Causality in the Dual-Fermion Approach

Abstract

Causality is a fundamental requirement for physical consistency in many-body approximations. In the context of Green's functions, causality requires a positive spectral function and analyticity in the upper half of the complex-frequency plane. These properties are not guaranteed by approximate methods, and formal proofs of causality for the underlying approximation schemes are often difficult to establish. The recently rediscovered Nevanlinna--Pick criterion provides a practical way to test causality directly from Matsubara-frequency data, without performing analytic continuation to the real-frequency axis. In this work, we use the Nevanlinna--Pick criterion to investigate the causality of the ladder dual-fermion approach, a diagrammatic extension of dynamical mean-field theory designed to incorporate nonlocal correlations. The analysis is performed for the spinless Falicov--Kimball model, which provides a useful testing ground because several quantities can be computed exactly or semi-analytically, avoiding Monte Carlo noise. This is particularly important since the Nevanlinna--Pick criterion is highly sensitive to noise in the input data. In addition, we derive several analytical properties of the Pick matrix, providing insight into the structure of the criterion when applied to Green's functions. For small systems, apparent causality violations are observed, but these are found to be associated with finite-size effects. For larger system sizes, no robust signs of causality violations are found, neither within the full ladder dual-fermion approach nor within the leading dual-fermion correction.

Popular Abstract

Suppose you are watching a horror movie with a friend. Suddenly, your friend leaps into the air, frightened by something on the screen. A few seconds later, the jump scare appears.

This would be rather strange. It would seem as if your friend had reacted before the jump scare occurred. Either your friend has violated the principle of causality, or they are deliberately trying to confuse you. Both possibilities would be equally surprising.

The principle of causality states that a cause must come before its effect. Intuitively, any physical theory should respect this principle to be taken seriously. However, as it turns out, causality can become a subtle issue in modern approaches to quantum-mechanical problems.

In quantum mechanics, exact solutions are rarely available, especially for systems with many interacting particles, such as electrons repelling each other inside a solid. To make progress, we must resort to approximations. One important approximation, introduced in the early 1990s, is called dynamical mean-field theory. The central idea is to focus on a single site inside a material and replace the influence of all other sites by an effective surrounding environment, often called a bath. By solving this simpler single-site problem, we obtain an approximate solution to the full system.

However, important physics can be missed when focusing only on a single site. For this reason, more advanced methods have been developed to include effects involving the interplay between multiple sites. One such method is the dual-fermion approach, which adds corrections on top of dynamical mean-field theory. Although adding corrections may seem like an obvious improvement, it also raises important questions of physical consistency, such as whether the corrected theory still preserves causality. Developing approximation methods that are both accurate and physically consistent is essential for improving our understanding of materials and, in the long term, for guiding the development of future technologies.

In this thesis, we investigate whether the dual-fermion approach respects causality. In other words, we ask whether this improved approximation ever predicts a world where effects appear before their causes. For small systems, we find indications of such causality violations. However, these warning signs disappear as the system size is increased, suggesting that they are finite-size effects rather than a fundamental flaw of the method.

So the next time your strange friend reacts to an event that has yet to occur, either try to take them out to buy the winning lottery ticket, or sit them down, provide a mathematical proof that what they are doing is impossible, and politely ask them to stop. It spoils the thrill of watching horror movies.

• author: Skoglund, William
• supervisor: Erik van Loon
• organization: Department of Physics
• course: FYSM34 20261
• year: 2026
• type: H2 - Master's Degree (Two Years)
• subject: Physics and Astronomy
• keywords: Causality, dual-fermion approach, dynamical mean-field theory, Green's functions, Nevanlinna-Pick criterion, Falicov-Kimball model
• language: English
• id: 9242481
• date added to LUP: 2026-07-03 14:50:08
• date last changed: 2026-07-03 14:50:08