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 state 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 exact-diagonalization based method for quantum impurity problems with more than 12 orbitals. The second idea is an improved Monte Carlo sampling scheme which is applicable to all variants of the hybridization expansion method. We show that this so-called sliding window sampling scheme speeds up the simulation by at least an order of magnitude for a broad range of model parameters, with the largest improvements at low temperature.
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 state 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 exact-diagonalization based method for quantum impurity problems with more than 12 orbitals. The second idea is an improved Monte Carlo sampling scheme which is applicable to all variants of the hybridization expansion method. We show that this so-called sliding window sampling scheme speeds up the simulation by at least an order of magnitude for a broad range of model parameters, with the largest improvements at low temperature.
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.
The Grassmann time-evolving matrix product operator method has shown great potential as a general-purpose quantum impurity solver, as its numerical errors can be well controlled and it is flexible to be applied on both the imaginary- and real-time axes. However, a major limitation of it is that its computational cost grows exponentially with the number of impurity flavors. In this work, we propose a multiflavor extension of it to overcome this limitation. The key insight is that to calculate multitime correlation functions on one or a few impurity flavors, one could integrate out the degrees of freedom of the rest flavors beforehand, which could greatly simplify the calculation. The idea is particularly effective for quantum impurity problems with diagonal hybridization function, i.e., each impurity flavor is coupled to an independent bath, a setting which is commonly used in the field. We demonstrate the accuracy and scalability of our method for the imaginary-time evolution of impurity problems with up to three impurity orbitals, i.e., six flavors, and benchmark our results against continuous-time quantum Monte Carlo calculations. Our method paves the way of scaling up tensor network algorithms to solve large-scale quantum impurity problems.
Due to the intrinsic complexity of the quantum many-body problem, quantum Monte Carlo algorithms and their corresponding Monte Carlo configurations can be defined in various ways. Configurations corresponding to few Feynman diagrams often lead to severe sign problems. On the other hand, computing the configuration weight becomes numerically expensive in the opposite limit in which many diagrams are grouped together. Here we show that for continuous-time quantum Monte Carlo in the hybridization expansion the efficiency can be substantially improved by dividing the local impurity trace into fragments, which are then sampled individually. For this technique, which also turns out to preserve the fermionic sign, a modified update strategy is introduced in order to ensure ergodicity. Our (super)state sampling is particularly beneficial to calculations with many d orbitals and general local interactions, such as full Coulomb interaction. For illustration, we reconsider the simple albeit well-known case of a degenerate three-orbital model at low temperatures. This allows us to quantify the coherence properties of the ``spin-freezing'' crossover, even close to the Mott transition.
An impurity model describes an atom or molecule embedded in some host or bath, with which it can exchange electrons. This exchange of electrons allows the impurity to make transitions between different quantum states, and leads to a non-trivial dynamics. Therefore, despite the zero dimensional nature (which makes impurity problems computationally much more tractable than fermionic lattice models), their numerical simulation remains a challenging task. Methods such as exact diagonalization or numerical RG, which explicitly treat a finite number of bath states, work well for single orbital models. However, because the number of bath states must be increased proportional to the number of orbitals, the computational effort grows exponentially with system size, and requires severe truncations of the bath already for two orbitals. Monte Carlo methods have the advantage that the bath is integrated out and thus the (infinite) size of the bath Hilbert space does not affect the simulation. While restricted to finite temperature, Monte Carlo methods are thus the method of choice for the solution of large multi-orbital or cluster impurity problems. Over the last few years, significant progress has been made (both in terms of efficiency and flexibility) with the development of diagrammatic Monte Carlo techniques. These notes provide an overview over two recently developed, complementary methods: (i) the weakcoupling approach, which scales favorably with system size and allows the efficient simulation of large impurity clusters, and (ii) the strong-coupling approach, which can handle impurity
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.
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.