- The paper derives Hall conductivity σ_xy = ρ/B in 2+1D CFTs by applying the AdS/CFT duality to a dyonic black hole model.
- The authors employ the Kubo formula and perturbative techniques to analyze current fluctuations without relying on weakly interacting quasiparticles.
- The study emphasizes the role of net charge density in achieving Hall response in strongly coupled systems, offering insights for both theoretical and experimental research.
Hall Conductivity from Dyonic Black Holes: An Analytical Study of Strongly Interacting CFTs
The paper "Hall Conductivity from Dyonic Black Holes" by Sean A. Hartnoll and Pavel K. Kovtun provides an intricate examination of the Hall conductivity in a class of 2+1 dimensional conformal field theories (CFTs) under a transverse magnetic field, utilizing the AdS/CFT correspondence. The research focuses on elucidating hydrodynamic response functions of the maximally supersymmetric SU(N) Yang-Mills theory at its conformal fixed point in the large N limit, where the dual gravitational description is modeled by dyonic black holes in anti-de Sitter (AdS) space.
Key Findings
This paper explores a highly technical analysis, leveraging the AdS/CFT duality to compute Hall conductivities in strongly interacting, non-quasiparticle systems. One of the main outcomes is that the Hall conductivity for the studied 2+1 dimensional CFTs is determined as σ_xy = ρ/B, with ρ being the charge density and B the magnetic field, in alignment with theoretical expectations from Lorentz invariance.
Theoretical Framework
The authors utilize the Kubo formula to calculate the DC conductivity based on the current-current retarded Green's functions. Importantly, this computation only depends on linear response theory, avoiding assumptions of weakly interacting quasiparticles. The use of dyonic black holes—objects possessing both electric and magnetic charges—in AdS_4 provides the necessary dual description to derive these resultants.
The framework also incorporates thermodynamic and hydrodynamic analyses of the dual CFT. The paper discusses how the presence of a net charge density is critical since, in a relativistic CFT, positive and negative excitations cancel each other's currents in a magnetic field, requiring a state with a net charge density for Hall conductivity to occur. This condition is satisfied by the dual representation of an electrically charged dyonic black hole.
Methodological Approach
The authors systematically derive equations governing the fluctuations within this AdS_4 black hole background using perturbative approaches to manage the low-frequency behavior. Solving these equations showcases how conserved current fluctuations in the boundary CFTs map to gauge field fluctuations within the bulk AdS space. Through these calculations, the paper derives both the Hall conductivity and the related momentum-current correlators at zero spatial momentum.
Practical and Theoretical Implications
From a practical perspective, understanding Hall conductivities in strongly coupled CFTs could provide insights into exotic material systems, potentially guiding experimental approaches in quantum critical systems and aiding in the modeling of such phenomena using gauge/gravity dualities. Theoretically, this paper offers substantial contributions by applying the AdS/CFT duality to characterize 2+1 dimensional systems, which previously had been less explored compared to their 3+1 dimensional counterparts.
Future Directions
While the paper presents conclusive findings in the ω/T << 1 regime, the authors suggest further exploration into the parameter space where crossover occurs between different regimes of small ω/T and small B/T. Additionally, an extension to account for nonzero spatial momenta could lend a more comprehensive understanding into the full linearized relativistic magnetohydrodynamics of the boundary theory.
In summary, this research lays the groundwork for further investigation into Hall conductivities within the context of strongly interacting CFTs, offering potential pathways for deeper exploration into the quantum critical phenomena using holographic methods. The findings contribute robustly to theoretical physics, especially in the understanding and application of the AdS/CFT correspondence, potentially impacting future theoretical and experimental studies dealing with quantum field theories in condensed matter systems.