[Paper Review] Holographic p-wave Josephson junction

Yong‐Qiang Wang, Yu-Xiao Liu+1 more|arXiv (Cornell University)|Sep 20, 2011

Black Holes and Theoretical Physics15 references10 citations

TL;DR

This paper constructs a holographic model for a p-wave Josephson junction using (3+1)-dimensional AdS gravity coupled to an SU(2) non-Abelian Yang-Mills gauge theory. Numerically solving the coupled nonlinear equations of motion, the study demonstrates that the DC supercurrent is proportional to the sine of the phase difference across the junction—mirroring the s-wave case and matching experimental BCS-like behavior. The maximum current decays exponentially with junction width, and the critical temperature dependence is consistent with condensed matter expectations.

ABSTRACT

In this work we generalized holographic model for s-wave DC Josephson junction constructed in arXiv:1101.3326[hep-th] to a holographic description for p-wave Josephson junction. By solving numerically the coupled equations of motion of Yang-Mills theory for a non-Abelian SU(2) gauge fields in (3+1)-dimensional AdS spacetimes, we shown that DC current of the p-wave Josephson junction is proportional to the sine of the phase difference across the junction like the s-wave case.

Motivation & Objective

• To extend holographic superconductivity to p-wave pairing by constructing a gravity dual for a p-wave Josephson junction.
• To investigate the Josephson effect in a non-Abelian gauge theory framework within asymptotically AdS spacetime.
• To determine whether the DC current-phase relationship in p-wave junctions matches the sine-law behavior observed in conventional superconductors.
• To analyze the dependence of the critical current on temperature, junction width, and chemical potential profile.
• To extract coherence length and condensate profiles from numerical solutions for comparison with condensed matter theory.

Proposed method

• Formulate a (3+1)-dimensional Einstein-Yang-Mills action in asymptotically AdS spacetime with a planar Schwarzschild-AdS black hole background.
• Impose an SU(2) gauge field ansatz with spatial dependence along the y-direction, including components for scalar-like fields and vector potentials.
• Derive the coupled nonlinear partial differential equations of motion from the Yang-Mills action, including field strength and gauge-covariant derivatives.
• Implement numerical boundary conditions: regularity at the black hole horizon and specific asymptotic behavior at the AdS boundary.
• Use spectral methods to solve the equations numerically, with coordinate transformations z = 1 - r_H/r and ỹ = tanh(y/(4σ)) to improve convergence.
• Extract the supercurrent from the boundary current and phase difference, and fit results to a sine function to determine J_max.

Experimental results

Research questions

• RQ1 Does the holographic p-wave Josephson junction exhibit a current-phase relationship proportional to sin(γ), as in conventional s-wave junctions?
• RQ2 How does the maximum critical current J_max depend on the width L of the junction and temperature T?
• RQ3 Can the coherence length ξ be extracted from the exponential decay of J_max and the condensate profile?
• RQ4 How does the chemical potential profile μ(y) affect the formation of the superconducting gap and Josephson current?
• RQ5 Is the holographic p-wave junction model consistent with the BCS-like behavior seen in real superconductors?

Key findings

• The DC supercurrent in the holographic p-wave Josephson junction is proportional to the sine of the phase difference across the junction, with J_max/T²_c ≈ 1.078.
• The maximum current J_max decays exponentially with increasing junction width L, fitting the form J_max/T²_c = A₀ e^(-ℓ/ξ) with ξ ≈ 1.11.
• The condensate at the center of the junction, ⟨𝒪⟩_x=0, also decays exponentially with width, fitting ⟨𝒪⟩/T²_c = A₁ e^(-ℓ/(2ξ)) with ξ ≈ 1.15.
• The two estimates of the coherence length ξ differ by only 4%, indicating internal consistency in the numerical results.
• The critical current J_max vanishes as temperature approaches the critical temperature T_c, consistent with BCS theory.
• The model reproduces key features of p-wave Josephson junctions in condensed matter systems, including phase-dependent current and exponential decay of supercurrent with junction size.