Holographic Dynamical Mean-Field Theory Reveals Relationships Between Many-Body Systems and Space-Time Geometry

Understanding the behaviour of electrons in complex materials remains a central challenge in condensed matter physics, and dynamical mean-field theory provides a powerful framework for tackling these strongly correlated electron systems. Kouichi Okunishi of Osaka Metropolitan University and Akihisa Koga of the Institute of Science Tokyo, along with their colleagues, now reveal a surprising connection between this theory and the concept of holography, originally developed in the study of black holes. Their work demonstrates that a holographic renormalization group, applied to the electronic structure of a specific model system, directly corresponds to the self-consistent equations at the heart of dynamical mean-field theory. This discovery not only deepens our theoretical understanding of these complex systems, but also provides a novel way to characterise phenomena like the Mott transition, where materials abruptly change their electrical properties, by linking them to the geometry of an effective two-dimensional space.

Researchers explore the holographic aspects of this theory, motivated by the potential to gain deeper insights into these complex systems through connections with concepts from holographic duality. This work investigates a specific mapping between dynamical mean-field theory and a gravitational system, aiming to establish a precise correspondence that allows for the translation of problems between the two frameworks. This mapping facilitates the application of techniques from quantum gravity to study the behaviour of electrons in materials where interactions between them are particularly strong, potentially revealing new phases of matter and providing a pathway to solve long-standing problems in condensed matter physics.

Mapping Many-Body Physics to Classical Gravity

Holography, developed in the field of quantum gravity, provides an intrinsic relationship between quantum many-body systems and space-time geometry. In this study, scientists demonstrate that these two theories are closely related by shedding light on holographic aspects of dynamical mean-field theory, particularly focusing on its application to strongly correlated materials. The method involves constructing a gravitational dual of the dynamical mean-field theory equations, allowing us to map the quantum many-body problem onto a problem in classical gravity. This dual description enables the calculation of various physical quantities using techniques from general relativity.

Researchers perform numerical simulations to solve the gravitational equations of motion and extract the corresponding quantum many-body properties. The results show a clear correspondence between the gravitational description and the original quantum problem, confirming the holographic principle in this context. Specifically, they investigate the behaviour of the system at different temperatures and doping levels, revealing novel insights into the emergence of correlated phenomena. The analysis demonstrates that the gravitational dual accurately captures the essential physics of the strongly correlated system, providing a powerful tool for understanding its behaviour.

DMFT Convergence via Conformal Mapping Stability

This research provides a detailed mathematical justification for the convergence of the dynamical mean-field theory iterative scheme when applied to the Bethe lattice model. Scientists leverage the connection between the dynamical mean-field theory recursion and conformal mapping to demonstrate the stability of the fixed point solution. This connection hints at links to holographic renormalization group ideas, as conformal mappings are central to the AdS/CFT correspondence. The analysis begins by establishing a baseline solution for the free electron gas on the Bethe lattice, providing a crucial starting point for understanding the more complex interacting case.

The team linearizes the recursive relation around the fixed-point solution to analyse the convergence behaviour. They demonstrate that the dynamical mean-field theory recursion can be represented in the form of a Möbius transformation, a powerful tool for analysing convergence. By analysing the eigenvalues of this transformation, they determine the convergence rate and rigorously justify the convergence of the dynamical mean-field theory iterative scheme. This work provides insights into the behaviour of strongly correlated electron systems and may be useful for developing new theoretical methods.

Holographic Renormalization Group Links Electron Systems

This research establishes a close relationship between dynamical mean-field theory and the concept of holography. Scientists demonstrated that these two seemingly distinct theories are connected by formulating a holographic renormalization group for electrons within a specific network structure called the Bethe lattice. This formulation reveals that the fixed point of this renormalization group corresponds directly to the self-consistent solution obtained through traditional dynamical mean-field theory calculations. Furthermore, the team introduced an effective two-dimensional anti-de Sitter space to characterize the scaling dimensions of the electron behaviour at the edge of the Bethe lattice.

Calculations on the Hubbard model show that these scaling dimensions accurately capture the Mott transition, a fundamental change in material properties driven by strong electron interactions. The findings suggest that information about the deep interior of the material can be systematically understood through the behaviour of electrons at its boundary. Researchers acknowledge that extending the holographic framework to magnetically ordered systems presents a challenge for future research and that applying this approach to more complex lattice models requires further investigation.