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Efficient real-frequency solvers for DMFT From optimal bath representation to exact results

Manuel Zingl, M. C. Nuss, Daniel Bauernfeind, Robert Triebl, Hans Gerd Evertz, Markus Aichhorn

Published Jul 20, 2017• 0 citations
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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.

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Truncated configuration interaction expansions as solvers for correlated quantum impurity models and dynamical mean-field theory

Dominika Zgid, Emanuel Gull, Garnet Kin‐Lic Chan

The development of polynomial cost solvers for correlated quantum impurity models, with controllable errors, is a central challenge in quantum many-body physics, where these models find applications ranging from nanoscience to the dynamical mean-field theory (DMFT). Here, we describe how configuration interaction (CI) approximations to exact diagonalization (ED) may be used as solvers in DMFT. CI approximations retain the main advantages of ED, such as the ability to treat general interactions and off-diagonal hybridizations and to obtain real spectral information, but are of polynomial cost. Furthermore, their errors can be controlled by monitoring the convergence of physical quantities as a function of the CI hierarchy. Using benchmark DMFT applications, such as single-site DMFT of the one-dimensional Hubbard model and $2\ifmmode\times\else\texttimes\fi{}2$ cluster DMFT of the two-dimensional Hubbard model, we show that CI approximations allow us to obtain near-exact ED results for a tiny fraction of the cost. This is true over the entire range of interaction strengths including ``difficult'' regimes, such as in the pseudogap phase of the two-dimensional Hubbard model. We use the ability of CI approximations to treat large numbers of orbitals to demonstrate convergence of the bath representation in the $2\ifmmode\times\else\texttimes\fi{}2$ cluster DMFT using a 24-bath orbital representation. CI approximations thus form a promising route to extend ED to problems that are currently difficult to study using other solvers such as continuous-time quantum Monte Carlo, including impurity models with large numbers of orbitals and general interactions.

arXiv:1203.1914•Physical Review B•Oct 22, 2012•134 citations
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.
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.
The development of polynomial cost solvers for correlated quantum impurity models, with controllable errors, is a central challenge in quantum many-body physics, where these models find applications ranging from nanoscience to the dynamical mean-field theory (DMFT). Here, we describe how configuration interaction (CI) approximations to exact diagonalization (ED) may be used as solvers in DMFT. CI approximations retain the main advantages of ED, such as the ability to treat general interactions and off-diagonal hybridizations and to obtain real spectral information, but are of polynomial cost. Furthermore, their errors can be controlled by monitoring the convergence of physical quantities as a function of the CI hierarchy. Using benchmark DMFT applications, such as single-site DMFT of the one-dimensional Hubbard model and $2\ifmmode\times\else\texttimes\fi{}2$ cluster DMFT of the two-dimensional Hubbard model, we show that CI approximations allow us to obtain near-exact ED results for a tiny fraction of the cost. This is true over the entire range of interaction strengths including ``difficult'' regimes, such as in the pseudogap phase of the two-dimensional Hubbard model. We use the ability of CI approximations to treat large numbers of orbitals to demonstrate convergence of the bath representation in the $2\ifmmode\times\else\texttimes\fi{}2$ cluster DMFT using a 24-bath orbital representation. CI approximations thus form a promising route to extend ED to problems that are currently difficult to study using other solvers such as continuous-time quantum Monte Carlo, including impurity models with large numbers of orbitals and general interactions.