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arXiv:1710.07773 · Physical review. B./Physical review. B

Dynamical mean-field theory, density-matrix embedding theory, and rotationally invariant slave bosons: A unified perspective

Thomas Ayral, Tsung-Han Lee, Gabriel Kotliar

Published Dec 26, 2017• 44 citations
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Abstract

We present a unified perspective on dynamical mean-field theory (DMFT), density-matrix embedding theory (DMET), and rotationally invariant slave bosons (RISB). We show that DMET can be regarded as a simplification of the RISB method where the quasiparticle weight is set to unity. This relation makes it easy to transpose extensions of a given method to another: For instance, a temperature-dependent version of RISB can be used to derive a temperature-dependent free-energy formula for DMET.

DOI
10.1103/physrevb.96.235139

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We present a unified perspective on dynamical mean-field theory (DMFT), density-matrix embedding theory (DMET), and rotationally invariant slave bosons (RISB). We show that DMET can be regarded as a simplification of the RISB method where the quasiparticle weight is set to unity. This relation makes it easy to transpose extensions of a given method to another: For instance, a temperature-dependent version of RISB can be used to derive a temperature-dependent free-energy formula for DMET.
We present detailed benchmark ground-state calculations of the one- and two-dimensional Hubbard model utilizing the cluster extensions of the rotationally invariant slave-boson mean-field theory and the density matrix embedding theory. Our analysis shows that the overall accuracy and the performance of these two methods are very similar. Furthermore, we propose a unified computational framework that allows us to implement both of these techniques on the same footing. This provides us with a different line of interpretation and paves the ways for developing systematically distinct generalizations of these complementary approaches.
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.
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.
Dynamical and variational frameworks have long been viewed as distinct paradigms. In particular, in quantum embedding (QE) frameworks, dynamical mean-field theory (DMFT) captures nonperturbative dynamical correlations through a frequency-dependent self-energy, while the Gutzwiller approximation (GA) is formulated in terms of a variationally optimized ground-state wavefunction. Here we bridge these perspectives, proving that the ghost-Gutzwiller approximation (ghost-GA), which also admits a density-matrix-matching QE formulation known as ghost density matrix embedding theory (ghost-DMET), becomes strictly equivalent to DMFT in the limit of infinitely many auxiliary bath modes. This formal unification has immediate consequences. In particular, it yields a rigorous finite-temperature extension of ghost-GA and shows that the physical Green's function can be determined from static expectation values of the embedding Hamiltonians, providing a route to computational studies of competing phases in strongly correlated matter with DMFT-level accuracy, while bypassing the need to calculate dynamical spectra with conventional impurity solvers. More broadly, it shows that the variational ghost-GA, the density-matrix-matching ghost-DMET formulation, and the dynamical DMFT description are not separate constructions, but complementary formulations of the same QE structure, thereby providing a concrete formal basis for future controlled extensions beyond DMFT.
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.