[Paper Review] Weyl point immersed in a continuous spectrum: an example from superconducting nanostructures

Y. Chen, Yuli V. Nazarov|arXiv (Cornell University)|Feb 7, 2021

Topological Materials and Phenomena52 references8 citations

TL;DR

This paper investigates a Weyl point in a superconducting nanostructure tunnel-coupled to normal leads, showing that the continuous spectrum induces a new energy scale Γ that smooths topological singularities. It demonstrates that tunnel currents sharply detect the Weyl point even at high temperatures and reveals that topological charge is spread over parameter space as a density, not localized at a point.

ABSTRACT

A Weyl point in a superconducting nanostructure is a generic minimum model of a topological singularity at low energies. We connect the nanostructure to normal leads thereby immersing the topological singularity in the continuous spectrum of the electron states in the leads. This sets another simple and generic model useful to comprehend the modification of low-energy singularity in the presence of continuous spectrum. The tunnel coupling to the leads gives rise to new low energy scale $\Gamma$ at which all topological features are smoothed. We investigate superconducting and normal currents in the nanostructure at this scale. We show how the tunnel currents can be used for detection of the Weyl point. Importantly, we find that the topological charge is not concentrated in a point but rather is spread over the parameter space in the vicinity of the point. We introduce and compute the resulting topological charge density. We also reveal that the pumping to the normal leads helps to detect and investigate the topological effects in the vicinity of the point.

Motivation & Objective

• To understand how immersion of a Weyl point in a continuous spectrum modifies its topological and spectral singularities.
• To investigate the role of tunnel coupling to normal leads in smoothing low-energy singularities associated with the Weyl point.
• To develop a generic analytical model for tunneling to multiple leads that captures isotropy breaking near the Weyl point.
• To explore experimental detection of Weyl points using tunnel currents at temperatures exceeding the level splitting.
• To redefine Berry curvature and compute topological charge density in the presence of continuous spectrum, showing it is no longer point-like.

Proposed method

• Formulates an effective Hamiltonian for a Weyl point in a superconducting nanostructure using three superconducting phase differences as control parameters.
• Applies non-equilibrium Green's function techniques and derives Heisenberg equations of motion for supercurrent and normal lead currents.
• Introduces a generic tunneling model to multiple normal leads, incorporating the new energy scale Γ from tunnel coupling.
• Evaluates supercurrents in equilibrium, stationary tunnel currents under applied voltages, and adiabatic pumping currents via frequency response functions.
• Redefines Berry curvature in the low-frequency limit to compute the topological charge density, revealing its spatial spread.
• Analyzes charge pumping to normal leads as a probe of topological features, showing quantized charge transfer dependent only on the contour encircling the Weyl point.

Experimental results

Research questions

• RQ1 How does coupling a Weyl point to a continuous spectrum via tunneling to normal leads modify its topological and spectral singularities?
• RQ2 What is the role of the tunneling rate Γ in smoothing the topological features of the Weyl point?
• RQ3 Can tunnel currents in the high-voltage and high-temperature regime be used to detect the Weyl point despite the absence of a gap?
• RQ4 How is the topological charge distributed in parameter space when the Weyl point is immersed in a continuous spectrum?
• RQ5 To what extent does adiabatic pumping to normal leads reveal the topological structure near the Weyl point, and how is this affected by Γ?

Key findings

• The tunnel coupling to normal leads introduces a new energy scale Γ, which governs the smoothing of topological singularities and sets the scale for parameter-space variations.
• The maximum derivative of the supercurrent with respect to control phases is set by Γ, indicating a finite resolution for detecting the Weyl point.
• Tunnel currents exhibit sharp features at high voltages and temperatures, enabling experimental detection of the Weyl point even when the level splitting is smaller than k_B T.
• The topological charge is not localized at a point but is spread over a finite region in parameter space, with the charge density explicitly computed and shown to be non-zero in the vicinity of the Weyl point.
• Adiabatic pumping to normal leads results in quantized charge transfer that depends only on the enclosed contour, providing a robust method to probe the Weyl point’s topology.
• The redefined Berry curvature in the low-frequency limit diverges at the Weyl point but yields a finite, non-singular topological charge density, confirming the spreading of the singularity.