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Chinese Physics B

Vortex bound states influenced by the Fermi surface anisotropy

Delong Fang

Published Sep 5, 2022• 6 citations
Publisher

Abstract

The spatial distribution of vortex bound states is often anisotropic, which is correlated with the underlying property of materials. In this work, we examine the effects of Fermi surface anisotropy on vortex bound states. The large-scale calculation of vortex bound states is introduced in the presence of fourfold or twofold Fermi surface by solving the Bogoliubov–de Gennes (BdG) equations. Two kinds of quasiparticles' behaviors can be extracted from the local density of states (LDOS) around a vortex. The angle-dependent quasiparticles will move from high energy to low energy when the angle varies from curvature maxima to minima of the Fermi surface, while the angle-independent quasiparticles tend to stay at a relatively higher energy. In addition, the weight of angle-dependent quasiparticles can be enhanced by the increasing anisotropy degree of Fermi surface.

DOI
10.1088/1674-1056/ac8f32

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The spatial distribution of vortex bound states is often anisotropic, which is correlated with the underlying property of materials. In this work, we examine the effects of Fermi surface anisotropy on vortex bound states. The large-scale calculation of vortex bound states is introduced in the presence of fourfold or twofold Fermi surface by solving the Bogoliubov–de Gennes (BdG) equations. Two kinds of quasiparticles' behaviors can be extracted from the local density of states (LDOS) around a vortex. The angle-dependent quasiparticles will move from high energy to low energy when the angle varies from curvature maxima to minima of the Fermi surface, while the angle-independent quasiparticles tend to stay at a relatively higher energy. In addition, the weight of angle-dependent quasiparticles can be enhanced by the increasing anisotropy degree of Fermi surface.
We have carried out a comprehensive investigation of the quasiparticle properties of a two-dimensional electron gas, interacting via the long-range Coulomb interaction in the presence of bare mass anisotropy (i.e., with an elliptic noninteracting Fermi surface) by calculating the self-energy, the spectral function, the scattering rate, and the effective mass within the leading-order dynamical self-energy approximation. Our theory is exact in the high-density limit. We find anisotropic features of quasiparticle properties that are not captured by the commonly used isotropic approximation where the anisotropic effective mass is replaced by the isotropic averaged density-of-states mass. Some of these interesting results are as follows: (1) The many-body renormalization of the quasiparticle spectrum becomes highly anisotropic as the quasiparticle energy increases away from the Fermi energy; (2) the interaction-induced inelastic-scattering rate features a strong anisotropy, exhibiting an abrupt jump at different injected energies depending on the momentum direction of the injected electron; (3) the effective-mass enhancement is larger (smaller) for the light (heavy) mass, showing that the anisotropy is reduced by interactions. Our results and analysis show that the unjustified neglect of the mass anisotropy can lead to an incorrect description of quasiparticle properties of the anisotropic system and inaccurate estimates of physical quantities of interest although the use of an equivalent isotropic approximation using the density-of-states effective mass as is commonly and uncritically performed in the literature, works as a reasonable approximation in many situations. In addition to the complete random phase approximation theory for the anisotropic quasiparticles, we also provide a theory using the simpler plasmon-pole approximation, commenting on its validity for anisotropic self-energy calculations. We comment also on the interaction effect on the Fermi-surface topology, finding that the elliptic shape of the bare Fermi surface is preserved with suppressed ellipticity in the interacting system to a high degree of accuracy except in the very strongly interacting limit (and for very high bare mass anisotropy). Our theory provides a complete generalization of the existing isotropic many-body theory of interacting electrons to the corresponding anisotropic systems.
A theoretical study on discrete vortex bound states is carried out near a vortex core in the presence of a van Hove singularity (VHS) near the Fermi level by solving Bogoliubov–de Gennes (BdG) equations. When the VHS lies exactly at the Fermi level and also at the middle of the band, a zero-energy state and other higher-energy states whose energy ratios follow integer numbers emerge. These discrete vortex bound state peaks undergo a splitting behavior when the VHS or Fermi level moves away from the middle of the band. Such splitting behavior will eventually lead to a new arrangement of quantized vortex core states whose energy ratios follow half-odd-integer numbers.
The temperature dependence of an isolated quantum vortex, embedded in an otherwise homogeneous fermionic superfluid of infinite extent, is determined via the Bogoliubov--de Gennes (BdG) equations across the BCS-BEC crossover. Emphasis is given to the BCS side of this crossover, where it is physically relevant to extend this study up to the critical temperature for the loss of the superfluid phase, such that the size of the vortex increases without bound. To this end, two techniques are introduced. The first one solves the BdG equations with ``free boundary conditions,'' which allows one to determine with high accuracy how the vortex profile matches its asymptotic value at a large distance from the center, thus avoiding a common practice of constraining the vortex in a cylinder with infinite walls. The second one improves on the regularization procedure of the self-consistent gap equation when the interparticle interaction is of the contact type, and permits us to considerably reduce the time needed for its numerical integration by drawing elements from the derivation of the Gross-Pitaevskii equation for composite bosons starting from the BdG equations.
Vortex mass in Fermi superfluids and superconductors and its influence on quantum tunneling of vortices are discussed. The vortex mass is essentially enhanced due to the fermion zero modes in the core of the vortex: the bound states of the Bogoliubov quasiparticles localized in the core. These bound states form the normal component, which is nonzero even in the low-temperature limit. In the collisionless regime ω 0 τ≫1 the normal component trapped by the vortex is unbound from the normal component in a bulk superfluid/superconductor and adds to the inertial mass of the moving vortex. In a d-wave superconductor the vortex mass has an additional factor of (B c2/B)1/2 due to the gap nodes.
Two-loop renormalization-group analysis for a two-dimensional electron system with a partially flat Fermi surface has been carried out. We found that, {\it irrespective of pairing mechanism}, the quasiparticle weight is anisotropically suppressed due to logarithmically singular processes in the flat regions of the Fermi surface. When the energy scale decreases, the quasiparticle weight is the most strongly suppressed around the center of the flat region, which qualitatively agrees with the anisotropic pseudogap behavior suggested through the angle-resolved photoemission spectroscopy experiments for underdoped Bi2_22​Sr2_22​CaCu2_22​O8_88​.
Motivated by understanding Majorana zero modes in topological superfluids in particle-number conserving framework beyond the present framework, we study the effect of particle number conservation on the Berry phase resulting from transport of a bound quasiparticle around a superfluid vortex. We find that particle-number non-conserving calculations based on Bogoliubov-de Gennes (BdG) equations are unable to capture the correct physics when the quasiparticle is within the penetration depth of the vortex core where the superfluid velocity is non-zero. Particle number conservation is crucial for deriving the correct Berry phase in this context, and the Berry phase takes non-universal values depending on the system parameters and the external trap imposed to bind the quasiparticle. Of particular relevance to Majorana physics are the findings that superfluid condensate affects the part of the Berry phase not accounted for in the standard BdG framework, and that the superfluid many-body ground state of odd number of fermions involves superfluid condensate deformation due to the presence of the bound quasiparticle - an effect which is beyond the description of the BdG equations.