Ginzburg-Landau Formalism in a Tilted Dirac Cone Metric

TL;DR

The paper addresses how to formulate a Ginzburg-Landau theory for superconductivity in tilted Dirac cone materials where the effective spacetime metric is curved and tunable by an external electric field. It derives the GL action from the microscopic BCS Hamiltonian in curved spacetime using the Painlevé–Gullstrand metric with tilt $\boldsymbol{\zeta}$, employing Hubbard–Stratonovich decoupling and integrating out fermions via $-\mathrm{tr}\ln \Upsilon^{-1}$. The resulting GL functional depends explicitly on spacetime curvature through covariant derivatives $\nabla_\rho$ and spin connections, yielding a curvature-sensitive description valid near the critical temperature $T_c$. This curvature-tunability suggests that tilt-induced metric control could extend GL applicability to higher temperatures and potentially realize high-$T_c$ superconductivity at lower pressures by adjusting the external electric field. Overall, the work links general-relativistic formalism with condensed-matter superconductivity in Dirac materials, offering a route to experimentally engineer superconducting states via metric engineering.

Abstract

Recent researches on tilted Dirac cone materials have unveiled an astonishing property, the metric of the spacetime can be altered in these materials by applying a perpendicular electric field. This phenomenon is observed near the Fermi velocity, which is significantly lower than the speed of light. According to this property, we derive the Ginzburg-Landau action from the microscopic Hamiltonian of the BCS theory for the tilted Dirac cone materials. This derivation is performed near the critical point within the framework of curved spacetime. The novelty of the present work lies in deriving a general Ginzburg-Landau action that depends on spacetime curvature, where the curvature is tuned by an external electric field. Furthermore, this finding enables us to apply the Ginzburg-Landau theory at high temperatures by changing the spacetime metric, potentially offering insights into achieving high-temperature superconductivity in these materials.

Figures (1)

• Figure 1: Interaction of electrons via exchange of a boson at the Fermi surface due to the exchange of crystal lattice phonons. $k$ and $\sigma$ are wave vector and spin of the electron, respectively.