Lectures on Holographic Superfluidity and Superconductivity (0904.1975v2)

Abstract: Four lectures on holography and the AdS/CFT correspondence applied to condensed matter systems. The first lecture introduces the concept of a quantum phase transition. The second lecture discusses linear response theory and Ward identities. The third lecture presents transport coefficients derived from AdS/CFT that should be applicable in the quantum critical region associated to a quantum phase transition. The fourth lecture builds in the physics of a superconducting or superfluid phase transition to the simple holographic model of the third lecture.

• The paper introduces holographic duality as a framework for analyzing superfluidity and superconductivity in strongly interacting systems.
• It employs the AdS/CFT correspondence and linear response theory to compute transport coefficients and predict quantum critical behavior.
• Herzog links numerical insights from holographic models to experimental observations in high-Tc superconductors, suggesting new avenues for research.

An Expert Overview of "Lectures on Holographic Superfluidity and Superconductivity"

The paper "Lectures on Holographic Superfluidity and Superconductivity" by C. P. Herzog encapsulates a series of lectures that delve into the application of the AdS/CFT correspondence within the field of condensed matter physics, particularly focusing on aspects of superfluidity and superconductivity. This approach marries concepts from high-energy physics and quantum gravity with strongly interacting field theories, illustrating the potential of holography as a tool for exploring quantum phase transitions.

Key Themes and Contributions

1. AdS/CFT Correspondence: The lectures initiate with an exposition on the AdS/CFT correspondence, emphasizing its twofold utility: a definition of quantum gravity in specific backgrounds and a method to paper strongly interacting field theories. Herzog suggests that this gravitational duality offers a window into phenomena such as superfluidity and superconductivity in condensed matter systems.
2. Quantum Phase Transitions: The discussion progresses to quantum phase transitions, pivotal in understanding the low-temperature phenomena inherent in condensed matter physics. Herzog outlines the conditions under which these transitions occur, typically driven by quantum fluctuations as opposed to thermal ones at absolute zero temperature.
3. Linear Response and Ward Identities: The lectures offer an intricate look into linear response theory and the computation of transport coefficients within strongly correlated systems. Herzog meticulously explains the role of Ward identities in constraining two-point functions of currents and stress-energy tensors, which are foundational in the holographic computation of conductivities.
4. Holographic Models and Applications: Central to Herzog's exposition is the application of holographic models to high temperature superconductors and other materials exhibiting quantum critical behavior. Through Einstein-Maxwell action modifications, he delineates how R-symmetry currents correspond to superfluid and superconducting phases in the boundary field theory.
5. Numerical Insights and Experimental Relevance: The third lecture highlights numerical results from holographic computations, showcasing the hydrodynamic limit and predicting transport behavior, such as conductivities, in strongly coupled regimes. The paper speculates on connections to experimental observations in high $T_c$ superconductors, emphasizing the role of holography in predicting non-trivial collective behaviors like the Nernst effect and superfluid transport.
6. Potential Extensions: Herzog concludes by posing questions about the limit and potential of AdS/CFT techniques in realistic condensed matter systems—highlighting both theoretical challenges and prospective synergies with empirical studies. The ambition is to construct models with gravity duals that replicate the peculiarities of actual materials, possibly even predicting novel phenomena.

Practical and Theoretical Implications

Herzog's lectures underscore several implications. Practically, they propose that quantum phase transitions relevant to material properties, like those seen in high $T_c$ superconductors, might be understood using holographic models. Theoretically, this work emphasizes how gravitational theories can encapsulate complex field-theoretic phenomena, such as superfluidity, which traditionally eludes perturbative approaches.

Furthermore, the paper sparks discussions on extending the AdS/CFT paradigm to non-relativistic situations, providing stiffer tests of the correspondence compared to relativistic settings. The introduction of transport calculations from gravity-side actions illuminates distinct dynamics in the strongly coupled regime, which could be relevant for other critical systems beyond high $T_c$ materials.

Future Developments

Looking ahead, this integration of holography into condensed matter physics may yield new insights into the dynamics of quantum critical points and unconventional superconductors. As computational techniques evolve and empirical tests become more feasible, Herzog's framework could bridge theoretical models and experimental reality, contributing to the development of novel superconductive technologies and the elucidation of elusive quantum phases.

In sum, Herzog's lectures provide a profound and technical framework that could serve as a cornerstone for ongoing research into the holographic depiction of complex condensed matter systems—a step towards merging high-energy theoretical concepts with practical condensed matter applications.