Zero Temperature Limit of Holographic Superconductors (0908.3677v1)

Abstract: We consider holographic superconductors whose bulk description consists of gravity minimally coupled to a Maxwell field and charged scalar field with general potential. We give an analytic argument that there is no "hard gap": the real part of the conductivity at low frequency remains nonzero (although typically exponentially small) even at zero temperature. We also numerically construct the gravitational dual of the ground state of some holographic superconductors. Depending on the charge and dimension of the condensate, the infrared theory can have emergent conformal or just Poincare symmetry. In all cases studied, the area of the horizon of the dual black hole goes to zero in the extremal limit, consistent with a nondegenerate ground state.

• The paper demonstrates that holographic superconductors exhibit non-zero conductivity at zero temperature by reformulating conductivity as a reflection coefficient problem.
• The numerical construction of extremal black hole solutions uncovers distinct ground state symmetries—conformal or Poincaré—depending on the scalar field parameters.
• The study confirms a uniquely non-degenerate ground state with a vanishing horizon area, ensuring consistency with theoretical expectations.

Overview of Zero Temperature Limit of Holographic Superconductors

In "Zero Temperature Limit of Holographic Superconductors," Horowitz and Roberts explore the behavior of holographic superconductors in the extremal, zero temperature limit. The paper utilizes the anti-de Sitter/conformal field theory (AdS/CFT) correspondence to model superconductivity via gravitational theories, specifically through gravity minimally coupled to a Maxwell field and a charged scalar field.

The research explores properties such as conductivity in the zero temperature limit, emergent symmetries, and the near-horizon geometries of extremal black holes that are dual to the superconductors. It leverages both analytical arguments and numerical constructions to reach its conclusions.

Key Contributions

1. Conductivity Behavior in Zero Temperature:
   • The paper provides an analytical argument that holographic superconductors do not have a "hard gap" at zero temperature. Specifically, the real part of the conductivity remains non-zero, although typically exponentially small, even as temperature approaches zero. This insight corrects any previous anticipations of a strictly zero conductivity at the zero temperature limit.
   • By recasting the conductivity in terms of a one-dimensional Schrödinger problem, they show that the conductivity is related to a reflection coefficient, clarifying the reasons behind the non-zero conductivity and the absence of a hard gap. This contributes to a deeper understanding of the frequency dependence and the low-temperature gaps numerically observed in finite-temperature studies.
2. Numerical Construction of Extremal Black Hole Solutions:
   • The extremal limits of charged black hole solutions (emergent from the gravitational dual description) were numerically constructed. This was done for cases with different masses and charges of the scalar field. The authors discovered that the behavior of the ground state varies based on the parameters, leading to either conformal or Poincaré symmetry in the infrared region.
   • Specifically, for $m^2 = 0$, the solution in the near-horizon limit restores conformal symmetry, with the extremal horizon being a simple AdS geometry. Whereas for $m^2 < 0$ and specific value ranges for the charge, they identified solutions with emergent Poincaré symmetry.
3. Implications of the Zero Temperature Ground State:
   • The ground state was found to be uniquely non-degenerate as indicated by the vanishing horizon area in the extremal limit. This feature assures the consistency of the constructed solution with expected physical behavior, mitigating concerns about degeneracy in previous theoretical models.

Implications and Future Directions

This paper extends the understanding of phase transitions and ground state properties in holographic superconductors. It particularly challenges previously held assumptions about the emergence of a hard gap at zero temperature and offers a unified theoretical framework for examining superconductive properties through holography. The transformation of conductivity analysis into a reflection coefficient problem introduces a method that could be applied to other related condensed matter systems.

Future developments in this area of research could further explore the ramifications of the absence of a hard gap on theoretical predictions of thermodynamic and transport properties in lower dimensional systems. Additionally, extending these results to understand real superconductive materials better or to other dimensions within the AdS/CFT framework remains a promising prospect.

By showing that the ground state exhibits either conformal or merely scale invariance, the paper opens avenues for examining how such dual descriptions might inform both classical gravity theories and condensed matter systems, especially under extremal conditions. Potential further studies could also delve into the role of more complex scalar potentials or non-minimal scalar couplings in shaping the low-temperature behavior and symmetry of holographic superconductors.