Building an AdS/BCFT Josephson junction within Horndeski gravity

Technical Summary: Building AdS/BCFT Josephson Junctions within Horndeski Gravity

Problem Statement
The paper addresses the challenge of modeling superconducting Josephson junctions—specifically constriction-type and Superconductor/Normal/Superconductor (SNS) junctions—within a holographic framework that extends beyond standard Einstein gravity. While the AdS/CFT correspondence has been successfully applied to holographic superconductors and Josephson junctions, and the AdS/BCFT (Boundary Conformal Field Theory) extension has been used to describe junctions with boundaries, these models typically rely on Einstein gravity. The authors propose investigating these systems within Horndeski gravity, the most general scalar-tensor theory with second-order equations of motion. The goal is to determine how Horndeski parameters (coupling constants $\alpha$ and $\gamma$) modify the gravitational dual, the formation of charged condensates, and the resulting Josephson current-phase relations.

Methodology
The authors employ the AdS/BCFT correspondence within the framework of Horndeski gravity. The methodology involves the following steps:

1. Holographic Setup: The bulk geometry is modeled as an asymptotically AdS$_4$ planar Schwarzschild black hole. The boundary theory is defined on a manifold with an additional boundary $\partial\Omega$, representing the interface of the Josephson junction.
2. Action and Fields: The total action includes the Horndeski gravity sector (involving a scalar field $\phi$ coupled to the Ricci scalar $R$ and Einstein tensor $G_{\mu\nu}$), a Maxwell field $A_\mu$, and a charged complex scalar field $\Psi$ (the order parameter). The authors work in the probe approximation ($q \to \infty$), where the backreaction of the matter fields on the metric is neglected, allowing the focus to remain on the dynamics of the scalar and gauge fields.
3. Boundary Conditions:
   • Dirichlet Boundary Conditions (DBC): Applied at the asymptotic AdS boundary ($M$).
   • Neumann Boundary Conditions (NBC): Applied at the end-of-the-world brane ($\partial\Omega$). This condition is crucial for the dynamical nature of the boundary and is derived from the variation of the generalized Gibbons–Hawking–York term modified by Horndeski terms.
4. Junction Profiles: Two distinct geometric configurations are constructed by solving the equations of motion for the induced metric on $\partial\Omega$:
   • Constriction Junction: Formed by two profiles meeting at a gap, controlled by boundary tension $\Sigma$.
   • Normal (SNS) Junction: Formed by gluing two AdS geometries where the scalar field represents the weak link between superconductors.
5. Analysis: The authors derive analytic expressions for the condensate expectation value $\langle \mathcal{O} \rangle$ and the maximum Josephson current $J_{max}$ as functions of the Horndeski parameters, temperature, and black hole horizon radius.

Key Contributions
The paper claims the following specific contributions:

• First Construction in Horndeski/AdS-BCFT: This is the first work to construct both constriction-type and SNS-type Josephson junctions within the AdS/BCFT framework using Horndeski gravity.
• Unified Analytic Framework: The authors derive unified analytic expressions for the condensate and the maximum Josephson current ($J_{max}$) that explicitly depend on the Horndeski coupling parameters ($\alpha, \gamma$).
• Control of Junction Geometry: The study demonstrates that Horndeski parameters provide a controlled mechanism to deform the Josephson relation and the coherence length ($\zeta$), effectively interpolating between different effective junction geometries from the gravitational side.
• Phase Transition Characterization: The work identifies a critical temperature ($T_c$) below which a charged condensate forms via a second-order phase transition in both junction types.

Results

• Condensate Formation: A charged condensate forms below a critical temperature $T_c$. The condensate is identified as comprising pairs of quasiparticles.
• Current-Phase Relation: The supercurrent $J$ follows the Josephson relation $J = J_{max} \sin(\Gamma)$, where $\Gamma$ is the phase difference. The magnitude of the supercurrent is modulated by the Horndeski parameters.
• Constriction Junctions:
  • The supercurrent exhibits an exponential decay with increasing gap width ($\Sigma$) and the parameter $\gamma$.
  • The coherence length $\zeta$ is derived as $\zeta \equiv \sqrt{(\alpha + \gamma\Lambda)/(6\alpha\gamma)}$. The authors note that large $\gamma$ or small $\alpha$ leads to a small coherence length.
  • The model reproduces the exponential behavior of $J_{max}$ observed in experimental planar junctions.
• SNS Junctions:
  • The scalar field profile allows for highly transparent superconductor-normal interfaces, facilitating Andreev reflection.
  • Analytic solutions for the condensate and current are obtained, showing that the condensate remains finite at zero temperature, ensuring the validity of the approximation.
  • The sensitivity of the tunneling current to the $\gamma$ parameter is significant, particularly at low temperatures. Large $\gamma$ values reduce the coherence length but allow the current to persist by reducing thermal noise.
• Transport Properties: The Horndeski parameters influence the transport properties, allowing the system to exhibit metallic or insulating characteristics depending on the sign and magnitude of the parameters.

Significance and Claims
The authors position this work as an extension of previous holographic Josephson junction studies (which were limited to Einstein gravity or lacked the specific Horndeski modifications). They claim that their model offers a "controlled way" to interpolate between junction geometries using gravitational parameters, which is of "independent interest for the holographic modeling of superconducting interfaces."

The paper emphasizes that the inclusion of Horndeski gravity allows for the avoidance of the "no-hair theorem" constraints in a specific manner (via gauge fixation and specific conditions on the scalar field) to support non-trivial solutions. The results align qualitatively with experimental observations of SNS and constriction junctions, particularly regarding the temperature dependence of resistance and the exponential decay of the critical current. The work suggests that Horndeski gravity provides a robust dual description for superconducting phase transitions and transport phenomena, offering new insights into how gravitational corrections might influence condensed matter systems.