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Accelerated impurity solver for DMFT and its diagrammatic extensions
Here, 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.
Revealing short- and long-range Li-ion diffusion in Li 2 MnO 3 from finite-temperature dynamical mean field theory
Li 2 MnO 3 is a key component of Li-excess layered cathodes of the form (1 − x), LiMO 2 + x, Li 2 MnO 3 (M = Mn, Ni, Co, …), yet its role in setting Li-ion transport limitations remains under debate. Here, in this study, we combine DFT+U, finite-temperature DFT+DMFT with a continuous-time quantum Monte Carlo impurity solver, and nudged-elastic-band (NEB) calculations to study Li + migration in paramagnetic Li 2 MnO 3 in the presence of a single Li vacancy. Evaluating DMFT total energies along the DFT+U NEB geometries reveals that dynamical correlations strongly renormalize the lowest-barrier processes, reducing the activation energies to E a = 0.18 eV for the shortest-range hop and E a = 0.50 eV for the next-lowest (transport-controlling) step. The 0.18 eV barrier quantitatively reproduces the short-range activation energy from µ+SR, while the 0.50 eV barrier is consistent with the long-range transport scale extracted from ac-impedance measurements. This single-vacancy, paramagnetic DMFT description thus provides a unified interpretation of local and macroscopic probes without invoking clustered vacancy configurations or strong extrinsic disorder, consistent with nearly stoichiometric Li 2 MnO 3 powders. More broadly, our results highlight finite-temperature dynamical correlations as an essential ingredient for predicting ionic migration energetics in correlated oxide electrodes.
Delocalized polaron and Burstein-Moss shift induced by Li in α–V 2 O 5 : A DFT + DMFT study
For this work, we performed density functional theory (DFT)+U and dynamical mean field theory (DMFT) calculations with a continuous-time quantum Monte Carlo impurity solver to investigate the electronic properties of V 2 O 5 and Li x V 2 O 5 (x = 0.125 and 0.25). Pristine V 2 O 5 is a charge-transfer insulator with strong O p–V d hybridization, and it exhibits a large band gap (E gap ) as well as a nonzero conduction-band (CB) gap.We show that the band gap, the number of d electrons of vanadium, N d , and the CB gap for V 2 O 5 obtained from our DMFT calculations are in excellent agreement with the experimental values. While the DFT +U approach replicates the experimental band gap, it overestimates the value of N d and underestimates the CB gap. In the presence of low Li doping, the electronic properties of V 2 O 5 are mainly driven by a polaronic mechanism, and electron spin resonance and electron nuclear double resonance spectroscopies observed the coexistence of free and bound polarons. Notably, our DMFT results identify both polaron types, with the bound polaron being energetically preferred, while the DFT +U method only predicts the free polaron. Our DMFT analysis also reveals that increased Li doping leads to electron filling in the conduction band, shifting the Fermi level. This result is consistent with the observed Burstein-Moss shift upon enhanced Li doping, and we thus demonstrate that the DFT + DMFT approach can be used for an accurate and realistic description of strongly correlated materials.
Effect of off-diagonal elements in the Wannier Hamiltonian on DFT + DMFT for low-symmetry materials: Study of Li 2 MnO 3
Here, we study the effect of the off-diagonal elements of the Wannier Hamiltonian on the electronic structure of the low-symmetry material Li 2 MnO 3 ( C2/m ), using dynamical mean field theory calculations with a continuous-time quantum Monte Carlo impurity solver. The presence of significant off-diagonal elements leads to a pronounced suppression of the energy gap. The off-diagonal elements are largest when the Wannier projection is used based on the global coordinate, and they remain substantial even with the projection using the local coordinate close to the direction of Mn-O bonds. We show that the energy gap is enhanced by the diagonalization of the Mn d block in the full p-d Hamiltonian with the application of a unitary rotation matrix. Additionally, the inclusion of small double counting energy is crucial for achieving the experimental gap by reducing p-d hybridization. Furthermore, we establish the efficiency of a low-energy (d-only basis) model for studying the electronic structure of Li 2 MnO 3 , as the Wannier basis represents a hybridized state of Mn d and O p orbitals. These findings suggest an appropriate approach for investigating low-symmetry materials using the density functional theory plus dynamical mean field theory (DFT + DMFT) method. We also find that the antiferromagnetic ground state $\Gamma$ 2u is stable with U ≤ 2 eV within density functional theory+$U$ calculations, which is much smaller than the widely used U = 5 eV.
First-principles study of the Kondo physics of a single Pu impurity in a Th host
Based on its condensed-matter properties, crystal structure, and metallurgy, which includes a phase diagram with six allotropic phases, plutonium is one of the most complicated pure elements in its solid state. Its anomalous properties, which are indicative of a very strongly correlated state, are related to its special position in the periodic table, which is at the boundary between the light actinides that have itinerant 5f electrons and the heavy actinides that have localized 5f electrons. As a foundational study to probe the role of local electronic correlations in Pu, we use the local-density approximation together with a continuous-time quantum Monte Carlo simulation to investigate the electronic structure of a single Pu atom that is either substitutionally embedded in the bulk and or adsorbed on the surface of a Th host. This is a simpler case than the solid phases of Pu metal. With the Pu impurity atom we have found a Kondo resonance peak, which is an important signature of electronic correlations, in the local density of states around the Fermi energy. We show that the peak width of this resonance is narrower for Pu atoms at the surface of Th than for those in the bulk due to a weakened Pu - 5f hybridization with the ligands at the surface.
Signatures of the Mott transition in the antiferromagnetic state of the two-dimensional Hubbard model
The properties of a phase with large correlation length can be strongly influenced by the underlying normal phase. Here, we illustrate this by studying the half-filled two-dimensional Hubbard model using cellular dynamical mean-field theory with continuous-time quantum Monte Carlo. Sharp crossovers in the mechanism that favors antiferromagnetic correlations and in the corresponding local density of states are observed. We found that these crossovers occur at values of the interaction strength U and temperature T that are controlled by the underlying normal-state Mott transition.
Exact diagonalization solver for extended dynamical mean-field theory
This paper presents an efficient exact diagonalization scheme for the extended dynamical mean-field theory and apply it to the extended Hubbard model on the square lattice with nonlocal charge-charge interactions. Our solver reproduces the phase diagram of this approximation with good accuracy. Details on the numerical treatment of the large Hilbert space of the auxiliary Holstein-Anderson impurity problem are provided. Benchmarks with a numerically exact strong-coupling continuous-time quantum Monte Carlo solver show better convergence behavior of the exact diagonalization in the deep insulator. Special attention is given to possible effects due to the discretization of the bosonic bath. We discuss the quality of real axis spectra and address the question of screening in the Mott insulator within extended dynamical mean-field theory.
Energy scales of the doped Anderson lattice model
This paper explores the energy scales of the doped Anderson lattice model using dynamical mean-field theory (DMFT), using a continuous-time Quantum Monte Carlo (CTQMC) impurity solver. We show that the low temperature properties of the lattice can not be scaled using the single ion local Kondo temperature T K but instead are governed by a doping-dependent coherence temperature T * which can be used to scale the temperature dependence of the spectral function, transport properties, and entropy. At half filling T * closely approximates the single ion T K , but as the filling n c is reduced to zero, T * also vanishes. The coherence temperature T * is shown to play a role of effective impurity Kondo temperature in the lattice model, and physical observables show significant evolution at T *. In the DMFT framework, we showed that the hybridization strength of the effective impurity model is qualitatively affected by the doping level, and determines T * in the lattice model.
Denoising of imaginary time response functions with Hankel projections
Imaginary-time response functions of finite-temperature quantum systems are often obtained with methods that exhibit stochastic or systematic errors. Reducing these errors comes at a large computational cost—in quantum Monte Carlo simulations, the reduction of noise by a factor of two incurs a simulation cost of a factor of four. In this paper, we relate certain imaginary-time response functions to an inner product on the space of linear operators on Fock space. We then show that data with noise typically does not respect the positive definiteness of its associated Gramian. The Gramian has the structure of a Hankel matrix. As a method for denoising noisy data, we introduce an alternating projection algorithm that finds the closest positive definite Hankel matrix consistent with noisy data. We test our methodology at the example of fermion Green's functions for continuous-time quantum Monte Carlo data and show remarkable improvements of the error, reducing noise by a factor of up to 20 in practical examples. We argue that Hankel projections should be used whenever finite-temperature imaginary-time data of response functions with errors is analyzed, be it in the context of quantum Monte Carlo, quantum computing, or in approximate semianalytic methodologies. Published by the American Physical Society 2024
DCA++: A software framework to solve correlated electron problems with modern quantum cluster methods
We present the first open release of the DCA++ project, a high-performance research software framework to solve quantum many-body problems with cutting edge quantum cluster algorithms. DCA++ implements the dynamical cluster approximation (DCA) and its DCA extension with a continuous self-energy. The algorithms capture nonlocal correlations in strongly correlated electron systems, thereby giving insight into high-$T_c$ superconductivity. The code’s scalability allows efficient usage of systems at all scales, from workstations to leadership computers. With regard to the increasing heterogeneity of modern computing machines, DCA++ provides portable performance on conventional and emerging new architectures, such as hybrid CPU–GPU, sustaining multiple petaflops on ORNL’s Titan and CSCS’ Piz Daint supercomputers. Moreover, we show how sustainable and scalable development of the code base has been achieved by adopting standard techniques of the software industry. These include employing a distributed version control system, applying test-driven development and following continuous integration.
Continuous-time Monte Carlo renormalization group
We implement the Monte Carlo renormalization group approach in the continuous-time Monte Carlo simulation of a quantum system. In this work, we demonstrate numerically the emergent isotropy between space and time at large distances for the systems that exhibit Lorentz invariance at quantum criticality. Here, this allows us to estimate accurately the sound velocity for these quantum systems. $\textit{Q}$-state Potts models in one and two space dimensions are used to illustrate the method.
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 a continuous-time Monte Carlo method with an 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. Furthermore, 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.
Quantum advantage for differential equation analysis
Quantum algorithms for differential equation solving, data processing, and machine learning potentially offer an exponential speedup over all known classical algorithms. However, there also exist obstacles to obtaining this potential speedup in useful problem instances. The essential obstacle for quantum differential equation solving is that outputting useful information may require difficult postprocessing, and the essential obstacle for quantum data processing and machine learning is that inputting the data is a difficult task just by itself. In this study, we demonstrate that, when combined, these difficulties solve one another. We show how the output of quantum differential equation solving can serve as the input for quantum data processing and machine learning, allowing dynamical analysis in terms of principal components, power spectra, and wavelet decompositions. To illustrate this, we consider continuous-time Markov processes on epidemiological and social networks. These quantum algorithms provide an exponential advantage over existing classical Monte Carlo methods.
A First-Passage Kinetic Monte Carlo method for reaction–drift–diffusion processes
Stochastic reaction–diffusion models are now a popular tool for studying physical systems in which both the explicit diffusion of molecules and noise in the chemical reaction process play important roles. The Smoluchowski diffusion-limited reaction model (SDLR) is one of several that have been used to study biological systems. Exact realizations of the underlying stochastic processes described by the SDLR model can be generated by the recently proposed First-Passage Kinetic Monte Carlo (FPKMC) method. This exactness relies on sampling analytical solutions to one and two-body diffusion equations in simplified protective domains. In this work we extend the FPKMC to allow for drift arising from fixed, background potentials. As the corresponding Fokker–Planck equations that describe the motion of each molecule can no longer be solved analytically, we develop a hybrid method that discretizes the protective domains. The discretization is chosen so that the drift–diffusion of each molecule within its protective domain is approximated by a continuous-time random walk on a lattice. New lattices are defined dynamically as the protective domains are updated, hence we will refer to our method as Dynamic Lattice FPKMC or DL-FPKMC. We focus primarily on the one-dimensional case in this manuscript, and demonstrate the numerical convergence and accuracy of our method in this case for both smooth and discontinuous potentials. We also present applications of our method, which illustrate the impact of drift on reaction kinetics.
An adaptive multi-level simulation algorithm for stochastic biological systems
Discrete-state, continuous-time Markov models are widely used in the modeling of biochemical reaction networks. Their complexity often precludes analytic solution, and we rely on stochastic simulation algorithms (SSA) to estimate system statistics. The Gillespie algorithm is exact, but computationally costly as it simulates every single reaction. As such, approximate stochastic simulation algorithms such as the tau-leap algorithm are often used. Potentially computationally more efficient, the system statistics generated suffer from significant bias unless tau is relatively small, in which case the computational time can be comparable to that of the Gillespie algorithm. The multi-level method [Anderson and Higham, “Multi-level Monte Carlo for continuous time Markov chains, with applications in biochemical kinetics,” SIAM Multiscale Model. Simul. 10(1), 146–179 (2012)] tackles this problem. A base estimator is computed using many (cheap) sample paths at low accuracy. The bias inherent in this estimator is then reduced using a number of corrections. Each correction term is estimated using a collection of paired sample paths where one path of each pair is generated at a higher accuracy compared to the other (and so more expensive). By sharing random variables between these paired paths, the variance of each correction estimator can be reduced. This renders the multi-level method very efficient as only a relatively small number of paired paths are required to calculate each correction term. In the original multi-level method, each sample path is simulated using the tau-leap algorithm with a fixed value of τ. This approach can result in poor performance when the reaction activity of a system changes substantially over the timescale of interest. By introducing a novel adaptive time-stepping approach where τ is chosen according to the stochastic behaviour of each sample path, we extend the applicability of the multi-level method to such cases. We demonstrate the efficiency of our method using a number of examples.
Multiorbital Quantum Impurity Solver for General Interactions and Hybridizations
Here we present a numerically exact inchworm Monte Carlo method for equilibrium multiorbital quantum impurity problems with general interactions and hybridizations. We show that the method, originally developed to overcome the dynamical sign problem in certain real-time propagation problems, can also overcome the sign problem as a function of temperature for equilibrium quantum impurity models. This is shown in several cases where the current method of choice, the continuous-time hybridization expansion, fails due to the sign problem. Our method therefore enables simulations of impurity problems as they appear in embedding theories without further approximations, such as the truncation of the hybridization or interaction structure or a discretization of the impurity bath with a set of discrete energy levels, and eliminates a crucial bottleneck in the simulation of ab initio embedding problems.