We discuss the efficient implementation of general impurity solvers for dynamical mean-field theory. We show that both Lanczos and quantum Monte Carlo in different flavors (Hirsch-Fye, continuous-time hybridization- and interaction-expansion) exhibit excellent scaling on massively parallel supercomputers. We apply these algorithms to simulate realistic model Hamiltonians including the full Coulomb vertex, crystal-field splitting, and spin-orbit interaction. We discuss how to remove the sign problem in the presence of non-diagonal crystal-field and hybridization matrices. We show how to extract the physically observable quantities from imaginary time data, in particular correlation functions and susceptibilities. Finally, we present benchmarks and applications for representative correlated systems.
We discuss the efficient implementation of general impurity solvers for dynamical mean-field theory. We show that both Lanczos and quantum Monte Carlo in different flavors (Hirsch-Fye, continuous-time hybridization- and interaction-expansion) exhibit excellent scaling on massively parallel supercomputers. We apply these algorithms to simulate realistic model Hamiltonians including the full Coulomb vertex, crystal-field splitting, and spin-orbit interaction. We discuss how to remove the sign problem in the presence of non-diagonal crystal-field and hybridization matrices. We show how to extract the physically observable quantities from imaginary time data, in particular correlation functions and susceptibilities. Finally, we present benchmarks and applications for representative correlated systems.
In this paper we review recent developments towards a realistic description of the electronic structure and magnetism of correlated nanosystems. A new class of so-called continuous-time solvers for the quantum impurity problem is discussed, which provides a numerically exact solution without systematic errors due to imaginary time discretization. These solvers are able to handle general interactions, like the full Coulomb vertex. We further show how four-point or higher-order correlation functions of the impurity problem can be computed. This allows the calculation of dynamical susceptibilities which provide information about spin excitations. Moreover, we discuss a principally new many-body scheme recently proposed for the description of non-local correlations in strongly correlated systems. This approach provides a basis for a many-body description of extended correlated nanostructures on a substrate.
The physics of strongly correlated materials poses one of the most challenging problems in condensed-matter sciences. Standard approximations applicable to wide classes of materials such as the local density approximation fail, due to the importance of the Coulomb repulsion between localized electrons. Instead, we resort to non-perturbative many-body methods. The calculations are, however, only feasible for rather small model systems. The full Hamiltonian of a real material is approximated by a model Hamiltonian comprising only the most important electronic degrees of freedom, while the effect of all other electrons is included in an average way by renormalizing the parameters. Realistic calculations of strongly correlated materials need to include sufficiently many of these electronic degrees of freedom. The new generation of massively parallel supercomputers allows for these realistic calculations. However, exploiting their computational power requires newly devised algorithms. As a solver we use the Lanczos method, which needs the full many-body state of the correlated system. It is thus limited by the available main memory. The foremost problem for a distributed-memory implementation is that the multiplication of the Hamiltonian to the many-body state leads to highly non-local memory access patterns. A simple yet important observation leads to an efficient solution: in the kinetic term of the Hamiltonian the electron-spin is conserved. Thus, writing the many-body vector as a matrix v(iup,idn), where the indices label spin-configurations, we find that the hopping term only connects vector elements that differ in one index. Hence, storing entire slices v(iup,:) on one node, the kinetic term for the spin-down electrons is local to that process. After transposing v, the same is true for the hopping of the spin-up electrons. With an efficient matrix transpose, implemented via MPI collectives, we thus obtain a version of the Lanczos method that efficiently well on JUBL, Jülich's Blue Gene/L. Using the code we treated two-orbital Hubbard Models which show interesting Mott-band insulator transitions. Kohn's criterium which determines the phase of a system by ground-state properties alone, namely the response of the ground-state energies to a vector potential, was employed. This vector potential can be equivalently regarded as a change in boundary conditions, requiring complex wave vectors. We found that half-filled systems become metallic in the region between a band and Mott insulating phase. To extrapolate to infinite system sizes and therefore getting access to an arbitrarily high resolution in reciprocal space, we use cluster perturbation theory (CPT). It works by diagonalizing a finite cluster exactly and then treating the hopping between identical clusters perturbatively in strong coupling perturbation theory. This leads to an effectively infinite chain. Due to our efficient Lanczos implementation it is possible to calculate relatively large clusters, minimizing finite-size effects. The combination of our Lanczos code and CPT allowed for a treatment of the the organic metal TTF-TCNQ. We were able to solve a long-standing problem in the interpretation of its experimental spectra. Being an interesting one-dimensional metal TTF-TCNQ has been studied thoroughly for more than 30 years. Theoretical estimates as well as experimental measurements suggest values for the parameters of the Hubbard model t and U. With these values Hubbard model calculations show qualitatively the same features as APRES experiments, namely the signatures of spin-charge separation at the Gamma-point. But in order to fit the experimental data to the numerical calculations t has to be doubled, which translates into a broadening of the spectra. This ad-hoc adjustment is not only unsatisfactory but also the temperature dependence of the spectral function is incorrectly reproduced. We resolved this problem by properly including nearest-neighbor Coulomb interactions. Recent DFT calculations show that indeed the commonly used values of t and U are correct. But Coulomb interactions between electrons on neighboring sites are also significant and must not be neglected. Thus, we studied the effect of the next-neighbor interaction parameter V being at about U/2. Including V in our Lanczos calculations for obtaining the angular-integrated spectral function shows that its effect is to broaden the spectra, comparable to increasing the value of t. Moreover, we found that doubling t indeed mimics the effect of Vapprox U/2. Surprisingly, the broadening of the spectrum as a result of V can be understood in first-order Rayleigh-Schrödinger perturbation theory.
The solution of an auxiliary quantum impurity system is the computationally expensive step in dynamical mean field theory simulations of lattice models and materials. In this review, we discuss Monte Carlo based impurity solvers, which are suitable for a wide range of applications. In particular, we present an efficient implementation of the hybridization expansion approach, which enables the simulation of multiorbital impurity problems with off-diagonal and complex hybridizations, and dynamically screened (retarded) density-density interactions. As a complementary approach, we discuss an impurity solver based on the determinant Monte Carlo method, which scales favorably with inverse temperature and hence provides access to the very low temperature regime. The usefulness of these state-of-the-art impurity solvers is demonstrated with applications to the downfolding problem, i.e., the systematic derivation of dynamically screened interactions for low-energy effective models, and to pyrochlore iridates, where the spin-orbit coupling leads to complex hybridization functions in a multi-orbital system. As a benchmark for cluster extensions of dynamical mean field theory, we also present results from lattice Monte Carlo simulations for the momentum dependence of the pseudo-gap in the half-filled two-dimensional Hubbard model.
This chapter, like the previous one, describes quantum Monte Carlo methods for the simulation of a system of interacting electrons at nonzero temperature. Here, we target a particular class of problems, characterized by a small number of correlated sites or orbitals, that we call impurity problems. While we could apply the methods of the previous chapter to such impurity models, the algorithms discussed here are more advantageous not only because they work in continuous imaginary time and hence lack the Trotter approximation error inherent to the previous methods, but also because they manipulate determinants of matrices of smaller size, which can be handled more efficiently. At the same time, the continuous-time approach is applicable to broader classes of impurity problems. Driving the development of these continuous-time impurity solvers was the desire to perform more efficient simulations of correlated lattice models within the framework of dynamical mean-field theory. This formalism maps the lattice problem onto an impurity problem, whose parameters are determined by a self-consistency condition. In some cases, the self-consistent solution gives a good approximation of the properties of the original lattice problem. When we combine this dynamical mean-field approximation with the local-density approximation for electronic structure calculations, we obtain a powerful scheme for electronic structure calculations of strongly correlated materials. In this chapter we discuss the most important classes of impurity models, sketch the basics of the dynamical mean-field approximation, and detail the different variants of continuous-time impurity solvers. Quantum impurity models A quantum impurity model describes an atom or molecule embedded in some host with which it exchanges electrons, spin, and energy. This exchange allows the impurity to make transitions between different quantum states, and in the presence of interactions within or between the impurity orbitals, these transitions lead to nontrivial dynamical properties. Quantum impurity models play a prominent role, for example, in the theoretical description of the magnetic and electric properties of dilute metal alloys and in theoretical studies of quantum dots and molecular conductors. These models also appear as an auxiliary problem whose solution yields the dynamical mean-field description of correlated lattice models.
Nanostructures with open shell transition metal or molecular constituents host often strong electronic correlations and are highly sensitive to atomistic material details. This tutorial review discusses method developments and applications of theoretical approaches for the realistic description of the electronic and magnetic properties of nanostructures with correlated electrons. First, the implementation of a flexible interface between density functional theory and a variant of dynamical mean field theory (DMFT) highly suitable for the simulation of complex correlated structures is explained and illustrated. On the DMFT side, this interface is largely based on recent developments of quantum Monte Carlo and exact diagonalization techniques allowing for efficient descriptions of general four fermion Coulomb interactions, reduced symmetries and spin-orbit coupling, which are explained here. With the examples of the Cr (001) surfaces, magnetic adatoms, and molecular systems it is shown how the interplay of Hubbard U and Hund’s J determines charge and spin fluctuations and how these interactions drive different sorts of correlation effects in nanosystems. Non-local interactions and correlations present a particular challenge for the theory of low dimensional systems. We present our method developments addressing these two challenges, i.e., advancements of the dynamical vertex approximation and a combination of the constrained random phase approximation with continuum medium theories. We demonstrate how non-local interaction and correlation phenomena are controlled not only by dimensionality but also by coupling to the environment which is typically important for determining the physics of nanosystems.
Predicting the physical properties of strongly correlated electron systems (SCES) requires the combined Density Functional Theory and Dynamical Mean-Field Theory (DFT+DMFT) framework. However, the computational bottle-neck lies in the impurity solver. This work focuses on rigorously benchmarking the Multi-Orbital Iterated Perturbation Theory (MO-IPT) impurity solver against the numerically exact Continuous-Time Quantum Monte Carlo (CTQMC) method, specifically through a detailed study of the three-orbital Hubbard model. The two-orbital model serves as a canonical system to test performance under realistic multi-band conditions. Our systematically derived MO-IPT overcomes the critical limitations of CTQMC, including exponential scaling with orbital number, the need for analytic continuation, and the severe fermionic sign problem. By demonstrating high accuracy across varying interaction strengths and temperatures, we validate MO-IPT as an inexpensive, yet reliable, replacement solver essential for future material-specific investigations.