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Chinese Journal of Physics

Renormalization Group Approach to Self-Consistent-Field Approximation for the Interacting Electrons

C.-H. Pao

Published Apr 1, 2000• 0 citations
OpenAlex

Abstract

Due to the retarded nature of the phonon-mediated electron-electron interaction, neither fast Fourier transform (FFT) nor previously developed NRG methods for Hubbard-type purely electronic models are applicable, while brute-force solutions are limited by the demands on computational time and storage which increase rapidly at low temperature T. A new numerical renormalization group (NRG) approach to the solution of self-consistent-field equations for correlated lattice electrons is illustrated for the two-dimensional Hubbard-type model. This approach allows us to handle the fermion frequency matrix inversions in the FLEX equations efficiently. Several orders of magnitude of CPU time and storage can be saved at the temperature scales relevant to electronic phase transitions.

Related papers

Numerical renormalization-group approach to fluctuation exchange in the presence of electron-phonon coupling: Pairing in the Holstein-Hubbard model

C.-H. Pao, H.-B. Schüttler

The fluctuation exchange (FLEX) approximation is applied to study the Holstein-Hubbard model. Due to the retarded nature of the phonon-mediated electron-electron interaction, neither fast Fourier transform nor previously developed numerical renormalization-group (NRG) methods for Hubbard-type purely electronic models are applicable, while brute force solutions are limited by the demands on computational time and storage which increase rapidly at low temperature T. Here, we describe a NRG technique to solve the FLEX equations efficiently. Several orders of magnitude of CPU time and storage can be saved at low T $(\ensuremath{\sim}80\mathrm{K})$. To test our approach, we compare our NRG results to brute force calculations on small lattices at elevated temperatures. Both s-wave and d-wave superconducting phase diagrams are then obtained by applying the NRG approach at low T. The isotope effect for s-wave pairing is BCS-like in a realistic phonon frequency range, but vanishes at unphysically large phonon frequency $(\ensuremath{\sim}\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h})$. For d-wave pairing, the isotope exponent is negative and small compared to the typical observed values in nonoptimally doped cuprates.

arXiv:cond-mat/9809321•Physical review. B, Condensed matter•Jul 1, 1999•5 citations

Renormalization-group acceleration of self-consistent field solutions: Two-dimensional Hubbard model

Chien-Hua Pao, N. E. Bickers

Momentum- and frequency-dependent self-consistent field theories provide a potentially powerful tool for analyzing condensed-matter models for interacting electrons. Such theories generally require the solution of multidimensional integral equations. Solutions are limited by demands on computational time and storage, which increase rapidly for calculation of low-temperature properties. A renormalization-group technique which allows the sequential elimination of regions of high frequency and momentum is presented in detail, and is then applied to the solution of the fluctuation exchange approximation for the two-dimensional Hubbard model. Effective meshes with as many as ${10}^{4}$ frequencies and ${512}^{2}$ k points can be analyzed on a vector supercomputer using this approach. As a sample application, it is demonstrated that the fluctuation exchange approximation exhibits non-Fermi liquid behavior near half filling for sufficiently large interaction strengths.

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Renormalization group approaches to strongly correlated electron systems

Fabian B. Kugler

Strongly correlated electron systems host a plethora of fascinating physical phenomena and pose formidable challenges in their theoretical analysis. The challenges originate from the inherent complexity of the quantum many-body problem - no classical computer will ever be able to fully simulate these systems - and the lack of an effective single-particle picture, as the strong mutual interactions make it impossible to regard the electrons as independent from each other. As a consequence, most systems of correlated electrons can only be tackled approximately and numerically. In this thesis, we develop a set of numerical methods for strongly correlated electrons, which are inspired by the renormalization group (RG) idea of including degrees of freedom successively from high to low energies. This enables an efficient organization of the diverse fluctuations and is key for an accurate treatment of interacting quantum systems, where collective behavior and composite objects emerge at energy scales far below those of the microscopic constituents. In a first part, we consider the functional renormalization group (fRG), a versatile framework to study the flow of correlation functions upon modulating the underlying action. Though widely used, it has often acted more as a qualitative rather than quantitative method, due a nontransparent approximation induced by truncating the hierarchy of flow equations. We develop an iterative multiloop fRG (mfRG) scheme, which ameliorates this approximation and eliminates many of the drawbacks of fRG experienced hitherto. In particular, it restores the independence of results on the choice of RG regulator and establishes a rigorous relation to the parquet formalism. Furthermore, we show how to derive the flow equations directly from self-consistent many-body relations. This establishes a form of diagrammatic resummations at the two-particle level which circumvents ill-behaved two-particle-irreducible vertices. An application to the prototypical two-dimensional Hubbard model illustrates how our multiloop scheme elevates the fRG approach to correlated electron systems to a quantitative level. Secondly, we employ the numerical renormalization group (NRG), based on the iterative diagonalization of impurity Hamiltonians, in conjunction with the dynamical mean-field theory (DMFT) to describe local correlations in multiorbital systems. Having access to arbitrarily low temperatures and energies, NRG is a unique, real-frequency impurity solver for DMFT. It has been pivotal to the understanding of Hund metals, where strong correlations arise from Hund's rules even at moderate Coulomb repulsion. Building on recent methodological advances, we extend the range of application of DMFT+NRG from orbital-degenerate models to more realistic setups: We first study orbital differentiation in a three-orbital Hund-metal model and unravel key effects of the orbital-selective Mott transition. In a real-materials setting, we then incorporate the bandstructure from density functional theory (DFT) and analyze the archetypal Hund-metal material Sr2RuO4. We particularly follow its RG flow to the Fermi-liquid regime at previously inaccessible low temperatures and generally present DFT+DMFT+NRG as a new computational paradigm for strongly correlated materials. As a side project of our fRG work, we develop an algorithm to count Feynman diagrams from closed many-body relations, which reveals the surprising outcome that totally irreducible contributions are responsible for the factorial growth in the number of diagrams. Additionally, we use NRG to study transport through three-level quantum dots and provide benchmark data for other RG methods, which aim at further describing these systems in nonequilibrium.

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The functional renormalization group has become a widely used tool for the analysis of the leading low-temperature correlations in weakly to moderately coupled many-fermion lattice systems. A bottleneck for quantitatively more precise results is that the treatment of the frequency dependence of the flowing interactions is numerically quite demanding. Yet the frequency dependence is needed to compute relevant self-energies and hence for controlled results on the energy scales for ordering or for the quasiparticle properties. Here we explore an approximate parametrization of the frequency dependence of the interaction vertex that is inspired by established simplifications in the theory of superconductivity and that keeps the numerical effort bounded. We demonstrate the validity of the approximation for Cooper pairing problems and apply it to the two-dimensional Hubbard model.

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RENORMALIZATION GROUP APPROACHES TO STRONGLY CORRELATED ELECTRON SYSTEMS

Alex C. Hewson, S. C. BRADLEY, R. Bulla +1 more

In recent years the numerical renormalization group (NRG) method has been extended to the calculation of dynamic response functions and transport properties of magnetic impurity models. The approach can now be applied more widely to lattice models of strongly correlated electron systems by the use of dynamical mean field theory (DMFT), in which the lattice problem is transformed into one for an effective impurity with an additional self-consistency constraint. We review these developments and assess the potential for further applications of this approach. We also discuss an alternative approach to renormalization, renormalized perturbation theory, in which the leading asymptotically exact results for the low temperature regime for a number of magnetic impurity models can be obtained within finite order perturbation theory.

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Improved treatment of retarded self-energy calculations in perturbation theory

S. Wermbter

I present a numerical treatment for a certain class of Feynman diagrams in perturbation theory with respect to the Coulomb correlation U. This method is formally exact for the frequency integrals occurring in the expression for the retarded Green's function. I specify this method to the self-consistent second-order perturbation theory (SOPT) for the single-impurity Anderson model and to the self-consistent fluctuation exchange approximation (FLEX) for the Hubbard model in infinite dimensions d\ensuremath{\rightarrow}\ensuremath{\infty}. The presented treatment is not only restricted to the limit d\ensuremath{\rightarrow}\ensuremath{\infty}. I present the resulting equations for the retarded self-energy, numerical results for the self-energy, density of states and the interaction spectrum in the FLEX for different scalings of the real frequency axis. The numerical solution of the SOPT and the FLEX could be obtained with very little numerical effort. In this form the calculation is nearly independent of the temperature and over a wide range nearly independent of the value of U. The computational time and the memory requirement for the solution of the self-energy equation are reduced to a minimum in this numerical treatment. \textcopyright{} 1996 The American Physical Society.

Physical review. B, Condensed matter•Apr 15, 1996•2 citations
Due to the retarded nature of the phonon-mediated electron-electron interaction, neither fast Fourier transform (FFT) nor previously developed NRG methods for Hubbard-type purely electronic models are applicable, while brute-force solutions are limited by the demands on computational time and storage which increase rapidly at low temperature T. A new numerical renormalization group (NRG) approach to the solution of self-consistent-field equations for correlated lattice electrons is illustrated for the two-dimensional Hubbard-type model. This approach allows us to handle the fermion frequency matrix inversions in the FLEX equations efficiently. Several orders of magnitude of CPU time and storage can be saved at the temperature scales relevant to electronic phase transitions.
The fluctuation exchange (FLEX) approximation is applied to study the Holstein-Hubbard model. Due to the retarded nature of the phonon-mediated electron-electron interaction, neither fast Fourier transform nor previously developed numerical renormalization-group (NRG) methods for Hubbard-type purely electronic models are applicable, while brute force solutions are limited by the demands on computational time and storage which increase rapidly at low temperature T. Here, we describe a NRG technique to solve the FLEX equations efficiently. Several orders of magnitude of CPU time and storage can be saved at low T $(\ensuremath{\sim}80\mathrm{K})$. To test our approach, we compare our NRG results to brute force calculations on small lattices at elevated temperatures. Both s-wave and d-wave superconducting phase diagrams are then obtained by applying the NRG approach at low T. The isotope effect for s-wave pairing is BCS-like in a realistic phonon frequency range, but vanishes at unphysically large phonon frequency $(\ensuremath{\sim}\mathrm{b}\mathrm{a}\mathrm{n}\mathrm{d}\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h})$. For d-wave pairing, the isotope exponent is negative and small compared to the typical observed values in nonoptimally doped cuprates.
Momentum- and frequency-dependent self-consistent field theories provide a potentially powerful tool for analyzing condensed-matter models for interacting electrons. Such theories generally require the solution of multidimensional integral equations. Solutions are limited by demands on computational time and storage, which increase rapidly for calculation of low-temperature properties. A renormalization-group technique which allows the sequential elimination of regions of high frequency and momentum is presented in detail, and is then applied to the solution of the fluctuation exchange approximation for the two-dimensional Hubbard model. Effective meshes with as many as 104{10}^{4}104 frequencies and 5122{512}^{2}5122 k points can be analyzed on a vector supercomputer using this approach. As a sample application, it is demonstrated that the fluctuation exchange approximation exhibits non-Fermi liquid behavior near half filling for sufficiently large interaction strengths.
Strongly correlated electron systems host a plethora of fascinating physical phenomena and pose formidable challenges in their theoretical analysis. The challenges originate from the inherent complexity of the quantum many-body problem - no classical computer will ever be able to fully simulate these systems - and the lack of an effective single-particle picture, as the strong mutual interactions make it impossible to regard the electrons as independent from each other. As a consequence, most systems of correlated electrons can only be tackled approximately and numerically. In this thesis, we develop a set of numerical methods for strongly correlated electrons, which are inspired by the renormalization group (RG) idea of including degrees of freedom successively from high to low energies. This enables an efficient organization of the diverse fluctuations and is key for an accurate treatment of interacting quantum systems, where collective behavior and composite objects emerge at energy scales far below those of the microscopic constituents. In a first part, we consider the functional renormalization group (fRG), a versatile framework to study the flow of correlation functions upon modulating the underlying action. Though widely used, it has often acted more as a qualitative rather than quantitative method, due a nontransparent approximation induced by truncating the hierarchy of flow equations. We develop an iterative multiloop fRG (mfRG) scheme, which ameliorates this approximation and eliminates many of the drawbacks of fRG experienced hitherto. In particular, it restores the independence of results on the choice of RG regulator and establishes a rigorous relation to the parquet formalism. Furthermore, we show how to derive the flow equations directly from self-consistent many-body relations. This establishes a form of diagrammatic resummations at the two-particle level which circumvents ill-behaved two-particle-irreducible vertices. An application to the prototypical two-dimensional Hubbard model illustrates how our multiloop scheme elevates the fRG approach to correlated electron systems to a quantitative level. Secondly, we employ the numerical renormalization group (NRG), based on the iterative diagonalization of impurity Hamiltonians, in conjunction with the dynamical mean-field theory (DMFT) to describe local correlations in multiorbital systems. Having access to arbitrarily low temperatures and energies, NRG is a unique, real-frequency impurity solver for DMFT. It has been pivotal to the understanding of Hund metals, where strong correlations arise from Hund's rules even at moderate Coulomb repulsion. Building on recent methodological advances, we extend the range of application of DMFT+NRG from orbital-degenerate models to more realistic setups: We first study orbital differentiation in a three-orbital Hund-metal model and unravel key effects of the orbital-selective Mott transition. In a real-materials setting, we then incorporate the bandstructure from density functional theory (DFT) and analyze the archetypal Hund-metal material Sr2RuO4. We particularly follow its RG flow to the Fermi-liquid regime at previously inaccessible low temperatures and generally present DFT+DMFT+NRG as a new computational paradigm for strongly correlated materials. As a side project of our fRG work, we develop an algorithm to count Feynman diagrams from closed many-body relations, which reveals the surprising outcome that totally irreducible contributions are responsible for the factorial growth in the number of diagrams. Additionally, we use NRG to study transport through three-level quantum dots and provide benchmark data for other RG methods, which aim at further describing these systems in nonequilibrium.
The functional renormalization group has become a widely used tool for the analysis of the leading low-temperature correlations in weakly to moderately coupled many-fermion lattice systems. A bottleneck for quantitatively more precise results is that the treatment of the frequency dependence of the flowing interactions is numerically quite demanding. Yet the frequency dependence is needed to compute relevant self-energies and hence for controlled results on the energy scales for ordering or for the quasiparticle properties. Here we explore an approximate parametrization of the frequency dependence of the interaction vertex that is inspired by established simplifications in the theory of superconductivity and that keeps the numerical effort bounded. We demonstrate the validity of the approximation for Cooper pairing problems and apply it to the two-dimensional Hubbard model.
In recent years the numerical renormalization group (NRG) method has been extended to the calculation of dynamic response functions and transport properties of magnetic impurity models. The approach can now be applied more widely to lattice models of strongly correlated electron systems by the use of dynamical mean field theory (DMFT), in which the lattice problem is transformed into one for an effective impurity with an additional self-consistency constraint. We review these developments and assess the potential for further applications of this approach. We also discuss an alternative approach to renormalization, renormalized perturbation theory, in which the leading asymptotically exact results for the low temperature regime for a number of magnetic impurity models can be obtained within finite order perturbation theory.
I present a numerical treatment for a certain class of Feynman diagrams in perturbation theory with respect to the Coulomb correlation U. This method is formally exact for the frequency integrals occurring in the expression for the retarded Green's function. I specify this method to the self-consistent second-order perturbation theory (SOPT) for the single-impurity Anderson model and to the self-consistent fluctuation exchange approximation (FLEX) for the Hubbard model in infinite dimensions d\ensuremath{\rightarrow}\ensuremath{\infty}. The presented treatment is not only restricted to the limit d\ensuremath{\rightarrow}\ensuremath{\infty}. I present the resulting equations for the retarded self-energy, numerical results for the self-energy, density of states and the interaction spectrum in the FLEX for different scalings of the real frequency axis. The numerical solution of the SOPT and the FLEX could be obtained with very little numerical effort. In this form the calculation is nearly independent of the temperature and over a wide range nearly independent of the value of U. The computational time and the memory requirement for the solution of the self-energy equation are reduced to a minimum in this numerical treatment. \textcopyright{} 1996 The American Physical Society.