Quantum impurity models describe an atom or molecule embedded in a host material with which it can exchange electrons. They are basic to nanoscience as representations of quantum dots and molecular conductors and play an increasingly important role in the theory of "correlated electron" materials as auxiliary problems whose solution gives the "dynamical mean field" approximation to the self energy and local correlation functions. These applications require a method of solution which provides access to both high and low energy scales and is effective for wide classes of physically realistic models. The continuous-time quantum Monte Carlo algorithms reviewed in this article meet this challenge. We present derivations and descriptions of the algorithms in enough detail to allow other workers to write their own implementations, discuss the strengths and weaknesses of the methods, summarize the problems to which the new methods have been successfully applied and outline prospects for future applications.
We present a continuous-time Monte Carlo method for quantum impurity models, which combines a weak-coupling expansion with an auxiliary-field decomposition. The method is considerably more efficient than Hirsch-Fye and free of time discretization errors, and is particularly useful as impurity solver in large cluster dynamical mean field theory (DMFT) calculations.
Multi-orbital quantum impurity models with general interaction and hybridization terms appear in a wide range of applications including embedding, quantum transport, and nanoscience. However, most quantum impurity solvers are restricted to a few impurity orbitals, discretized baths, diagonal hybridizations, or density-density interactions. Here, we generalize the inchworm quantum Monte Carlo method to the interaction expansion and explore its application to typical single- and multi-orbital problems encountered in investigations of impurity and lattice models. Our implementation generically ou…
We explore two complementary modifications of the hybridization-expansion continuous-time Monte Carlo method, aiming at large multi-orbital quantum impurity problems. One idea is to compute the imaginary-time propagation using a matrix product states representation. We show that bond dimensions considerably smaller than the dimension of the Hilbert space are sufficient to obtain accurate results, and that this approach scales polynomially, rather than exponentially with the number of orbitals. Based on scaling analyses, we conclude that a matrix product state implementation will outperform the…
We develop a projective quantum Monte Carlo algorithm of the Hirsch-Fye type for obtaining ground state properties of the Anderson impurity model. This method is employed to solve the self-consistency equations of dynamical mean field theory. It is shown that the approach converges rapidly to the ground state so that reliable zero-temperature results are obtained. As a first application, we study the Mott-Hubbard metal-insulator transition of the one-band Hubbard model, reconfirming the numerical renormalization group results.
We develop a continuous-time quantum Monte Carlo method based on a strong-coupling expansion for Anderson impurity models with phonon-assisted hybridizations for arbitrary number of phonon modes. As a benchmark, we investigate the two-channel Anderson model with a single phonon, and numerically demonstrate that an SO(5) susceptibility composed of localized-electron charge and phonon-parity operators diverges logarithmically at the non-Fermi liquid critical point in the model, which verifies the predictions by the boundary conformal field theory[K. Hattori: Phys. Rev. B {\bf 85} (2012) 214411].
In the present paper, we present an efficient continuous-time quantum Monte Carlo impurity solver with high acceptance rate at low temperature for multi-orbital quantum impurity models with general interaction. In this hybridization expansion impurity solver, the imaginary time evolution operator for the high energy multiplets, which decays very rapidly with the imaginary time, is approximated by a probability normalized $δ$-function. As the result, the virtual charge fluctuations of $f^{n}\rightarrow f^{n\pm1}$ are well included on the same footing without applying Schrieffer-Wolff transforma…
Since the inception of the dynamical mean-field theory, numerous numerical studies have relied on the Hirsch-Fye quantum Monte Carlo (HF-QMC) method for solving the associated impurity problem. Recently developed continuous-time algorithms (CT-QMC) avoid the Trotter discretization error and allow for faster configuration updates, which makes them candidates for replacing HF-QMC. We demonstrate, however, that a state-of-the-art implementation of HF-QMC (with extrapolation of discretization delta_tau -> 0) is competitive with CT-QMC. A quantitative analysis of Trotter errors in HF-QMC estimates …
In our reply, we show that the objections put forward in cond-mat/0508763 concerning our paper, Phys. Rev. Lett. 93, 136405 (2004), are not valid: (i) There is no orthogonality catastrophe (OC) for our calculations, and it is also generally not ``unpractical'' to avoid it. (ii) The OC does not affect our results.
Optimized measurements for the susceptibility, self-energy, as well as three-leg and four-leg vertex functions are introduced for the continuous-time hybridization expansion quantum Monte Carlo solver for the impurity model in presence of a retarded interaction. The self-energy and vertex functions are computed from impurity averages which involve time integrals over the retarded interaction. They can be evaluated efficiently within the segment representation. These quantities are computed within dynamical mean-field theory in presence of plasmonic screening. In the antiadiabatic regime, the s…
We present TRIQS/CTHYB, a state-of-the art open-source implementation of the continuous-time hybridisation expansion quantum impurity solver of the TRIQS package. This code is mainly designed to be used with the TRIQS library in order to solve the self-consistent quantum impurity problem in a multi-orbital dynamical mean field theory approach to strongly-correlated electrons, in particular in the context of realistic calculations. It is implemented in C++ for efficiency and is provided with a high-level Python interface. The code is ships with a new partitioning algorithm that divides the loca…
In order to investigate the effects of nonmagnetic impurities in strongly correlated systems, Quantum Monte Carlo (QMC) simulations have been carried out for the doped two-dimensional Hubbard model with one nonmagnetic impurity. Using a bare impurity potential which is onsite and attractive, magnetic and single-particle properties have been calculated. The QMC results show that giant oscillations develop in the Knight shift response around the impurity site due to the short-range antiferromagnetic correlations. These results are useful for interpreting the NMR data on Li and Zn substituted lay…
We propose that a combination of the semiclassical approximation with Monte Carlo simulations can be an efficient and reliable impurity solver for dynamical mean field theory equations and their cluster extensions with large cluster sizes. In order to show the reliability of the method, we consider two test cases: (i) the single-band Hubbard model within the dynamical cluster approximation with 4- and 8-site clusters and (ii) the anisotropic two-orbital Hubbard model with orbitals of different band width within the single-site dynamical mean field theory. We compare our results with those obta…
We extend the recently developed real-time Diagrammatic Monte Carlo method, in its hybridization expansion formulation, to the full Kadanoff-Baym-Keldysh contour. This allows us to study real-time dynamics in correlated impurity models starting from an arbitrary, even interacting, initial density matrix. As a proof of concept we apply the algorithm to study the non equilibrium dynamics after a local quantum quench in the Anderson Impurity Model. Being a completely general approach to real-time dynamics in quantum impurity models it can be used as a solver for Non Equilibrium Dynamical Mean Fie…
We present a diagrammatic Monte Carlo method for quantum impurity problems with general interactions and general hybridization functions. Our method uses a recursive determinant scheme to sample diagrams for the scattering amplitude. Unlike in other methods for general impurity problems, an approximation of the continuous hybridization function by a finite number of bath states is not needed, and accessing low temperature does not incur an exponential cost. We test the method for the example of molecular systems, where we systematically vary temperature, interatomic distance, and basis set siz…
We study the impurity-induced phase transitions in a quasi-one-dimensional Heisenberg antiferromagnet doped with magnetic spin-1/2 impurities and non-magnetic ones. The impurity-induced transition temperature determined by the quantum Monte Carlo method with the continuous-time loop algorithm is monotonically increasing as a function of the magnitude of the impurity spin. To these results, we give discussions based on the valence-bond solid-like picture for the pure system and the inspection of the local magnetic structure around the impurities.
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 $i$QIST). This package contains several highly optimized quantum impurity solvers which are based on the hybridization expansion continuous-time q…
The conductance of two Anderson impurity models, one with two-fold and another with four-fold degeneracy, representing two types of quantum dots, is calculated using a world-line quantum Monte Carlo (QMC) method. Extrapolation of the imaginary time QMC data to zero frequency yields the linear conductance, which is then compared to numerical renormalization group results in order to assess its accuracy. We find that the method gives excellent results at low temperature (T<Tk) throughout the mixed valence and Kondo regimes, but it is unreliable for higher temperature.
We present a quantum impurity solver based on a pseudo-particle framework, which combines diagrammatic resummations for a three-point vertex with diagrammatic Monte Carlo sampling of a four-point vertex. This recently proposed approach [A. J. Kim et al., arXiv:2112.15549] is generalized here to fermionic impurity problems and we discuss the technical details of the implementation, including the time-stepping approach, the Monte Carlo updates, and the routines for checking the two-particle irreducibility of the four-point vertex. We also explain how the vertex information can be efficiently sto…
We generalized the recently introduced new impurity solver based on the diagrammatic expansion around the atomic limit and Quantum Monte Carlo summation of the diagrams. We present generalization to the cluster of impurities, which is at the heart of the cluster Dynamical Mean-Field methods, and to realistic multiplet structure of a correlated atom, which will allow a high precision study of actinide and lanthanide based compounds with the combination of the Dynamical Mean-Field theory and band structure methods. The approach is applied to both, the two dimensional Hubbard and t-J model within…
The negative sign problem in quantum Monte Carlo (QMC) simulations of cluster impurity problems is the major bottleneck in cluster dynamical mean field calculations. In this paper we systematically investigate the dependence of the sign problem on the single-particle basis. We explore both the hybridization-expansion and the interaction-expansion variants of continuous-time QMC for three-site and four-site impurity models with baths that are diagonal in the orbital degrees of freedom. We find that the sign problem in these models can be substantially reduced by using a non-trivial single-parti…
A Monte Carlo sampling of diagrammatic corrections to the non-crossing approximation is shown to provide numerically exact estimates of the long-time dynamics and steady state properties of nonequilibrium quantum impurity models. This `bold' expansion converges uniformly in time and significantly ameliorates the sign problem that has heretofore limited the power of real-time Monte Carlo approaches to strongly interacting real-time quantum problems. The new approach enables the study of previously intractable problems ranging from generic long time nonequilibrium transport characteristics in sy…
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 $0-π$ (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 …
We present a very efficient solver for the general Anderson impurity problem. It is based on the perturbation around a solution obtained from exact diagonalization using a small number of bath sites. We formulate a perturbation theory which is valid for both weak and strong coupling and interpolates between these limits. Good agreement with numerically exact quantum Monte-Carlo results is found for a single bath site over a wide range of parameters. In particular, the Kondo resonance in the intermediate coupling regime is well reproduced for a single bath site and the lowest order correction. …
We present a 31P NMR study of the coupled spin 1/2 ladder compound BiCu2PO6. In the pure material, intrinsic susceptibility and dynamics show a spin gap of about . Substitution of non magnetic Zn or magnetic Ni impurity at Cu site induces a staggered magnetization which results in a broadening of the 31P NMR line, while susceptibility far from the defects is unaffected. The effect of Ni on the NMR line broadening is twice that of Zn, which is consistent with Quantum Monte Carlo (QMC) calculations assuming that Ni couples ferromagnetically to its adjacent Cu. The induced moment follows a 1/T te…
We describe an open-source implementation of the continuous-time hybridization-expansion quantum Monte Carlo method for impurity models with general instantaneous two-body interactions and complex hybridization functions. The code is built on the ALPS libraries (Applications and Libraries for Physics Simulations).
In this supplementary material, we investigate further the impurity-induced freezing mechanism in a doped system of 3D weakly coupled ladders resembling Bi(Cu$_{1-x}$Zn$_x$)$_2$ZnPO$_6$ using large scale Quantum Monte Carlo simulations.
We use two different methods, the Hirsch-Fye quantum Monte Carlo simulation and the slave-boson mean field analysis to estimate the effect of spin-orbit coupling on the Kondo temperature in a quantum wire. In quantum Monte Carlo simulation, we calculate the product of spin susceptibility and temperature for different spin-orbit couplings and impurity energies. The variation of the Kondo temperature is estimated via the low temperature universal curves. In the mean field analysis, the Kondo temperature is estimated as the condensation temperature of slave-boson. Both the two methods demonstrate…
We introduce a quantum Monte Carlo technique to calculate exactly at finite temperatures the Green function of a fermionic quantum impurity coupled to a bosonic field. While the algorithm is general, we focus on the single impurity Anderson model coupled to a Holstein phonon as a schematic model for a molecular transistor. We compute the density of states at the impurity in a large range of parameters, to demonstrate the accuracy and efficiency of the method. We also obtain the conductance of the impurity model and analyze different regimes. The results show that even in the case when the effe…
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…
The nonequilibrium spectral properties of the Anderson impurity model with a chemical potential bias are investigated within a numerically exact real time quantum Monte Carlo formalism. The two-time correlation function is computed in a form suitable for nonequilibrium dynamical mean field calculations. Additionally, the evolution of the model's spectral properties are simulated in an alternative representation, defined by a hypothetical but experimentally realizable weakly coupled auxiliary lead. The voltage splitting of the Kondo peak is confirmed and the dynamics of its formation after a co…
We analyze the spectral function of the single-impurity two-terminal Anderson model at finite voltage using the recently developed diagrammatic quantum Monte Carlo technique as well as perturbation theory. In the (particle-hole-)symmetric case we find an excellent agreement of the numerical data with the perturbative results of second order up to interaction strengths $U/Γ\approx 2$, where $Γ$ is the transparency of the impurity-electrode interface. The analytical results are obtained in form of the nonequilibrium self-energy for which we present explicit formulas in the closed form at arbitra…
In this paper, we present a new version of the $i$QIST software package, which is capable of solving various quantum impurity models by using the hybridization expansion (or strong coupling expansion) continuous-time quantum Monte Carlo algorithm. In the revised version, the software architecture is completely redesigned. New basis (intermediate representation or singular value decomposition representation) for the single-particle and two-particle Green's functions is introduced. A lot of useful physical observables are added, such as the charge susceptibility, fidelity susceptibility, Binder …
Motivated by recent Monte-Carlo simulations of Hoglund and Sandvik (arXiv:0808.0408), we study edge response in square lattice quantum antiferromagnets. We use the O(3) non-linear sigma-model to compute the decay asymptotics of the staggered magnetization, energy density and local magnetic susceptibility away from the edge. We find that the total edge susceptibility is negative and diverges logarithmically as the temperature vanishes. We confirm the predictions of the continuum theory by performing a 1/S expansion of the microscopic Heisenberg model with the edge. We propose a qualitative expl…
We investigate the cluster size convergence of the energy and observables using two forms of density matrix embedding theory (DMET): the original cluster form (CDMET) and a new formulation motivated by the dynamical cluster approximation (DCA-DMET). Both methods are applied to the half-filled one- and two-dimensional Hubbard models using a sign-problem free auxiliary-field quantum Monte Carlo (AFQMC) impurity solver, which allows for the treatment of large impurity clusters of up to 100 sites. While CDMET is more accurate at smaller impurity cluster sizes, DCA- DMET exhibits faster asymptotic …
An impurity solver based on a continuous-time quantum Monte Carlo method is developed for the Coqblin-Schrieffer model. The Monte Carlo simulation does not encounter a sign problem for antiferromagnetic interactions, and accurately reproduces the Kondo effect. Our algorithm can deal with an arbitrary number N of local degrees of freedom, becomes more efficient for larger values of N, and is hence suitable for models with orbital degeneracy. The dynamical susceptibility and the impurity t-matrix are derived with the aid of the Padé approximation for various values of N, and good agreement is fo…
We propose a Monte Carlo method, which is a hybrid method of the quantum Monte Carlo method and variational Monte Carlo theory, to study the Hubbard model. The theory is based on the off-diagonal and the Gutzwiller type correlation factors which are taken into account by a Monte Carlo algorithm. In the 4x4 system our method is able to reproduce the exact results obtained by the diagonalization. An application is given to investigate the half-filled band case of two-dimensional square lattice. The energy is favorably compared with quantum Monte Carlo data.
An investigation is presented of the utility of semiclassical approximations for solving the quantum-impurity problems arising in the dynamical-mean-field approach to the correlated-electron models. The method is based on performing a exact numerical integral over the zero-Matsubara-frequency component of the spin part of a continuous Hubbard-Stratonovich field, along with a spin-field-dependent steepest descents treatment of the charge part. We test this method by applying it to one or two site approximations to the single band Hubbard model with different band structures, and comparing the r…
Dynamical mean-field theory (DMFT) is one of the most widely-used methods to treat accurately electron correlation effects in ab-initio real material calculations. Many modern large-scale implementations of DMFT in electronic structure codes involve solving a quantum impurity model with a Continuous-Time Quantum Monte Carlo (CT-QMC) solver. The main advantage of CT-QMC is that, unlike standard quantum Monte Carlo approaches, it is able to generate the local Green's functions of the correlated system on an arbitrarily fine imaginary time grid, and is free of any systematic errors. In this work,…
We use a quantum Monte Carlo method (stochastic series expansion) to study the effects of a magnetic or nonmagnetic impurity on the magnetic susceptibility of the two-dimensional Heisenberg antiferromagnet. At low temperatures, we find a log-divergent contribution to the transverse susceptibility. We also introduce an effective few-spin model that can quantitatively capture the differences between magnetic and nonmagnetic impurities at high and intermediate temperatures.
We implement the Lanczos algorithm on natural orbital basis to solve the zero-temperature Green's function of Anderson impurity models, following the work of Y. Lu, M. Höppner, O. Gunnarsson, and M. W. Haverkort, Phys. Rev. B {\bf 90} (2014) 085102. We present the technical details, generalize the algorithm to the cases of particle-hole asymmetry, with local magnetic field, and of two impurities. The results are benchmarked with conventional Lanczos, quantum Monte Carlo, and numerical renormalization group methods, demonstrating its potential as a powerful impurity solver for the dynamical mea…
We present an impurity solver based on adaptively truncated Hilbert spaces. The solver is particularly suitable for dynamical mean-field theory in circumstances where quantum Monte Carlo approaches are ineffective. It exploits the sparsity structure of quantum impurity models, in which the interactions couple only a small subset of the degrees of freedom. We further introduce an adaptive truncation of the particle or hole excited spaces, which enables computations of Green functions with an accuracy needed to avoid unphysical (sign change of imaginary part) self-energies. The method is benchma…
The impurity Green's function Gf in the local non-Fermi liquid state is evaluated by means of the continuous-time quantum Monte Carlo method extended to the multichannel Anderson model. For N=M (where N and M are numbers of spin components and channels, respectively), Gf is expressed as -Im Gf(w+i0) = c - b |w|^{1/2}, and the zero-frequency value c depends only on N (=M). A corresponding impurity self-energy at low frequencies is composed of two parts: a resonance term related to c, and a non-Fermi liquid term proportional to |w|^{1/2}. The characteristic energy scale is discussed in terms of …
We develop a continuous-time quantum Monte Carlo (CTQMC) method for quantum impurities coupled to interacting quantum wires described by a Tomonaga-Luttinger liquid. The method is negative-sign free for any values of the Tomonaga-Luttinger parameter, which is rigorously proved, and thus, efficient low-temperature calculations are possible. Duality between electrons and bosons in one dimensional systems allows us to construct a simple formula for the CTQMC algorithm in these systems. We show that the CTQMC for Tomonaga-Luttinger liquids can be implemented with only minor modifications of previo…
We present a Quantum Monte Carlo study for Heisenberg spin-1/2 two leg ladder systems doped with non magnetic impurities. The simulations are applied to the doped spin ladder compound SrCu2O3 doped with Zn, where a large broadening of the 65Cu NMR lines has been observed in experiment at low temperatures but far above the Neel temperature. We find that interladder-couplings with a sizeable coupling in stacking direction are required to describe the line broadening, which can not be explained by considering a single ladder only. Around a single impurity, spin correlations cause an exponentially…
We study the electronic structure of a spherical jellium in the presence of a central Gaussian impurity. We test how well the resulting inhomogeneity effects beyond spherical jellium are reproduced by several approximations of density functional theory (DFT). Four rungs of Perdew's ladder of DFT functionals, namely local density approximation (LDA), generalized gradient approximation (GGA), meta-GGA and orbital-dependent hybrid functionals are compared against our quantum Monte Carlo (QMC) benchmarks. We identify several distinct transitions in the ground state of the system as the electronic …
We present a realistic study for electronic and magnetic properties in dilute magnetic semiconductor (Ga,Mn)As. A multi-orbital Haldane-Anderson model parameterized by density-functional calculations is presented and solved with the Hirsch-Fye quantum Monte Carlo algorithm. Results well reproduce experimental results in the dilute limit. When the chemical potential is located between the top of the valence band and an impurity bound state, a long-range ferromagnetic correlations between the impurities, mediated by antiferromagnetic impurity-host couplings, are drastically developed. We observe…
We discuss an alternative accurate Monte Carlo method to calculate the ground-state energy and related quantities for Laughlin states of the fractional quantum Hall effect in a disk geometry. This alternative approach allows us to obtain accurate bulk regime (thermodynamic limit) values for various quantities from Monte Carlo simulations with a small number of particles (much smaller than that needed with standard Monte Carlo approaches).
Quantum Monte Carlo (QMC) methods are some of the most accurate methods for simulating correlated electronic systems. We investigate the compatibility, strengths and weaknesses of two such methods, namely, diffusion Monte Carlo (DMC) and auxiliary-field quantum Monte Carlo (AFQMC). The multi-determinant trial wave functions employed in both approaches are generated using the configuration interaction using a perturbative selection made iteratively (CIPSI) technique. Complete basis set full configuration interaction (CBS-FCI) energies estimated with CIPSI are used as a reference in this compara…
We describe an open-source implementation of the continuous-time interaction-expansion quantum Monte Carlo method for cluster-type impurity models with onsite Coulomb interactions and complex Weiss functions. The code is based on the ALPS libraries.
