[Paper Review] A Holographic model for Non-Relativistic Superconductor

Shi Pu, Sang-Jin Sin+1 more|ArXiv.org|Mar 25, 2009

Black Holes and Theoretical Physics25 references4 citations

TL;DR

This paper proposes a holographic model for non-relativistic superconductors using gauge/gravity duality in a 4+1 dimensional Einstein gravity background with Schrödinger symmetry. By coupling a complex scalar field to a U(1) gauge field in a black hole spacetime, it demonstrates a second-order phase transition below a critical temperature, leading to charged condensation and infinite DC conductivity. The non-relativistic parameter $\beta$ enhances condensation observability and shifts peak frequencies in the optical conductivity away from $\omega=0$, confirming a robust non-relativistic superconducting phase transition.

ABSTRACT

We build a holographic description of non-relativistic system for superconductivity in strongly interacting condensed matter via gauge/gravity duality. We focus on the phase transition and give an example to show that a simple gravitational theory can provide a non-relativistic holographical dual description of a superconductor. There is also a critical temperature like the relativistic case, below which a charged condensation field appears by a second order phase transition and the (DC) conductivity becomes infinite. We also calculated the frequency dependent conductivity.

Motivation & Objective

• To develop a holographic dual description of strongly correlated non-relativistic superconductors using gauge/gravity duality.
• To investigate whether a second-order phase transition with charged condensation and infinite DC conductivity can emerge in a non-relativistic setting, analogous to the relativistic case.
• To analyze the role of the non-relativistic parameter $\beta$ in modulating the condensation and optical conductivity features.
• To compute the frequency-dependent conductivity and study its dependence on $\beta$ and temperature.

Proposed method

• Construct a 4+1 dimensional Einstein gravity background with a black hole dual to a 2+1 dimensional non-relativistic conformal field theory, using a null Melvin twist of the planar Schwarzschild-AdS black hole.
• Implement an Abelian-Higgs model with a complex scalar field (charged condensate) and a U(1) gauge field in the bulk to model superconductivity.
• Use the Schrödinger symmetry of the background to ensure non-relativistic scaling in the dual field theory.
• Solve the coupled equations of motion for the gauge field and scalar field numerically, imposing infalling boundary conditions at the horizon and regularity at infinity.
• Compute the retarded two-point function for the current-current correlation using the Kubo formula and extract the conductivity from $\sigma(\omega) = \frac{1}{i\omega} G^{R}_{xx}(\omega)$.
• Analyze the behavior of the scalar condensate and conductivity as functions of temperature and the non-relativistic parameter $\beta$.

Experimental results

Research questions

• RQ1 Can a holographic model realize a second-order phase transition to a superconducting phase in a non-relativistic, strongly correlated system?
• RQ2 How does the non-relativistic parameter $\beta$ affect the onset of the charged condensate and the critical temperature?
• RQ3 What is the frequency-dependent conductivity in the non-relativistic superconducting phase, and how does it differ from the relativistic case?
• RQ4 How does $\beta$ influence the position and structure of peaks in the optical conductivity spectrum?

Key findings

• A second-order phase transition occurs below a critical temperature $T_c$, where a charged scalar condensate forms, leading to infinite DC conductivity, analogous to the relativistic case.
• As the non-relativistic parameter $\beta$ increases, the condensate becomes more pronounced, indicating a more robust superconducting phase.
• The frequency-dependent conductivity shows peaks whose positions shift away from $\omega=0$ as $\beta$ increases, indicating enhanced dynamics at finite frequency.
• The near-horizon behavior of the gauge field $A_x \propto (r^4 - r_+^4)^{-i\omega\beta/(4r_+)}$ confirms the hydrodynamic nature of the response and supports the existence of a gapless mode at low frequency.
• The conductivity diverges at $\omega=0$ below $T_c$, confirming the superconducting nature of the phase, with the divergence driven by the non-zero scalar condensate.
• The model demonstrates that non-relativistic superconductivity can be consistently described via holography in a Schrödinger-symmetric background, extending the AdS/CFT framework beyond relativistic systems.