The paper by Hartnoll, Herzog, and Horowitz explores the holographic superconductivity in a gravitational dual system. They extend their earlier analysis by considering all values of the scalar field charge and adding a perpendicular magnetic field. Key findings include: 1. **Backreaction on the Metric**: Away from the large charge limit, the backreaction of the scalar field on the spacetime metric becomes significant. Despite this, the qualitative behavior of the dual superconductor remains similar for all charges. However, a new type of black hole instability is observed in the limit of arbitrarily small charge. 2. **London Equation and Magnetic Penetration Depth**: By adding a perpendicular magnetic field \( B \), they derive the London equation and calculate the magnetic penetration depth. They show that these holographic superconductors are Type II, meaning they start in a normal phase at large \( B \) and low temperatures, evolving into superconducting droplets as \( B \) is reduced. 3. **Transport Properties**: The authors compute the electrical, thermal, and thermoelectric conductivities. They find that the system exhibits infinite DC conductivity due to translational invariance, which is distinct from superconductivity. This infinite conductivity is observed in the normal phase above the critical temperature \( T_c \). 4. **Phase Diagram**: They present a phase diagram showing the critical temperature \( T_c \) as a function of the scalar field charge \( q \). The critical temperature is suppressed compared to the probe limit everywhere except at very small \( q \). Even neutral operators can condense at low temperatures. 5. **New Instability**: A new type of instability is identified, where a near-extremal charged black hole becomes unstable to forming neutral scalar hair. This instability is driven by the near horizon geometry of the black hole, which is \( AdS_2 \times \mathbb{R}^2 \). 6. **Euclidean Action**: The Euclidean action for the hairy black hole is computed, showing that it reduces to a simple surface term at infinity. This action is regulated using standard counter terms. 7. **Transport Phenomena**: They study the linear response of the system to fluctuations of the bulk fields, leading to the calculation of the electric, thermal, and thermoelectric conductivities. The results show a minimum in the real part of the conductivity at \( \omega = 0 \) for \( T = T_c \) and above, indicating infinite DC conductivity in the normal phase. Overall, the paper provides a comprehensive analysis of the holographic superconductivity, highlighting the unique properties and mechanisms that arise in this gravitational dual system.The paper by Hartnoll, Herzog, and Horowitz explores the holographic superconductivity in a gravitational dual system. They extend their earlier analysis by considering all values of the scalar field charge and adding a perpendicular magnetic field. Key findings include: 1. **Backreaction on the Metric**: Away from the large charge limit, the backreaction of the scalar field on the spacetime metric becomes significant. Despite this, the qualitative behavior of the dual superconductor remains similar for all charges. However, a new type of black hole instability is observed in the limit of arbitrarily small charge. 2. **London Equation and Magnetic Penetration Depth**: By adding a perpendicular magnetic field \( B \), they derive the London equation and calculate the magnetic penetration depth. They show that these holographic superconductors are Type II, meaning they start in a normal phase at large \( B \) and low temperatures, evolving into superconducting droplets as \( B \) is reduced. 3. **Transport Properties**: The authors compute the electrical, thermal, and thermoelectric conductivities. They find that the system exhibits infinite DC conductivity due to translational invariance, which is distinct from superconductivity. This infinite conductivity is observed in the normal phase above the critical temperature \( T_c \). 4. **Phase Diagram**: They present a phase diagram showing the critical temperature \( T_c \) as a function of the scalar field charge \( q \). The critical temperature is suppressed compared to the probe limit everywhere except at very small \( q \). Even neutral operators can condense at low temperatures. 5. **New Instability**: A new type of instability is identified, where a near-extremal charged black hole becomes unstable to forming neutral scalar hair. This instability is driven by the near horizon geometry of the black hole, which is \( AdS_2 \times \mathbb{R}^2 \). 6. **Euclidean Action**: The Euclidean action for the hairy black hole is computed, showing that it reduces to a simple surface term at infinity. This action is regulated using standard counter terms. 7. **Transport Phenomena**: They study the linear response of the system to fluctuations of the bulk fields, leading to the calculation of the electric, thermal, and thermoelectric conductivities. The results show a minimum in the real part of the conductivity at \( \omega = 0 \) for \( T = T_c \) and above, indicating infinite DC conductivity in the normal phase. Overall, the paper provides a comprehensive analysis of the holographic superconductivity, highlighting the unique properties and mechanisms that arise in this gravitational dual system.