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arXiv:2608.08948

Real-Time Approach to the Dynamical Bethe-Salpeter Equation for Finite Systems

Tucker Allen, Barry Yangtao Li, Daniel Neuhauser

Published Aug 9, 2026• 0 citations
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Abstract

We present a real-time linear-response approach to solving the dynamical Bethe-Salpeter equation (BSE). The polarization part of the screened interaction is obtained from time-dependent Hartree propagation of the one-particle density matrix within an orbital basis-set representation. The frequency convolution defining the dynamical kernel is evaluated as a product in time, avoiding numerical integration and storage of the full screened Coulomb operator. The resulting nonlinear eigenvalue problem is then solved directly, going beyond the static screening approximation with full-frequency dependence in the screened interaction. We validate the method by calculating the low-lying singlet and triplet excitation energies of water, formaldehyde, and acetylene. While the deterministic scaling remains $\mathcal{O}(N^6)$, the time propagation formulation is readily compatible with grid-based stochastic sampling methods, which will open the possibility for dynamical BSE calculations of very large systems.

DOI
10.48550/arxiv.2608.08948

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•Jan 1, 2016•4 citations
We present a real-time linear-response approach to solving the dynamical Bethe-Salpeter equation (BSE). The polarization part of the screened interaction is obtained from time-dependent Hartree propagation of the one-particle density matrix within an orbital basis-set representation. The frequency convolution defining the dynamical kernel is evaluated as a product in time, avoiding numerical integration and storage of the full screened Coulomb operator. The resulting nonlinear eigenvalue problem is then solved directly, going beyond the static screening approximation with full-frequency dependence in the screened interaction. We validate the method by calculating the low-lying singlet and triplet excitation energies of water, formaldehyde, and acetylene. While the deterministic scaling remains O(N6)\mathcal{O}(N^6)O(N6), the time propagation formulation is readily compatible with grid-based stochastic sampling methods, which will open the possibility for dynamical BSE calculations of very large systems.
In a many-body perturbation treatment of electronic excitations, one has to solve the so-called Bethe–Salpeter equation (BSE) for the kernel (vertex) describing the interaction between electrons. In general, the BSE exhibits a non-local and frequency-dependent screened interaction and consequently it is extremely difficult to solve. We have developed a scheme that solves the BSE equation iteratively, including dynamically screening, by considering the change in the electron self-energy upon applying a time-dependent field. The BSE is obtained from the self-energy in the GW approximation (GWA) using the Schwinger functional technique. We apply our formalism to a simple model system and discuss briefly changes in the charge response, when dynamical screening is taken into account.
The helium atom is the simplest many-body electronic system provided by nature. The exact solution to the Schrödinger equation is known for helium ground and excited states, and it represents a benchmark for any many-body methodology. Here, we check the ab initio many-body GW approximation and the Bethe-Salpeter equation (BSE) against the exact solution for helium. Starting from the Hartree-Fock method, we show that the GW and the BSE yield impressively accurate results on excitation energies and oscillator strength, systematically improving the time-dependent Hartree-Fock method. These findings suggest that the accuracy of the BSE and GW approximations is not significantly limited by self-interaction and self-screening problems even in this few electron limit. We further discuss our results in comparison to those obtained by time-dependent density-functional theory.
Self-consistent Hartree-Fock approximation combined with solutions of the Bethe-Salpeter equation offers a powerful tool for studies of strong correlation effects arising in condensed matter models, nuclear physics, quantum field theories, and real materials. The standard finite-temperature approach would be to first solve the problem in the Matsubara representation and then apply numerical analytic continuation to the real-frequency axis to link theoretical results with experimental probes, but this ill-conditioned procedure often distorts important spectral features even for very accurate imaginary-frequency data. We demonstrate that the ladder-type finite-temperature Bethe-Salpeter equation in the Hartree-Fock basis for the three-point vertex function and, ultimately, system's polarization can be accurately solved directly on the real frequency axis using the diagrammatic Monte Carlo technique and series resummation. We illustrate the method by applying it to the homogeneous electron gas model and demonstrate how multiple scattering events renormalize Landau damping.
We develop an improved stochastic formalism for the Bethe-Salpeter equation (BSE), based on an exact separation of the effective-interaction W into two parts, W = (W - vW) + vW, where the latter is formally any translationally invariant interaction, vW(r - r'). When optimizing the fit of the exchange kernel vW to W, using a stochastic sampling W, the difference W - vW becomes quite small. Then, in the main BSE routine, this small difference is stochastically sampled. The number of stochastic samples needed for an accurate spectrum is then largely independent of system size. While the method is formally cubic in scaling, the scaling prefactor is small due to the constant number of stochastic orbitals needed for sampling W.
A formally exact Bethe-Salpeter-like equation for the linear-response function is introduced with a kernel which depends only on the one frequency of the applied field. This is in contrast with the standard Bethe-Salpeter equation (BSE) which involves multiple-frequency integrals over the kernel and response functions. From the one-frequency kernel, known approximations are straightforwardly recovered. However, the present formalism lends itself to more powerful approximations. This is demonstrated with the exact analytical solution of the Hubbard molecule. Similarities and differences of the GW + BSE approach with the self-consistent random-phase approximation are also discussed.
In this thesis, an approximate method introduced to solve time-dependent many-body problems known as time-dependent many-body perturbation theory is studied. Many-body perturbation theory for interacting electrons and phonons is reviewed. In particular, the electron propagator G and an unconventional two-component phonon propagator, which satisfy coupled integral Dyson equations, are introduced. In practice, the associated integral kernels known as the electron Σ and phonon self-energies need to be approximated. The conserving approximations known as the Hartree (-Fock) and the first and second Born approximations, which respect the continuity equation between the electron density and current, are considered in this work. Time-dependent systems of interest are studied in this thesis by using the integro-differential forms of the Dyson equations referred to as the Kadanoff-Baym Equations (KBE). The Kadanoff-Baym equations are introduced for the electron and phonon propagators unconventionally as coupled firstorder integro-differential equations. It is reviewed how these equations are solved numerically by describing an integration rule, a class of practical methods and a parallel implementation of the numerical method. In addition, documentation of how the Kadanoff-Baym equations allow to solve the Bethe-Salpeter Equation (BSE) with the kernel δΣ/δG for the density response function, is provided. In two of the enclosed publications, we benchmarked observables obtained by using the Hartree and partially and fully self-consistent Born approximations against numerically exact results for the two- site, two-electron Holstein model. In this model, the two electrons which are constrained to move between two lattice sites interact indirectly via the electron-phonon coupling. It was observed that only the fully self-consistent Born approximation could cope qualitatively correctly with the competition between the delocalizing and localizing effects of the kinetic and interaction energies. For the other two approximations, spurious bifurcative symmetry breaking with an associated unbounded density response was observed. Nevertheless, also the self-consistent Born approximation was concluded to fail in describing the bipolaronic behavior of the true system. In the third publication, we benchmarked the Hartree-Fock and second Born approximations against an exact method for the few-site Hubbard and Pariser-Parr-Pople models in which the underlying lattice is inert and the electrons interact amongst themselves directly. It was found that the second Born approximation is capable of describing the so-called correlation induced doubly-excited states. This is not possible for time-local approximations such as Hartree-Fock. In addition to the qualitative results, which highlight successes of the applied simple self-energy approximations, the approximate and exact results were also compared on a more quantitative level. It is the quantitative and qualitative results combined which are used in this thesis to assess the quality of the many-body approximations used, with the aim to better understand when these methods are predictive.