[Paper Review] Magic angles and correlations in twisted nodal superconductors

Pavel A. Volkov, Justin H. Wilson+2 more|arXiv (Cornell University)|Dec 14, 2020

Topological Materials and Phenomena4 citations

TL;DR

This paper proposes that twisted bilayers of two-dimensional nodal superconductors exhibit a 'magic angle' where quasiparticle velocity vanishes due to interlayer hopping, leading to a quadratic band touching. At this angle, interactions induce a time-reversal-symmetry-breaking superconducting state, tunable via displacement field, magnetic field, or current, offering a new route to engineer correlated quantum phases in nodal superconductors like cuprates and heavy fermions.

ABSTRACT

Motivated by recent advances in the fabrication of twisted bilayers of 2D materials, we consider the low-energy properties of a twisted pair of two-dimensional nodal superconductors. We study both the cases of singlet and triplet superconductors. It is demonstrated that the Bogoliubov-de Gennes (BdG) quasiparticle dispersion undergoes dramatic reconstruction due to the twist. In particular, the velocity of the neutral massless Dirac excitations near the gap nodes is strongly renormalized by the interlayer hopping and vanishes at a ``magic angle'' where in the limit of a circular Fermi surface a quadratic band touching is formed. In addition, it is shown that the BdG disperion can be tuned with an interlayer displacement field, magnetic field, and current, which can suppress the velocity renormalization, create finite BdG Fermi surfaces, or open a gap, respectively. Finally, interactions between quasiparticles are shown to lead to the emergence of a correlated superconducting state breaking time-reversal symmetry in the vicinity of the magic angle. Estimates of the magic angle in a variety of nodal superconductors are presented, ranging from the cuprates to the organic and heavy fermion superconductors, all of which are shown to be promising for the experimental realization of our proposal.

Motivation & Objective

• To understand the low-energy quasiparticle spectrum of twisted bilayers of 2D nodal superconductors, particularly near gap nodes.
• To investigate how interlayer hopping and twist angle affect the Bogoliubov-de Gennes (BdG) quasiparticle dispersion and velocity.
• To explore tunability of the system via external fields (displacement, magnetic field, current) and their impact on quasiparticle properties.
• To examine the role of electron-electron interactions in driving a correlated superconducting state that breaks time-reversal symmetry near the magic angle.
• To estimate the magic angle in real materials, including cuprates, organic, and heavy fermion superconductors, for experimental feasibility.

Proposed method

• Formulate a low-energy effective model for twisted nodal superconductors using the Bogoliubov-de Gennes (BdG) formalism to describe quasiparticle excitations.
• Analyze the BdG quasiparticle dispersion under interlayer hopping, showing velocity renormalization and the emergence of a quadratic band touching at a critical 'magic angle'.
• Use a continuum model with a circular Fermi surface to analytically derive the condition for vanishing quasiparticle velocity at the magic angle.
• Apply external perturbations—displacement field, magnetic field, and current—to tune the BdG spectrum, including suppressing velocity renormalization or opening a gap.
• Solve the gap equation with vertex corrections to include electron-electron interactions, showing the emergence of a time-reversal-symmetry-breaking superconducting state near the magic angle.
• Compute the Josephson current-phase relation to assess the impact of the modified spectrum on supercurrent transport, particularly near the magic angle.

Experimental results

Research questions

• RQ1 What is the effect of twist angle on the velocity of neutral Dirac quasiparticles in nodal superconductors?
• RQ2 At what twist angle does the quasiparticle velocity vanish, leading to a quadratic band touching, and how is this 'magic angle' determined?
• RQ3 How do external fields—displacement, magnetic field, and current—tune the BdG quasiparticle spectrum in twisted nodal superconductors?
• RQ4 Do electron-electron interactions lead to a time-reversal-symmetry-breaking superconducting state near the magic angle?
• RQ5 What are the estimates of the magic angle in experimentally relevant nodal superconductors such as cuprates, organic, and heavy fermion materials?

Key findings

• The quasiparticle velocity in twisted nodal superconductors is strongly renormalized by interlayer hopping and vanishes at a 'magic angle' where a quadratic band touching forms, signaling a topological transition.
• The magic angle is analytically derived in the limit of a circular Fermi surface, with the condition that the interlayer hopping energy matches the Fermi velocity and gap scale.
• External displacement fields can suppress velocity renormalization, magnetic fields can open a gap in the BdG spectrum, and applied current can induce a finite Fermi surface in the quasiparticle dispersion.
• Electron-electron interactions lead to the emergence of a correlated superconducting state that breaks time-reversal symmetry in the vicinity of the magic angle.
• Estimates of the magic angle range from ~1° to ~5° across various nodal superconductors, including cuprates, organic, and heavy fermion systems, indicating experimental feasibility.
• The Josephson current-phase relation remains approximately sinusoidal near the magic angle, with logarithmic corrections at very low phase differences, but these are negligible for typical experimental conditions.