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arXiv:1510.03330 · Physical review. B./Physical review. B

Efficient implementation of the parquet equations: Role of the reducible vertex function and its kernel approximation

Gang Li, Nils Wentzell, Petra Pudleiner, Patrik Thunström, Karsten Held

Published Apr 4, 2016• 95 citations
arXivHTMLPDF

Abstract

We present an efficient implementation of the parquet formalism that respects the asymptotic structure of the vertex functions at both single- and two-particle levels in momentum and frequency space. We identify the two-particle reducible vertex as the core function that is essential for the construction of the other vertex functions. This observation stimulates us to consider a two-level parameter reduction for this function to simplify the solution of the parquet equations. The resulting functions, which depend on fewer arguments, are coined ``kernel functions.'' With the use of the kernel functions, the open boundary of various vertex functions in Matsubara-frequency space can be faithfully satisfied. We justify our implementation by accurately reproducing the dynamical mean-field theory results from momentum-independent parquet calculations. The high-frequency asymptotics of the single-particle self-energy and the two-particle vertex are correctly reproduced, which turns out to be essential for the self-consistent determination of the parquet solutions. The current implementation is also feasible for the dynamical vertex approximation.

DOI
10.1103/physrevb.93.165103

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We present an efficient implementation of the parquet formalism that respects the asymptotic structure of the vertex functions at both single- and two-particle levels in momentum and frequency space. We identify the two-particle reducible vertex as the core function that is essential for the construction of the other vertex functions. This observation stimulates us to consider a two-level parameter reduction for this function to simplify the solution of the parquet equations. The resulting functions, which depend on fewer arguments, are coined ``kernel functions.'' With the use of the kernel functions, the open boundary of various vertex functions in Matsubara-frequency space can be faithfully satisfied. We justify our implementation by accurately reproducing the dynamical mean-field theory results from momentum-independent parquet calculations. The high-frequency asymptotics of the single-particle self-energy and the two-particle vertex are correctly reproduced, which turns out to be essential for the self-consistent determination of the parquet solutions. The current implementation is also feasible for the dynamical vertex approximation.
We exploit the parquet formalism to derive exact flow equations for the two-particle-reducible four-point vertices, the self-energy, and typical response functions, circumventing the reliance on higher-point vertices. This includes a concise, algebraic derivation of the multiloop flow equations, which have previously been obtained by diagrammatic considerations. Integrating the multiloop flow for a given input of the totally irreducible vertex is equivalent to solving the parquet equations with that input. Hence, one can tune systems from solvable limits to complicated situations by variation of one-particle parameters, staying at the fully self-consistent solution of the parquet equations throughout the flow. Furthermore, we use the resulting differential form of the Schwinger-Dyson equation for the self-energy to demonstrate one-particle conservation of the parquet approximation and to construct a conserving two-particle vertex via functional differentiation of the parquet self-energy. Our analysis gives a unified picture of the various many-body relations and exact renormalization group equations.
The parquet equations are a self-consistent set of equations for the effective two-particle vertex of an interacting many-fermion system. The application of these equations to bulk models is, however, demanding due to the complex emergent momentum and frequency structure of the vertex. Here, we show how a channel decomposition by means of truncated unities, which was developed in the context of the functional renormalization group to efficiently treat the momentum dependence, can be transferred to the parquet equations. This leads to a significantly reduced complexity and memory consumption scaling only linearly with the number of discrete momenta. We apply this technique to the half-filled repulsive Hubbard model on the square lattice and present approximate solutions for the channel-projected vertices and the full reducible vertex.
We exploit the parquet formalism to derive exact flow equations for the two-particle-reducible four-point vertices, the self-energy, and typical response functions, circumventing the reliance on higher-point vertices. This includes a concise, algebraic derivation of the multiloop flow equations, which have previously been obtained by diagrammatic considerations. Integrating the multiloop flow for a given input of the totally irreducible vertex is equivalent to solving the parquet equations with that input. Hence, one can tune systems from solvable limits to complicated situations by variation of one-particle parameters, staying at the fully self-consistent solution of the parquet equations throughout the flow. Furthermore, we use the resulting differential form of the Schwinger–Dyson equation for the self-energy to demonstrate one-particle conservation of the parquet approximation and to construct a conserving two-particle vertex via functional differentiation of the parquet self-energy. Our analysis gives a unified picture of the various many-body relations and exact renormalization group equations.
{\it Victory}, i.e. \underline{vi}enna \underline{c}omputational \underline{to}ol deposito\underline{ry}, is a collection of numerical tools for solving the parquet equations for the Hubbard model and similar many body problems. The parquet formalism is a self-consistent theory at both the single- and two-particle levels, and can thus describe individual fermions as well as their collective behavior on equal footing. This is essential for the understanding of various emergent phases and their transitions in many-body systems, in particular for cases in which a single-particle description fails. Our implementation of {\it victory} is in modern Fortran and it fully respects the structure of various vertex functions in both momentum and Matsubara frequency space. We found the latter to be crucial for the convergence of the parquet equations, as well as for the correct determination of various physical observables. In this release, we thoroughly explain the program structure and the controlled approximations to efficiently solve the parquet equations, i.e. the two-level kernel approximation and the high-frequency regulation.
Vertex functions are a crucial ingredient of several forefront many-body algorithms in condensed matter physics. However, the full treatment of their frequency and momentum dependence severely restricts numerical calculations. A significant advancement requires an efficient treatment of the high-frequency asymptotic behavior of the vertex functions. In this work, we first provide a detailed diagrammatic analysis of the high-frequency structures and their physical interpretation. Based on these insights, we propose a parametrization scheme, which captures the whole high-frequency domain for arbitrary values of the Coulomb interaction and electronic density, and we discuss the details of its algorithmic implementation in many-body solvers based on parquet equations as well as functional renormalization group schemes. Finally, we assess its validity by comparing our results for a single impurity Anderson model with exact diagonalization calculations. The proposed parametrization is pivotal for the algorithmic development of all quantum many-body methods based on vertex functions arising from both local and nonlocal static microscopic interactions as well as effective dynamic interactions which uniformly approach a static value for large frequencies. In this way, our present technique can substantially improve vertex-based diagrammatic approaches including spatial correlations beyond dynamical mean-field theory.
Abstract The parquet approach to vertex corrections is unbiased but computationally demanding. Most applications are therefore restricted to small cluster sizes or rely on various simplifying approximations. We have recently shown that the bosonization of the parquet diagrams provides interpretative and algorithmic advantages over the original purely fermionic formulation. Here, we present first results of the numerical implementation of this method by applying it to the half-filled Hubbard model on the square lattice at weak coupling. The improved algorithmic performance allows us to evaluate the parquet approximation for a 16×1616\times 1616×16 16 × 16 lattice, retaining the full momentum and frequency structure of the various vertex functions. We discuss their symmetries and consider parametrizations of their momentum dependence using the truncated-unity approximation. Graphical abstract