Emergent quantum criticality, Fermi surfaces, and AdS2 (0907.2694v2)

Abstract: Gravity solutions dual to d-dimensional field theories at finite charge density have a near-horizon region which is AdS_2 x R^{d-1}. The scale invariance of the AdS_2 region implies that at low energies the dual field theory exhibits emergent quantum critical behavior controlled by a (0+1)-dimensional CFT. This interpretation sheds light on recently-discovered holographic descriptions of Fermi surfaces, allowing an analytic understanding of their low-energy excitations. For example, the scaling behavior near the Fermi surfaces is determined by conformal dimensions in the emergent IR CFT. In particular, when the operator is marginal in the IR CFT, the corresponding spectral function is precisely of the "Marginal Fermi Liquid" form, postulated to describe the optimally doped cuprates.

• The paper demonstrates that emergent scale invariance in the AdS2 region yields quantum critical behavior, with spectral functions governed by IR CFT conformal dimensions.
• It employs analytic and numerical methods to solve the Dirac equation for bulk spinors, uncovering log-periodic behavior in the spectral density indicative of complex scaling exponents.
• The paper connects holographic principles with condensed matter phenomena, providing insights into Fermi surface instabilities and non-Fermi liquid behavior in strongly correlated systems.

Emergent Quantum Criticality, Fermi Surfaces, and AdS$_2$

The paper "Emergent Quantum Criticality, Fermi Surfaces, and AdS$_2$" by Faulkner et al. explores the intriguing connections between quantum field theories at finite charge density and classical gravity systems through the AdS/CFT correspondence. The main focus is on understanding the nature of emergent quantum critical behavior and the properties of Fermi surfaces within this holographic framework.

Overview

The AdS$_2$ region, which emerges in the near-horizon geometry of certain black holes, plays a central role in controlling the low-energy excitations of the dual field theories. This understanding builds on the observation that $d$-dimensional field theories at finite charge density have AdS$_2 \times R^{d-1}$ regions that lead to quantum critical behavior. Specifically, the AdS$_2$ geometry induces emergent scale invariance and a low-energy description in terms of a $(0+1)$-dimensional conformal field theory (CFT), allowing for analytic access to previously intractable problems such as non-Fermi liquid behavior.

The paper emphasizes the importance of the inner AdS$_2$ region in revealing the intricacies of Fermi surfaces in strongly interacting systems. The holographic description provides a handle on the scaling behavior near Fermi surfaces, which is shown to be governed by the conformal dimensions in the emergent IR CFT. This aspect is tantalizingly reminiscent of the "Marginal Fermi Liquid" conceptual framework, proposed to explain the properties of cuprates, and suggests a theoretical mechanism for its emergence.

Technical Insights and Results

1. Retarded Green Functions: The authors derive that at low frequencies, the retarded Green functions exhibit quantum critical behavior, deeply linked to the geometry of the AdS$_2$. The scaling exponents of the spectral function are directly tied to the conformal dimensions of operators in the IR CFT. This critical insight underscores the power of the AdS/CFT duality in unraveling the characteristics of strongly-coupled quantum systems.
2. Numerical Solutions and Analytic Structure: Utilizing solutions of the Dirac equation for bulk spinor fields in the black hole geometry, the work numerically explores the spectral functions. An intriguing finding is the log-periodic behavior in the spectral density function, indicative of an underlying complex scaling exponent.
3. Fermi Surfaces and Instabilities: A notable result is that the inner AdS$_2$ region is responsible for the non-Fermi liquid behavior observed in the dual theory. The existence of sharp quasi-particle peaks is a smoking gun for Fermi surfaces, with the work analytically confirming that their properties, including instabilities, are well accounted for by the holographic description.

Implications and Future Directions

The research articulated in the paper carries significant implications beyond the immediate results concerning Fermi surfaces. On a theoretical level, it enriches the understanding of holographic dualities at finite densities and their connection to condensed matter physics. Practically, it opens avenues for exploring a diverse array of quantum phases that exhibit non-conventional behaviors like those postulated in high-$T_c$ superconductors.

These insights can stimulate further research into the application of holography to real-world materials, particularly those exhibiting strongly-correlated electronic phenomena. Furthermore, the techniques and methodologies developed herein point to a promising future in investigating the interplay between gravity, quantum criticality, and non-Fermi liquids in higher dimensions.

Conclusion

The paper effectively bridges significant gaps between gravitational theories and emergent phenomena in condensed matter physics. By focusing on the root causes of behavior in the boundary field theory, the authors have illuminated paths to understanding quantum criticality and its holographic duals. As theoretical tools continue to advance, studies like these will be pivotal in pioneering the broader application of string theory and holographic principles in practical and observable physical systems.