A recently developed continuous time solver based on an expansion in hybridization about an exactly solved local limit is reformulated in a manner appropriate for general classes of quantum impurity models including spin exchange and pair hopping terms. The utility of the approach is demonstrated via applications to the dynamical mean field theory of the Kondo lattice and two-orbital models. The algorithm can handle low temperatures and strong couplings without encountering a sign problem.
A recently developed continuous time solver based on an expansion in hybridization about an exactly solved local limit is reformulated in a manner appropriate for general classes of quantum impurity models including spin exchange and pair hopping terms. The utility of the approach is demonstrated via applications to the dynamical mean field theory of the Kondo lattice and two-orbital models. The algorithm can handle low temperatures and strong couplings without encountering a sign problem.
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.
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.
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.
We present a new continuous-time solver for quantum impurity models such as those relevant to dynamical mean field theory. It is based on a stochastic sampling of a perturbation expansion in the impurity-bath hybridization parameter. Comparisons with Monte Carlo and exact diagonalization calculations confirm the accuracy of the new approach, which allows very efficient simulations even at low temperatures and for strong interactions. As examples of the power of the method we present results for the temperature dependence of the kinetic energy and the free energy, enabling an accurate location of the temperature-driven metal-insulator transition.
We present an implementation of the hybridization expansion impurity solver which employs sparse-matrix exact-diagonalization techniques to compute the time evolution of the local Hamiltonian. This method avoids computationally expensive matrix-matrix multiplications and becomes advantageous over the conventional implementation for models with five or more orbitals. In particular, this method will allow the systematic investigation of 7-orbital systems (lanthanide and actinide compounds) within single-site dynamical mean-field theory. We illustrate the power and usefulness of our approach with dynamical mean-field results for a 5-orbital model which captures some aspects of the physics of the iron-based superconductors.
Multiorbital 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 we explore its application to typical single- and multiorbital problems encountered in investigations of impurity and lattice models. Our implementation generically outperforms bare and bold-line quantum Monte Carlo algorithms in the interaction expansion. For the systems studied here, our implementation remains inferior to the more specialized hybridization expansion and auxiliary field algorithms. The problem of convergence to unphysical fixed points, which hampers so-called bold-line methods, is not encountered in inchworm Monte Carlo.