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arXiv:1311.5801 · Physical Review B

Self-energy and vertex functions from hybridization-expansion continuous-time quantum Monte Carlo for impurity models with retarded interaction

Hartmut Hafermann

Published Jun 24, 2014• 52 citations
arXivHTMLPDF

Abstract

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 the 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 the presence of plasmonic screening. In the antiadiabatic regime, the self-energy is strongly renormalized but retains features of the low energy scale set by the screened interaction. An explicit expression for its high-frequency behavior is provided. Across the screening driven and interaction driven metal-insulator transitions, the vertex functions are found to exhibit similar structural changes, which are hence identified as generic features of the Mott transition.

DOI
10.1103/physrevb.89.235128

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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 the 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 the presence of plasmonic screening. In the antiadiabatic regime, the self-energy is strongly renormalized but retains features of the low energy scale set by the screened interaction. An explicit expression for its high-frequency behavior is provided. Across the screening driven and interaction driven metal-insulator transitions, the vertex functions are found to exhibit similar structural changes, which are hence identified as generic features of the Mott transition.
We propose efficient measurement procedures for the self-energy and vertex function of the Anderson impurity model within the hybridization expansion continuous-time quantum Monte Carlo algorithm. The method is based on the measurement of higher-order correlation functions related to the quantities being sought through the equation of motion, a technique previously introduced in the numerical renormalization-group context. For the case of interactions of density-density type, the additional correlators can be obtained at essentially no additional computational cost. In combination with a recently introduced method for filtering the Monte Carlo noise using a representation in terms of orthogonal polynomials, we obtain data with unprecedented accuracy. This leads to an enhanced stability in analytical continuations of the self-energy or in two-particle-based theories such as the dual fermion approach. As an illustration of the method we reexamine the previously reported spin-freezing and high-spin to low-spin transitions in a two-orbital model with density-density interactions. In both cases, the vertex function undergoes significant changes, which suggests significant corrections to the dynamical mean-field solutions in dual fermion calculations.
In this paper, we present an efficient and stable method to determine the one-particle Green's function in the hybridization-expansion continuous-time (CT-HYB) quantum Monte Carlo method within the framework of the dynamical mean-field theory. The high-frequency tail of the impurity self-energy is replaced with a noise-free function determined by a dual-expansion around the atomic limit. This method does not depend on the explicit form of the interaction term. More advantageous, it does not introduce any additional numerical cost to the CT-HYB simulation. We discuss the symmetries of the two-particle vertex, which can be used to optimize the simulation of the four-point correlation functions in the CT-HYB. Here, we adopt it to accelerate the dual-expansion calculation, which turns out to be especially suitable for the study of material systems with complicated band structures. As an application, a two-orbital Anderson impurity model with a general on-site interaction form is studied. The phase diagram is extracted as a function of the Coulomb interactions for two different Hund's coupling strengths. In the presence of the hybridization between different orbitals, for smaller interaction strengths, this model shows a transition from metal to band-insulator. Increasing the interaction strengths, this transition is replaced by a crossover from Mott-insulator to band-insulator behavior.
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.
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.
Dynamical screening of the Coulomb interactions in correlated electron systems results in a low-energy effective problem with a dynamical Hubbard interaction $\mathcal{U}(\ensuremath{\omega})$. We propose a Green's function ansatz for the Anderson impurity problem with retarded interactions, in which the Green's function factorizes into a contribution stemming from an effective static-UUU problem and a bosonic high-energy part introducing collective plasmon excitations. Our approach relies on the scale separation of the low-energy properties, related to the instantaneous static UUU, from the intermediate- to high-energy features originating from the retarded part of the interaction. We argue that for correlated materials where retarded interactions arise from downfolding higher-energy degrees of freedom, the characteristic frequencies are typically in the antiadiabatic regime. In this case, accurate approximations to the bosonic factor are relatively easy to construct, with the most simple being the boson factor of the dynamical atomic-limit problem. We benchmark the quality of our method against numerically exact continuous-time quantum Monte Carlo results for the Anderson-Holstein model both at half- and quarter-filling. Furthermore, we study the Mott transition within the Hubbard-Holstein model within extended dynamical mean field theory. Finally, we apply our technique to a realistic three-band Hamiltonian for SrVO3{}_{3}3​. We show that our approach reproduces both the effective mass renormalization and the position of the lower Hubbard band by means of a dynamically screened UUU, previously determined ab initio within the constrained random phase approximation. Our approach could also be used within schemes beyond dynamical mean field theory, opening a quite general way of describing satellites and plasmon excitations in correlated materials.
We extend the continuous-time interaction-expansion quantum Monte Carlo method with respect to measuring observables for fermion-boson lattice models. Using generating functionals, we express expectation values involving boson operators, which are not directly accessible because simulations are done in terms of a purely fermionic action, as integrals over fermionic correlation functions. We also demonstrate that certain observables can be inferred directly from the vertex distribution, and present efficient estimators for the total energy and the phonon propagator of the Holstein model. Furthermore, we generalize the covariance estimator of the fidelity susceptibility, an unbiased diagnostic for phase transitions, to the case of retarded interactions. The new estimators are applied to half-filled spinless and spinful Holstein models in one dimension. The observed renormalization of the phonon mode across the Peierls transition in the spinless model suggests a soft-mode transition in the adiabatic regime. The critical point is associated with a minimum in the phonon kinetic energy and a maximum in the fidelity susceptibility.