including on-site repulsive interaction is investigated by combining the cellular dynamical mean-field theory and the continuous-time quantum Monte Carlo algorithm. The effect of temperature and on-site repulsive interaction on the metallic–insulating phase transition in ruby lattice with fermions is discussed based on the density of states and double occupancy. In addition, the magnetic property of each phase is discussed by defining certain magnetic order parameters. Our results show that the antiferromagnetic metal is found at the low temperature and weak interaction region and the antiferromagnetic insulating phase is found at the low temperature and strong interaction region. The paramagnetic metal appears in whole on-site repulsive interaction region when the temperature is higher than a certain value and the paramagnetic insulator appears at the middle scale of temperature and on-site repulsive interaction.

Improved Hubbard-I approximation impurity solver for quantum impurity models

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.

Finite-temperature phase diagram of (111) nickelate bilayers

We report a density functional theory plus dynamical mean field theory (DFT+DMFT) study of an oxide heterostructure of LaNiO$_3$ (LNO) bilayers in [111] direction interleaved with four atomic monolayers of LaAlO$_3$. DFT+$U$ optimizations yield two stable solutions: a uniform structure with equivalent NiO$_6$ octahedra, as well as a bond-disproportionated (BD) structure featuring a breathing distortion. For both structures, we construct the low-energy models describing the Ni $e_g$ states by means of Wannier projections supplemented by the Kanamori interaction, and solve them by DMFT. Using the continuous-time quantum Monte Carlo algorithm in the hybridization expansion, we study the temperature range between 145 and 450 K. For the uniform and the BD structure, we find similar phase diagrams that comprise four phases: a ferromagnetic metal (FM), a paramagnetic metal (PM), an antiferro-orbitally-ordered insulator (AOI), as well as a paramagnetic insulator (PI). By calculating momentum-resolved spectral functions on a torus and a cylinder, we demonstrate that the FM phase is not a Dirac metal, while both insulating phases are topologically trivial. By a comparison with available experimental data and model DMFT studies for the two-orbital Hubbard model, we suggest that LNO bilayers are in the AOI phase at room temperature.

Self-learning Monte Carlo method: Continuous-time algorithm

The recently-introduced self-learning Monte Carlo method is a general-purpose numerical method that speeds up Monte Carlo simulations by training an effective model to propose uncorrelated configurations in the Markov chain. We implement this method in the framework of continuous time Monte Carlo method with auxiliary field in quantum impurity models. We introduce and train a diagram generating function (DGF) to model the probability distribution of auxiliary field configurations in continuous imaginary time, at all orders of diagrammatic expansion. By using DGF to propose global moves in configuration space, we show that the self-learning continuous-time Monte Carlo method can significantly reduce the computational complexity of the simulation.

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

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

Anderson-Holstein model in two flavors of the noncrossing approximation

The dynamical interplay between electron-electron interactions and electron-phonon coupling is investigated within the Anderson-Holstein model, a minimal model for open quantum systems that embody these effects. The influence of phonons on spectral and transport properties is explored in equilibrium, for nonequilibrium steady state and for transient dynamics after a quench. Both the particle-hole symmetric and the more generic particle-hole asymmetric cases are studied. The treatment is based on two complementary noncrossing approximations, the first of which is constructed around the weak-coupling limit and the second around the polaron limit. In general, the two methods disagree in nontrivial ways, indicating that more reliable approaches to the problem are needed. The frameworks used here can form the starting point for numerically exact methods based on bold-line continuous-time quantum Monte Carlo algorithms capable of treating open systems simultaneously coupled to multiple fermionic and bosonic baths.

Quantum phase transitions in two-dimensional strongly correlated fermion systems

In this article, we review our recent work on quantum phase transition in two-dimensional strongly correlated fermion systems. We discuss the metal–insulator transition properties of these systems by calculating the density of states, double occupancy, and Fermi surface evolution using a combination of the cellular dynamical mean-field theory (CDMFT) and the continuous-time quantum Monte Carlo algorithm. Furthermore, we explore the magnetic properties of each state by defining magnetic order parameters. Rich phase diagrams with many intriguing quantum states, including antiferromagnetic metal, paramagnetic metal, Kondo metal, and ferromagnetic insulator, were found for the two-dimensional lattices with strongly correlated fermions. We believe that our results would lead to a better understanding of the properties of real materials.

Efficient continuous-time quantum Monte Carlo algorithm for fermionic lattice models

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.

Formula omitted]: An open source continuous-time quantum Monte Carlo impurity solver toolkit

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 iQIST). 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 iQIST are also explained. Finally, several simple tutorials are presented in order to demonstrate the usage and power of iQIST. Program summaryProgram title:iQISTCatalogue identifier: AEWQ_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEWQ_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU General Public License, version 3No. of lines in distributed program, including test data, etc.: 226270No. of bytes in distributed program, including test data, etc.: 5263144Distribution format: tar.gzProgramming 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 OpenMPRAM: Depends on the complexity of the problemClassification: 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 iQIST 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.

Multiorbital cluster dynamical mean-field theory with an improved continuous-time quantum Monte Carlo algorithm

We implement a multi-orbital cluster dynamical mean-field theory (DMFT), by improving a sample-update algorithm in the continuous-time quantum Monte Carlo method based on the interaction expansion. The proposed sampling scheme for the spin-flip and pair-hopping interactions in the two-orbital systems mitigates the sign problem, giving an efficient way to deal with these interactions. In particular, in the single-site DMFT, we see that the negative signs vanish. We apply the method to the two-dimensional two-orbital Hubbard model at half filling, where we take into account the short-range spatial correlation effects within a four-site cluster. We show that, compared to the single-site DMFT results, the critical interaction value for the metal-insulator transition decreases and that the effects of the spin-flip and pair-hopping terms are less significant in the parameter region we have studied. The present method provides a firm starting point for the study of inter-site correlations in multi-orbital systems. It also has a wide applicable scope in terms of realistic calculations in conjunction with density functional theory.

Quantum phase transitions of fermionic atoms in an anisotropic triangular optical lattice

The effect of anisotropy caused by a confining potential on the properties of fermionic cold atoms in a triangular optical lattice is systematically investigated by using the dynamical cluster approximation combined with the continuous time quantum Monte—Carlo algorithm. The quantum phase diagrams which reflect the temperature—interaction relation and the competition between the anisotropic parameter and the interaction are presented with full consideration of the anisotropy of the system. Our results show that the system undergoes a transition from Fermi liquid to Mott insulator when the repulsive interaction reaches a critical value. The Kondo effect also can be observed in this system and the pseudogap is suppressed at low temperatures due to the Kondo effect. A feasible experiment protocol to observe these phenomena in an anisotropic triangular optical lattice with cold atoms is proposed, in which the hopping terms are closely related to the lattice confining potential and the atomic interaction can be adjusted via the Feshbach resonance.

Efficient implementation of the continuous-time hybridization expansion quantum impurity solver

Strongly correlated quantum impurity problems appear in a wide variety of contexts ranging from nanoscience and surface physics to material science and the theory of strongly correlated lattice models, where they appear as auxiliary systems within dynamical mean-field theory. Accurate and unbiased solutions must usually be obtained numerically, and continuous-time quantum Monte Carlo algorithms, a family of algorithms based on the stochastic sampling of partition function expansions, perform well for such systems. With the present paper we provide an efficient and generic implementation of the hybridization expansion quantum impurity solver, based on the segment representation. We provide a complete implementation featuring most of the recently developed extensions and optimizations. Our implementation allows one to treat retarded interactions and provides generalized measurement routines based on improved estimators for the self-energy and for vertex functions. The solver is embedded in the ALPS-DMFT application package. Program summaryProgram title: ct-hybCatalogue identifier: AEOL_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEOL_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Use of the hybridization expansion impurity solvers requires citation of this paper. Use of any ALPS program requires citation of the ALPS [1] paper.No. of lines in distributed program, including test data, etc.: 650044No. of bytes in distributed program, including test data, etc.: 20553265Distribution format: tar.gzProgramming language: C++/Python.Computer: Desktop PC, high-performance computers.Operating system: Unix, Linux, OSX, Windows.Has the code been vectorized or parallelized?: Yes, MPI parallelized.RAM: 1 GBClassification: 7.3.External routines: ALPS [1, 2, 3], BLAS [4, 5], LAPACK [6], HDF5 [7]Nature of problem:Quantum impurity models were originally introduced to describe a magnetic transition metal ion in a non-magnetic host metal. They are widely used today. In nanoscience they serve as representations of quantum dots and molecular conductors. In condensed matter physics, they are playing an increasingly important role in the description of strongly correlated electron materials, where the complicated many-body problem is mapped onto an auxiliary quantum impurity model in the context of dynamical mean-field theory, and its cluster and diagrammatic extensions. They still constitutes a non-trivial many-body problem, which takes into account the (possibly retarded) interaction between electrons occupying the impurity site. Electrons are allowed to dynamically hop on and off the impurity site, which is described by a time-dependent hybridization function.Solution method:The quantum impurity model is solved using a continuous-time quantum Monte Carlo algorithm which is based on a perturbation expansion of the partition function in the impurity–bath hybridization. Monte Carlo configurations are represented as segments on the imaginary time interval and individual terms correspond to Feynman diagrams which are stochastically sampled to all orders using a Metropolis algorithm. For a detailed review on the method, we refer the reader to [8].Running time:1–8 h.

Conserved quantities ofSU(2)-invariant interactions for correlated fermions and the advantages for quantum Monte Carlo simulations

In the context of realistic calculations for strongly-correlated materials with $d$- or $f$-electrons the efficient computation of multi-orbital models is of paramount importance. Here we introduce a set of invariants for the SU(2)-symmetric Kanamori Hamiltonian which allows to massively speed up the calculation of the fermionic trace in hybridization-expansion continuous-time quantum Monte Carlo algorithms. As an application, we show that, exploiting this set of good quantum numbers, the study of the orbital-selective Mott-transition in systems with up to seven correlated orbitals becomes feasible.

Improved estimators for the self-energy and vertex function in hybridization-expansion continuous-time quantum Monte Carlo simulations

We propose efficient measurement procedures for the self-energy and vertex function of the Anderson impurity model within the hybridization expansion continuous-time quantum Monte Carlo algorithm. The method is based on the measurement of higher-order correlation functions related to the quantities being sought through the equation of motion, a technique previously introduced in the NRG context. For the case of interactions of density-density type, the additional correlators can be obtained at essentially no additional computational cost. In combination with a recently introduced method for filtering the Monte Carlo noise using a representation in terms of orthogonal polynomials, we obtain data with unprecedented accuracy. This leads to an enhanced stability in analytical continuations of the self-energy or in two-particle based theories such as the dual fermion approach. As an illustration of the method we reexamine the previously reported spin-freezing and high-spin to low-spin transitions in a two-orbital model with density-density interactions. In both cases, the vertex function undergoes significant changes, which suggests significant corrections to the dynamical mean-field solutions in dual fermion calculations.

Quantum Monte Carlo study of the transverse-field Ising model on a frustrated checkerboard lattice

We present the numerical results for low temperature behavior of the transverse-field Ising model on a frustrated checkerboard lattice, with focus on the effect of both quantum and thermal fluctuations. Applying the recently-developed continuous-time quantum Monte Carlo algorithm, we compute the magnetization and susceptibility down to extremely low temperatures while changing the magnitude of both transverse and longitudinal magnetic fields. Several characteristic behaviors are observed, which were not inferred from the previously-studied quantum order from disorder at zero temperature, such as a horizontal-type stripe ordering at a substantial longitudinal field and a persistent critical behavior down to low temperature in a weak longitudinal field region.

Diagrammatic quantum Monte Carlo solution of the two-dimensional cooperon-fermion model

We investigate the two-dimensional cooperon-fermion model in the correlated regime with a new continuous-time diagrammatic determinant quantum Monte Carlo (DDQMC) algorithm. We estimate the transition temperature $T_{c}$, examine the effectively reduced band gap and cooperon mass, and find that delocalization of the cooperons enhances the diamagnetism. When applied to diamagnetism of the pseudogap phase in high-$T_{c}$ cuprates, we obtain results in a qualitative agreement with recent torque magnetization measurements.