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arXiv:2207.01681

Merging numerical renormalization group and intermediate representation to compactify two- and three-point correlators

Huber, Sebastian, Markus Wallerberger, Paul Worm, Karsten Held

Published Jul 4, 2022• 0 citations
arXivHTMLPDF

Abstract

The vanguard of many-body theory is nowadays dealing with the full frequency dynamics of n-point Green's functions for n higher than two. Numerically, these objects easily become a memory bottleneck, even when working with discrete imaginary-time Matsubara frequencies. Here, we use the intermediate representation (IR) to compactify the two-point Green's function and three-point Fermion-Bose vertex directly on the real frequency axis, on the basis of numerical renormalization group (NRG) data. We empirically observe an upper bound of the relative error when comparing the IR reconstructed signal with the original NRG data, and demonstrate that a IR compacification is possible.

DOI
10.48550/arxiv.2207.01681

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The vanguard of many-body theory is nowadays dealing with the full frequency dynamics of n-point Green's functions for n higher than two. Numerically, these objects easily become a memory bottleneck, even when working with discrete imaginary-time Matsubara frequencies. Here, we use the intermediate representation (IR) to compactify the two-point Green's function and three-point Fermion-Bose vertex directly on the real frequency axis, on the basis of numerical renormalization group (NRG) data. We empirically observe an upper bound of the relative error when comparing the IR reconstructed signal with the original NRG data, and demonstrate that a IR compacification is possible.
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Model-independent compact representations of imaginary-time data are presented in terms of the intermediate representation (IR) of analytical continuation. We demonstrate the efficiency of the IR through continuous-time quantum Monte Carlo calculations of an Anderson impurity model. We find that the IR yields a significantly compact form of various types of correlation functions. This allows the direct quantum Monte Carlo measurement of Green's functions in a compressed form, which considerably reduces the computational cost and memory usage. Furthermore, the present framework will provide general ways to boost the power of cutting-edge diagrammatic/quantum Monte Carlo treatments of many-body systems.
The imaginary-time Green's function is a building block of various numerical methods for correlated electron systems. Recently, it was shown that a model-independent compact orthogonal representation of the Green's function can be constructed by decomposing its spectral representation. We investigate the performance of this so-called intermediate representation (IR) from several points of view. First, we develop an efficient algorithm for computing the IR basis functions of arbitrary high degree. Second, for two simple models, we study the number of coefficients required to represent the Green's function within a given tolerance. We show that the number of coefficients grows only as $O(ln\ensuremath{\beta})$ for fermions and converges to a constant for bosons as temperature $T=1/\ensuremath{\beta}$ decreases. Third, we show that this remarkable feature is ascribed to the properties of the physically constructed basis functions. The fermionic basis functions on the real-frequency axis have features whose width is scaled as O(T)O(T)O(T), which is consistent with the low-TTT properties of quasiparticles in a Fermi liquid state. On the other hand, the properties of the bosonic basis functions are consistent with those of spin/orbital susceptibilities at low TTT. These results demonstrate the potential wide applications of the IR to calculations of correlated systems.
The many-body problem is usually approached from one of two perspectives: the first originates from an action and is based on Feynman diagrams, the second is centered around a Hamiltonian and deals with quantum states and operators.The connection between results obtained in either way is made through spectral (or Lehmann) representations, well known for two-point correlation functions.Here, we complete this picture by deriving generalized spectral representations for multipoint correlation functions that apply in all of the commonly used many-body frameworks: the imaginary-frequency Matsubara and the realfrequency zero-temperature and Keldysh formalisms.Our approach separates spectral from time-ordering properties and thereby elucidates the relation between the three formalisms.The spectral representations of multipoint correlation functions consist of partial spectral functions and convolution kernels.The former are formalism independent but system specific; the latter are system independent but formalism specific.Using a numerical renormalization group method described in the accompanying paper, we present numerical results for selected quantum impurity models.We focus on the four-point vertex (effective interaction) obtained for the single-impurity Anderson model and for the dynamical mean-field theory solution of the one-band Hubbard model.In the Matsubara formalism, we analyze the evolution of the vertex down to very low temperatures and describe the crossover from strongly interacting particles to weakly interacting quasiparticles.In the Keldysh formalism, we first benchmark our results at weak and infinitely strong interaction and then reveal the rich real-frequency structure of the dynamical mean-field theory vertex in the coexistence regime of a metallic and insulating solution.
The multipoint numerical renormalization group (mpNRG) is a powerful impurity solver that provides accurate spectral data useful for computing local, dynamic correlation functions in imaginary or real frequencies nonperturbatively across a wide range of interactions and temperatures. It gives access to a local, nonperturbative four-point vertex in imaginary and real frequencies, which can be used as input for subsequent computations such as diagrammatic extensions of dynamical mean-field theory. However, computing and manipulating the real-frequency four-point vertex on large, dense grids quickly becomes numerically challenging when the density and/or the extent of the frequency grid is increased. In this paper, we compute four-point vertices in a strongly compressed quantics tensor train format using quantics tensor cross interpolation, starting from discrete partial spectral functions obtained from mpNRG. This enables evaluations of the vertex on frequency grids with resolutions far beyond the reach of previous implementations. We benchmark this approach on the four-point vertex of the single-impurity Anderson model across a wide range of physical parameters, both in its full form and in its asymptotic decomposition. For imaginary frequencies, the full vertex can be represented to an accuracy on the order of 2 × 10 − 3 with maximum bond dimensions not exceeding 120. The more complex full real-frequency vertex requires maximum bond dimensions not exceeding 170 for an accuracy of ≲ 2 % . Our work marks another step toward tensor-train-based diagrammatic calculations for correlated electronic lattice models starting from a local, nonperturbative mpNRG vertex.
Two-particle Green's functions and the vertex functions play a critical role in theoretical frameworks for describing strongly correlated electron systems. However, numerical calculations at the two-particle level often suffer from large computation time and massive memory consumption. We derive a general expansion formula for the two-particle Green's functions in terms of an overcomplete representation based on the recently proposed ``intermediate representation'' basis. The expansion formula is obtained by decomposing the spectral representation of the two-particle Green's function. We demonstrate that the expansion coefficients decay exponentially, while all high-frequency and long-tail structures in the Matsubara-frequency domain are retained. This representation therefore enables efficient treatment of two-particle quantities and opens a route to the application of modern many-body theories to realistic strongly correlated electron systems.