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arXiv:2412.01848 · Physical Review Research

Ladder equation for the three-particle vertex and its approximate solution

Patrick Kappl, T. Ribic, Anna Kauch, Karsten Held

Published Sep 10, 2025• 0 citations
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Abstract

We generalize the three two-particle Bethe-Salpeter equations to ten three-particle ladders. These equations are exact and yield the exact three-particle vertex, if we knew the three-particle vertex irreducible in one of the ten channels. However, as we do not have this three-particle irreducible vertex at hand, we approximate this building block for the ladder by the sum of two-particle irreducible vertices, each connecting two fermionic lines. The comparison to the exact solution shows that this approximation is only good for rather weak interactions and even then only qualitatively—at least for the nonlinear response function analyzed.

DOI
10.1103/fqmb-h9lb

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We generalize the three two-particle Bethe-Salpeter equations to ten three-particle ladders. These equations are exact and yield the exact three-particle vertex, if we knew the three-particle vertex irreducible in one of the ten channels. However, as we do not have this three-particle irreducible vertex at hand, we approximate this building block for the ladder by the sum of two-particle irreducible vertices, each connecting two fermionic lines. The comparison to the exact solution shows that this approximation is only good for rather weak interactions and even then only qualitatively—at least for the nonlinear response function analyzed.
The dynamical vertex approximation and related methods represent a great leap forward in the description of nonlocal electronic correlations. It is based on a local two-particle irreducible vertex, from which the momentum-dependent self-energy is constructed via Bethe-Salpeter and Schwinger-Dyson equations. In this paper, the hitherto used one-shot calculations are improved by repeating these equations with feedback from the updated self-energy until self-consistency. The validity of the method is benchmarked by several numerical examples and, prominently, the formation of a pseudogap is demonstrated.
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A Bethe-Salpeter equation for the three-particle correlation function is derived, representing a second-order response generalization of the usual Bethe-Salpeter equation. The equation can be solved formally, giving the three-particle correlation function in terms of the two-particle correlation function and many-body interaction kernels. The similarity to second-order response time-dependent density-functional theory can be used to improve the understanding of the higher-order exchange and correlation kernels of this theory. An exact expression for such a kernel is derived.
The functional renormalization group (fRG) approach has the property that, in general, the flow equation for the two-particle vertex generates O(N4)\mathcal{O}({N}^{4})O(N4) independent variables, where NNN is the number of interacting states (e.g., sites of a real-space discretization). In order to include the flow equation for the two-particle vertex, one needs to make further approximations if NNN becomes too large. We present such an approximation scheme, called the coupled-ladder approximation, for the special case of an onsite interaction. Like the generic third-order-truncated fRG, the coupled-ladder approximation is exact to second order and is closely related to a simultaneous treatment of the random phase approximation in all channels, i.e., summing up parquet-type diagrams. The scheme is applied to a one-dimensional model describing a quantum point contact.
Extends the author's work (1979) on solution of the Bethe-Salpeter equation for the ladder approximation, Gamma L , to the effective interaction in a Fermi liquid. This permits treatment of the case of bare potentials having finite range, with a spatial dependence which is either exponential or is representable as the Laplace transform of another function, and an exponential decay in time. The integral equation is transformed to real space and the kernel replaced by a differential operator, in a generalisation of an approach developed by Hahne, Heiss, and Engelbrecht (1979) for interactions depending only on time. This procedure simplifies calculation of the terms in the iterative solution of the integral equation, as is demonstrated by explicit calculation, for several cases, of the first two iterative terms. It is found, in agreement with earlier results, that going from a zero-range or Dirac delta interaction to one of finite range can change markedly the analytic character of Gamma L and its calculation.