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arXiv:1909.07713 · Physical review. B./Physical review. B

Frequency-dependent and algebraic bath states for a dynamical mean-field theory with compact support

Max Nusspickel, George H. Booth

Published Jan 27, 2020• 11 citations
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Abstract

We demonstrate an algebraic construction of frequency-dependent bath orbitals which can be used in a robust and rigorously self-consistent DMFT-like embedding method, here called $\ensuremath{\omega}$-DMFT, suitable for use with Hamiltonian-based impurity solvers. These bath orbitals are designed to exactly reproduce the hybridization of the impurity to its environment, while allowing for a systematic expansion of this bath space as impurity interactions couple frequency points. In this way, the difficult nonlinear fit of bath parameters necessary for many Hamiltonian-formulation impurity solvers in DMFT is avoided, while the introduction of frequency dependence in this bath space is shown to allow for more compact bath sizes. This has significant potential use with a number of new, emerging Hamiltonian solvers which allow for the embedding of large impurity spaces within a DMFT framework. We present results of the $\ensuremath{\omega}$-DMFT approach for the Hubbard model on the Bethe lattice, a one-dimensional chain, and the two-dimensional square lattice, which show excellent agreement with standard DMFT results, with fewer bath orbitals and more compact support for the hybridization representation in the key impurity model of the method.

DOI
10.1103/physrevb.101.045126

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This thesis develops a theoretical framework for quantum embedding based on compact sum-over-poles (SOP) representations of Green's functions, with a particular focus on dynamical mean-field theory (DMFT) formulated close to the real-frequency axis. The work starts from the algorithmic inversion method (AIM-SOP) and generalizes it to many-body lattice systems, proving that any interacting fragment embedded in an arbitrary bath can be represented by an auxiliary Hamiltonian with a finite set of non-interacting bath sites. The author embeds this formalism into a DMFT self-consistency loop formulated on complex contours close to the real axis, complemented by a SOP-based fitting strategy and exact-diagonalization (ED) impurity solvers, constructing a DMFT scheme that does not rely on approximate analytic continuation techniques. In particular, the framework enables us to work on compact (SOP) representations of Green's functions and self-energies alone, and leads to a frequency-independent formulation of DMFT, grounded in effective free energy (Baym-Kadanoff and Klein) functionals. Applications to a 100010001000-site one-dimensional Hubbard ring demonstrate that compact SOP forms reproduce state-of-the-art ED-DMFT results on the Matsubara frequencies and retain key spectral features as the contour approaches the real axis, while also revealing convergence trends and multiple-solution behavior near challenging correlation regimes. Beyond its numerical validation, this approach paves the way for novel variational implementations of DMFT and realistic applications to materials, harnessing its promising ability to resolve fine spectral features.
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We present a time-domain iteration scheme for solving the Dynamical Mean-Field Theory (DMFT) self-consistent equations using retarded Green's functions in real time. Unlike conventional DMFT approaches that operate in imaginary time or frequency space, our scheme operates directly with real-time quantities. This makes it particularly suitable for near-term quantum computing hardware with limited Hilbert spaces, where real-time propagation can be efficiently implemented via Trotterization or variational quantum algorithms. We map the effective impurity problem to a finite one-dimensional chain with a small number of bath sites, solved via exact diagonalization as a proof-of-concept. The hybridization function is iteratively updated through time-domain fitting until self-consistency. We demonstrate stable convergence across a wide range of interaction strengths for the half-filled Hubbard model on a Bethe lattice, successfully capturing the metal-to-insulator transition. Despite using limited time resolution and a minimal bath discretization, the spectral functions clearly exhibit the emergence of Hubbard bands and the suppression of spectral weight at the Fermi level as interaction strength increases. This overcomes major limitations of two-site DMFT approximations by delivering detailed spectral features while preserving efficiency and compatibility with quantum computing platforms through real-time dynamics.
We introduce a real-frequency framework for the quantum embedding of solids, based on sum-over-poles (SOP) representations. We partition the lattice into periodically repeated interacting fragments and derive an embedded Dyson equation for the fragment, to be solved self-consistently. The resulting scheme generalizes the algorithmic-inversion method of our previous work to interacting Hamiltonians and is closely related to dynamical mean-field theory (DMFT), especially when an extended impurity problem is introduced. By exploiting the compact pole expansion of the embedding potential, we carry out the self-consistent procedure close to the real axis, which avoids ill-posed analytic continuation. In addition, the SOP representation allows us to introduce a pole-and-residue dynamics, which we use to reach the self-consistent solution. After deriving the method, we benchmark it against exact-diagonalization implementations of DMFT on the one-dimensional Hubbard ring at half filling. The framework offers a natural route to real-space dynamical quantum embedding, for periodic systems as well as for strongly correlated defects in materials.
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 here present how a self-consistent solution of the dynamical mean field theory equations can be obtained using exact diagonalization of an Anderson impurity model with accuracies comparable to those found using renormalization group or quantum Monte Carlo methods. We show how one can solve a correlated quantum impurity coupled to several hundred uncorrelated bath sites, using a restricted active basis set. The number of bath sites determines the resolution of the obtained spectral function, which consists of peaks roughly spaced by the band width divided by the number of bath sites. The self-consistency cycles are fully performed on the real frequency axis and expressed as numerical stable matrix operations. The same impurity solver has been used on Ligand Field and finite size cluster calculations and is capable of treating involved Hamiltonians including the full rotational invariant Coulomb interaction, spin-orbit coupling and low symmetry crystal-fields. The proposed method allows for the calculation of a variety of correlation functions at little extra cost.