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Chinese Physics B

Discrete vortex bound states with a van Hove singularity in the vicinity of the Fermi level

Delong Fang, Yunkang Cui

Published Nov 29, 2022• 1 citation
Publisher

Abstract

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.

DOI
10.1088/1674-1056/aca6d6

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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.
In order to clarify the structure of a singly quantized vortex in a superfluid fermion gas near the Feshbach resonance, we numerically solve the generalized Bogoliubov-de Gennes equation in the boson-fermion model. The superfluid gap, which contains contributions from both condensed fermion pairs and condensed bosons, is self-consistently determined, and the quasiparticle excitation levels bound in the vortex core are explicitly shown. We find that the boson condensate contributes to enhance the matter density depletion and the discreteness of localized quasiparticle spectrum inside the core. It is predicted that the matter density depletion and the discrete core levels are detectable in the vicinity of the BCS-Bose-Einstein condensation crossover point.
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 examine the effects of a van Hove singularity (vHs) in the density of states (DOS) of a two dimensional, t−t′−Ut-t'-Ut−t′−U Hubbard model within the local approximation. Within our approximation the non-interacting Fermi surface is always retained away from half-filling; and we find that the vHs in the DOS survives, and for hole doping is nearly "pinned" to the Fermi level due to the enhancement of the effective mass, m∗m^*m∗. We discuss the electron-hole asymmetry of this phenomena and its relevance to the electron and hole doped cuprates. At optimal (hole) doping, with the Fermi level at the vHs, we find that the single particle damping rate ΓΓΓ, and the resistivity ρdcρ_{dc}ρdc​, scale as TTT for moderately low temperatures (T≳100T \gtrsim 100T≳100K), while for the electron doped system ΓΓΓ and ρdc∼T2ρ_{dc} \sim T^2ρdc​∼T2.
The structures of multiply quantized vortices (MQVs) of an equal-population atomic Fermi superfluid in a rotating spherical bubble trap approximated as a thin shell are analyzed by solving the Bogoliubov--de Gennes (BdG) equation throughout the Bardeen-Cooper-Schrieffer (BCS)--Bose-Einstein condensation (BEC) crossover. Consistent with the Poincar\'e-Hopf theorem, a pair of vortices emerge at the poles of the rotation axis in the presence of azimuthal symmetry and the compact geometry provides confinement for the MQVs. While the single-vorticity vortex structure is similar to that in a planar geometry, higher-vorticity vortices exhibit interesting phenomena at the vortex center, such as a density peak due to accumulation of a normal Fermi gas and reversed circulation of current due to in-gap states carrying angular momentum in the BCS regime but not the BEC regime because of the subtle relations between the order parameter and density. The energy spectrum shows the number of the in-gap state branches corresponds to the vorticity of a vortex, and an explanation based on a topological correspondence is provided.
Controlling the ordered phases of correlated electron systems remains a central challenge in quantum materials design. Divergences in the electronic density of states, known as Van Hove singularities (VHSs), are one obvious route to such control. It is clear from recent work that the exact functional form of these divergences can profoundly affect which phases are realized; a full picture, however, remains elusive. In this work, we use both the hot-spot parquet renormalization group and the truncated-unity functional renormalization group to theoretically study the emergent correlated states of a two-dimensional square-lattice Hubbard model with VHSs at or near the Fermi level. By varying a single hopping parameter, t3t_3t3​, we are able to change the strength of the VHS divergence in the density of states from logarithmic (for t3<t3ct_3 < t_{3c}t3​<t3c​) to power-law (for t3=t3ct_3 = t_{3c}t3​=t3c​). Further increase of t3t_3t3​ (t3>t3ct_3 > t_{3c}t3​>t3c​) causes each original Van Hove point to split into two, both of the conventional logarithmic type. We show that which of these regimes we are in strongly influences the predicted ordered states. We also study the dependence on doping, and find that the ferromagnetic state that occurs at Van Hove filling in these models is unstable to very small shifts in the Fermi level, often giving way to distinct ordered states depending on whether the model is electron- or hole-doped. These results highlight the importance of tuning VHS properties to control ordered states in correlated materials, and offer design rules to engineer these phases in novel systems.
We devise a computational approach to the Hubbard model that captures the strong coupling dynamics arising when the Fermi level is at a Van Hove singularity in the density of states. We rely on an approximate degeneracy among the many-body states accounting for the main instabilities of the system (antiferromagnetism, d-wave superconductivity). The Fermi line turns out to be deformed in a manner consistent with the pinning of the Fermi level to the Van Hove singularity. For a doping rate $\ensuremath{\delta}\ensuremath{\sim}0.2,$ the ground state is characterized by d-wave symmetry, quasiparticles gapped only at the saddle points of the band, and a large peak at zero momentum in the d-wave pairing correlations.