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arXiv:2305.11128 · Physical review. B./Physical review. B

Accuracy of ghost-rotationally-invariant slave-boson theory for multiorbital Hubbard models and realistic materials

Tsung-Han Lee, Corey Melnick, Ran Adler, Nicola Lanatà, Gabriel Kotliar

Published Dec 19, 2023• 14 citations
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Abstract

We assess the accuracy of ghost-rotationally-invariant slave-boson (g-RISB) theory in multiorbital systems by applying it to both the three-orbital degenerate Hubbard model and a realistic ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ model extracted from first-principle simulations and comparing the results to those obtained using dynamical mean-field theory (DMFT). Our findings indicate that g-RISB's accuracy can be systematically improved toward the exact DMFT limit in infinite-dimensional multiorbital models by increasing the number of ghost orbitals. This allows for a more precise description of aspects of Hund metal physics and Mott physics compared with the original RISB approach. We also demonstrate that g-RISB reliably captures the quasiparticle weights, Fermi surface, and low-energy spectral function for the realistic ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ model compared with DMFT. Moreover, we showcase the potential of using the density matrix renormalization group method as an impurity solver within the g-RISB framework to study systems with a larger number of ghost orbitals. These results show the potential of g-RISB as a reliable tool for simulating correlated materials. The connection between the g-RISB and DMFT self-energy is also discussed.

DOI
10.1103/physrevb.108.245147

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We assess the accuracy of ghost-rotationally-invariant slave-boson (g-RISB) theory in multiorbital systems by applying it to both the three-orbital degenerate Hubbard model and a realistic Sr2RuO4{\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}Sr2​RuO4​ model extracted from first-principle simulations and comparing the results to those obtained using dynamical mean-field theory (DMFT). Our findings indicate that g-RISB's accuracy can be systematically improved toward the exact DMFT limit in infinite-dimensional multiorbital models by increasing the number of ghost orbitals. This allows for a more precise description of aspects of Hund metal physics and Mott physics compared with the original RISB approach. We also demonstrate that g-RISB reliably captures the quasiparticle weights, Fermi surface, and low-energy spectral function for the realistic Sr2RuO4{\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}Sr2​RuO4​ model compared with DMFT. Moreover, we showcase the potential of using the density matrix renormalization group method as an impurity solver within the g-RISB framework to study systems with a larger number of ghost orbitals. These results show the potential of g-RISB as a reliable tool for simulating correlated materials. The connection between the g-RISB and DMFT self-energy is also discussed.
We compare the accuracy of the ghost rotationally invariant slave-boson (g-RISB) theory and dynamical mean field theory (DMFT) on the single-band Hubbard model, as a function of the number of bath sites in the embedding impurity Hamiltonian. Our benchmark calculations confirm that the accuracy of g-RISB can be systematically improved by increasing the number of bath sites, similar to DMFT. With a few bath sites, we observe that g-RISB is systematically more accurate than DMFT for the ground-state observables. On the other hand, the relative accuracy of these methods is generally comparable for the quasiparticle weight and the spectral function. As expected, we observe that g-RISB satisfies the variational principle in infinite dimensions, as the total energy decreases monotonically towards the exact value as a function of the number of bath sites, suggesting that the g-RISB wave function may approach the exact ground state in infinite dimensions. Our results suggest that the g-RISB is a promising method for first-principles simulations of strongly correlated matter, which can capture the behavior of both static and dynamical observables, at a relatively low computational cost.
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We develop a generalization of the ghost-rotationally-invariant slave-boson (ghost-RISB) method that incorporates local phonon modes coupled to arbitrary on-site electronic degrees of freedom, enabling a nonperturbative treatment of electron-electron and electron-phonon interactions within an efficient variational framework. The method extends the ghost-orbital construction to capture dynamical self-energy effects and phonon-induced renormalizations beyond static slave-boson approaches. Benchmarking against dynamical mean-field theory (DMFT) results for the Hubbard-Holstein model, we find excellent quantitative agreement for electron quasiparticle weights and phonon properties across a wide range of coupling strengths, and it captures accurately the competition between electron-electron and electron-phonon interactions. We show that the inclusion of the ghost orbitals is crucial to accurately describe the regime of low-frequency, strongly dynamical, phonons. The extended ghost-RISB achieves this accuracy at a fraction of the computational cost of DMFT, due to its self-consistency rooted in static observables instead of dynamical ones, enabling rapid exploration of correlated electron-phonon phase diagrams. We exploit this advantage to characterize the most demanding regime of strong coupling and adiabatic phonons. Our analysis shows a suppression of the superconducting order parameter, which is interpreted as a Franck-Condon-like reduction of the overlap between the phonon wavefunctions associated with empty and doubly-occupied sites in the bipolaronic regime.
The theoretical and numerical description of correlated electron systems on a lattice proves notoriously complicated because of the exponential growth of Hilbert space, paired with the limitations of Monte-Carlo techniques due to the fermion-sign problem. Mean-field approaches such as dynamical-mean field theory (DMFT) provide valuable insight when the self-energy has a dominant local structure. But especially for larger orbital manifolds and complicated local Hamiltonians, also the DMFT performance has still its limitations. Furthermore, the generalized many-body representation renders the extraction of efficient low-energy theories often difficult. The rotational-invariant slave boson (RISB) approach in its mean-field version enables a simplified alternative access to correlated lattice electrons that may bridge these gaps. We present a thorough symmetry-adapted advancement of RISB theory, suited to deal with manifest multi-orbital challenges. Albeit the focus of this work is on the method, illustrative examples in view of Hund's physics in 3- and 5-orbital problems, including crystal-field terms as well as spin-orbit interaction, are enclosed.
We propose a finite-temperature Gaussian-fluctuation formalism for response functions within ghost rotationally invariant slave-boson (gRISB) theory and apply it to the two-dimensional Hubbard model. The gRISB magnetic phase boundary and static spin and charge susceptibilities capture the trends of dynamical mean-field theory (DMFT), significantly improving upon standard RISB and the random phase approximation (RPA). At the RISB level, our formalism recovers the Kotliar-Ruckenstein response. The dynamical susceptibilities resolve paramagnons, low-energy sound modes, and high-energy Hubbard excitations. Because they are evaluated directly on the real-frequency axis, the results avoid Matsubara truncation, stochastic sampling, and numerical analytic continuation. We also introduce a conserving quasiparticle approximation that expresses the response in terms of dressed Green's functions and vertex corrections.
Dynamical mean-field theory (DMFT) is a cornerstone technique for studying strongly correlated electronic systems. However, each DMFT step is computationally demanding, and many iterations can be required to achieve convergence. Here, we accelerate the convergence of DMFT by initializing its self-consistent cycle with solutions from computationally cheaper and more approximate methods. We compare the initialization with the non-interacting solution to a range of quantum embedding compatible approaches: Hartree-Fock, the Hubbard-I approximation, rotationally invariant slave bosons (RISB), and its ghost extension (g-RISB). We find that these initializations can reduce the number of DMFT iterations by up to an order of magnitude, with g-RISB providing the most effective and reliable benefits. In most regimes, initializing with g-RISB and performing a single DMFT iteration suffices to recover the full dynamical structure. The improvement in convergence is controlled by the initial solution's accuracy in the low-energy part of the self-energy, on the scale of the non-interacting bandwidth. This strategy is especially effective at the Mott insulator-metal transition, where an initialization from the non-interacting limit can lead to a breakdown of DMFT due to the sign problem. Our results establish the usage of accurate yet cheaper quantum embedding methods as a powerful means to substantially reduce the computational cost of DMFT, particularly in regimes where convergence is slow or prone to failure.