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Rigorous multi-orbital implementation of rotational-invariant slave-boson theory and its application to Hund's physics

Christoph Piefke, Frank Lechermann

Published Aug 10, 2017• 0 citations
OpenAlex

Abstract

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.

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Rigorous symmetry adaptation of multiorbital rotationally invariant slave-boson theory with application to Hund's rules physics

Christoph Piefke, Frank Lechermann

The theory of correlated electron systems on a lattice proves notoriously complicated because of the exponential growth of Hilbert space. Mean-field approaches provide valuable insight when the self-energy has a dominant local structure. Additionally, the extraction of effective low-energy theories from the generalized many-body representation is highly desirable. In this respect, the rotational-invariant slave-boson (RISB) approach in its mean-field formulation enables versatile access to correlated lattice problems. However, in its original form, due to numerical complexity, the RISB approach is limited to about three correlated orbitals per lattice site. We thus present a thorough symmetry-adapted advancement of RISB theory, suited to efficiently deal with multiorbital Hubbard Hamiltonians for complete atomic-shell manifolds. It is utilized to study the intriguing problem of Hund's physics for three- and especially five-orbital manifolds on the correlated lattice, including crystal-field terms as well as spin-orbit interaction. The well-known Janus-face phenomenology, i.e., strengthening of correlations at smaller-to-intermediate Hubbard $U$ accompanied by a shift of the Mott transition to a larger $U$ value, has a stronger signature and more involved multiplet resolution for five-orbital problems. Spin-orbit interaction effectively reduces the critical local interaction strength and weakens the Janus-face behavior. Application to the realistic challenge of Fe chalcogenides underlines the subtle interplay of the orbital degrees of freedom in these materials.

arXiv:1708.03191•Physical review. B./Physical review. B•Mar 30, 2018•21 citations

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

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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 ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ 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.

arXiv:2305.11128•Physical review. B./Physical review. B•Dec 19, 2023•14 citations

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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.

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Strongly correlated electrons on the decorated honeycomb lattice studied with rotationally invariant slave-boson mean-field theory

H. L. Nourse

The single-orbital Hubbard model is often a good description of the low energy dynamics of strongly correlated electron systems. Importantly, the single-orbital Hubbard model at half-filling is able to capture the Mott metal-insulator transition that occurs in many materials. However, there are insulating phases (as well as other exotic phases) that occur due to strong electronic correlations in materials that are away from half-filling. In many cases, these additional phases occur because the constituent electrons of the material come from multi-orbital atomic environments that require more complicated descriptions such as the multi-orbital Hubbard model. In other cases, as we show in this thesis, the rich phase diagram arises from the underlying lattice because of special geometric features.For example, decorated lattices, which often occur in organometallics (e.g., metal-organic frameworks (MOFs)), have geometrical complexity due to the molecular structures that decorate the lattice. We argue that the effective description for many of these materials is the single-orbital Hubbard model. Yet, these materials still have rich phase diagrams because the molecular structures of the material cause effective multi-orbital interactions generated from the local Coulomb interaction.We investigate the single-orbital Hubbard model on the decorated honeycomb lattice at zero temperature using rotationally invariant slave-boson (RISB) theory in the mean-field approximation, at a range of fillings. Numerous metal-insulator transitions occur - the insulating phases include a real-space Mott insulator, a dimer Mott insulator, a trimer Mott insulator, flat-band ferromagnets, and a spin-1 Mott insulator, dominated by spin-triplet fluctuations. This menagerie of phases is caused by effective multi-orbital interactions arising from the interplay of electronic correlations, lattice structure, and band filling effects. For example, with only on-site Coulomb interactions, the spin-1 Mott insulator is driven by an effective Hund's coupling on a three-site ring that favors the formation of triplets when the three-site ring has four electrons. We argue that this exotic phase may be realized in layers of the organometallic molecular crystal Mo3S7(dmit)3.

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Operatorial formulation of the ghost rotationally invariant slave-boson theory

Nicola Lanatà

We propose an operatorially exact formalism for describing the equilibrium and quantum-dynamical properties of many-electron systems interacting locally on a lattice, called ``ghost rotationally invariant slave-boson theory'' (g-RISB). We demonstrate that our theoretical framework reduces to the recently developed ghost Gutzwiller approximation (g-GA) at the mean-field level. Furthermore, we introduce the time-dependent mean-field g-RISB action, generalizing the time-dependent GA theory. Since the g-RISB is based on exact reformulation of the many-body problem, it may pave the way to the development of practical implementations for adding systematically quantum-fluctuation corrections towards the exact solution, in arbitrary dimension.

arXiv:2109.10320•Physical review. B./Physical review. B•Jan 5, 2022•10 citations
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.
The theory of correlated electron systems on a lattice proves notoriously complicated because of the exponential growth of Hilbert space. Mean-field approaches provide valuable insight when the self-energy has a dominant local structure. Additionally, the extraction of effective low-energy theories from the generalized many-body representation is highly desirable. In this respect, the rotational-invariant slave-boson (RISB) approach in its mean-field formulation enables versatile access to correlated lattice problems. However, in its original form, due to numerical complexity, the RISB approach is limited to about three correlated orbitals per lattice site. We thus present a thorough symmetry-adapted advancement of RISB theory, suited to efficiently deal with multiorbital Hubbard Hamiltonians for complete atomic-shell manifolds. It is utilized to study the intriguing problem of Hund's physics for three- and especially five-orbital manifolds on the correlated lattice, including crystal-field terms as well as spin-orbit interaction. The well-known Janus-face phenomenology, i.e., strengthening of correlations at smaller-to-intermediate Hubbard UUU accompanied by a shift of the Mott transition to a larger UUU value, has a stronger signature and more involved multiplet resolution for five-orbital problems. Spin-orbit interaction effectively reduces the critical local interaction strength and weakens the Janus-face behavior. Application to the realistic challenge of Fe chalcogenides underlines the subtle interplay of the orbital degrees of freedom in these materials.
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 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 single-orbital Hubbard model is often a good description of the low energy dynamics of strongly correlated electron systems. Importantly, the single-orbital Hubbard model at half-filling is able to capture the Mott metal-insulator transition that occurs in many materials. However, there are insulating phases (as well as other exotic phases) that occur due to strong electronic correlations in materials that are away from half-filling. In many cases, these additional phases occur because the constituent electrons of the material come from multi-orbital atomic environments that require more complicated descriptions such as the multi-orbital Hubbard model. In other cases, as we show in this thesis, the rich phase diagram arises from the underlying lattice because of special geometric features.For example, decorated lattices, which often occur in organometallics (e.g., metal-organic frameworks (MOFs)), have geometrical complexity due to the molecular structures that decorate the lattice. We argue that the effective description for many of these materials is the single-orbital Hubbard model. Yet, these materials still have rich phase diagrams because the molecular structures of the material cause effective multi-orbital interactions generated from the local Coulomb interaction.We investigate the single-orbital Hubbard model on the decorated honeycomb lattice at zero temperature using rotationally invariant slave-boson (RISB) theory in the mean-field approximation, at a range of fillings. Numerous metal-insulator transitions occur - the insulating phases include a real-space Mott insulator, a dimer Mott insulator, a trimer Mott insulator, flat-band ferromagnets, and a spin-1 Mott insulator, dominated by spin-triplet fluctuations. This menagerie of phases is caused by effective multi-orbital interactions arising from the interplay of electronic correlations, lattice structure, and band filling effects. For example, with only on-site Coulomb interactions, the spin-1 Mott insulator is driven by an effective Hund's coupling on a three-site ring that favors the formation of triplets when the three-site ring has four electrons. We argue that this exotic phase may be realized in layers of the organometallic molecular crystal Mo3S7(dmit)3.
We propose an operatorially exact formalism for describing the equilibrium and quantum-dynamical properties of many-electron systems interacting locally on a lattice, called ``ghost rotationally invariant slave-boson theory'' (g-RISB). We demonstrate that our theoretical framework reduces to the recently developed ghost Gutzwiller approximation (g-GA) at the mean-field level. Furthermore, we introduce the time-dependent mean-field g-RISB action, generalizing the time-dependent GA theory. Since the g-RISB is based on exact reformulation of the many-body problem, it may pave the way to the development of practical implementations for adding systematically quantum-fluctuation corrections towards the exact solution, in arbitrary dimension.