Space-Time Geometry and Many-Body Systems: Holographic Dynamical Mean-Field Theory Discovers Links
Determining electron behaviour in complex materials remains a condensed matter physics challenge. Dynamic mean field theory (DMFT) is a reliable paradigm for highly coupled electron systems. Kouichi Okunishi of Osaka Metropolitan University and Akihisa Koga of Tokyo Institute of Science identified a strong link between DMFT and quantum gravity-era holography.
H-DMFT is a theoretical reformulation that illustrates a profound, intrinsic link between holography, which derives from black holes and quantum gravity, and DMFT, a standard framework for understanding complex materials.
This method's "precise correspondence" lets scientists simplify a quantum many-body problem, the behaviour of strongly connected electrons, to classical gravity and spacetime geometry.
The fundamental concepts of H-Dynamical Mean Field Theory are explained below:
Reformulated Holographic Renormalisation Group as DMFT
Using a holographic renormalisation group (RG) to reinterpret dynamical mean field theory's process is its main achievement.
Target system: Electron systems with a semicircle density of states can be placed on the Bethe lattice network. The Bethe lattice, a tree-like structure, has exponentially more nodes at its edge.
Researchers created a recursive RG transformation for the branch Green's function, a key indicator of electron activity, to methodically handle information. From the Bethe lattice's outer edge boundary, this recursive process reaches the material's deep interior. This flow from the boundary to the bulk is seen as a holographic spacetime's depth dimension evolution.
Through broad network penetration, this recursive RG flow converges to a fixed point. Conventional dynamical mean field theory calculations yield the same self-consistent local Green's function solution as this fixed point solution.
Geometric Interpretation and Spacetime Duality
The relationship offers closely linked electrons a geometric meaning by transferring to a dual spacetime.
Effective Geometry: The researchers conceptually smoothed the Bethe network's discontinuous lattice nodes to create an effective coordinate system for a two-dimensional Anti-de Sitter space.
Geometric Dimensions: The Bethe lattice's branching number determines its properties, notably the AdS space's effective Poincar├й radius.
Scaling Dimensions: Geometry can be used to generate scaling dimensions that describe electron correlation functions at the system's outer border. These scaling dimensions are crucial to the fixed-point Green's function.
Importantly, this paradigm indicates that electron activity at the material's edge can be used to understand its deep interior.
Mathematical Consistency as Physical Use
By providing theoretical rigour and empirical results, the H-DMFT framework helps study material phase transitions.
The work provides a detailed mathematical reason for the convergence of the dynamical mean field theory iterative approach. A representation of the recursive relation was found in the M├╢bius transformation, a conformal mapping. The stability of the fixed point solution is rigorously tested by investigating the properties of this transformation, which suggests links to holographic renormalisation groups, which are key to the Anti-de Sitter/Conformal Field Theory relationship
This captures the Mott Transition: Numerical computations on the Hubbard model for electron interactions indicated that this holographic technique's scaling dimensions accurately capture the Mott transition. This transition occurs when strong electron interactions abruptly turn a metal into an insulator. Metallic and insulating phases have differing scaling dimensions due to their physics.
Consistency with Holography: This work's scaling dimensions follow a simple link that matches fields' effective AdS space behaviour. According to p-adic AdS/CFT, this simple connection captures the Bethe lattice's tree network nature.
In summary, H-DMFT provides a geometric vocabulary for understanding complex electron processes by connecting condensed matter physics (dynamical mean field theory) and quantum gravity (holography).
