We study the vortex zero-energy bound states in presence of pairing among the low-energy Dirac fermions on the surface of a topological insulator. The pairing symmetries considered include the $s$-wave, $p$-wave, and, in particular, the mixed-parity symmetry, which arises in absence of the inversion symmetry on the surface. The zero-mode is analyzed within the generalized Jackiw-Rossi-Dirac Hamiltonian that contains a momentum-dependent mass-term, and includes the effects of the electromagnetic gauge field and the Zeeman coupling as well. At a finite chemical potential, as long as the spectrum without the vortex is fully gapped, the presence of a single Fermi surface with a definite helicity always leads to one Majorana zero-mode, in which both electron's spin projections participate. In particular, the critical effects of the Zeeman coupling on the zero-mode are discussed.
The pairing symmetry of interacting Dirac fermions on the $π$-flux lattice is studied with the determinant quantum Monte Carlo and numerical linked cluster expansion methods. The extended $s^*$- (i.e. extended $s$-) and d-wave pairing symmetries, which are distinct in the conventional square lattice, are degenerate under the Landau gauge. We demonstrate that the dominant pairing channel at strong interactions is an exotic $ds^*$-wave phase consisting of alternating stripes of $s^*$- and d-wave phases. A complementary mean-field analysis shows that while the $s^*$- and d-wave symmetries individ…
Layered kagome metals AV3Sb5 (A=K, Rb, Cs) exhibit diverse correlated electron phenomena. It includes charge density wave formation and superconductivity the pairing symmetry of which, however, is controversial due to contradictory experimental evidence. Through calculations based on real-space lattice models at the mean-field level, we investigate the vortex and surface spectra of all competitive pairing propensities suggested for AV3Sb5 from a weak coupling analysis of unconventional superconductivity. Chiral p-wave pairing emerges as the only option to host Majorana bound states in the vort…
We study the superconducting pairing symmetry in twisted bilayer graphene by solving the Bogoliubov-de Gennes equation for all electrons in moiré supercells. With increasing the pairing potential, the system evolves from the mixed nontopological $d+id$ and $p+ip$ phase to the $s+p+d$ phase via the first-order phase transition. In the time-reversal symmetry breaking $d+id$ and $p+ip$ phase, vortex and antivortex lattices accompanying spontaneous supercurrent are induced by the twist. The superconducting order parameter is nonuniform in the moiré unit cell. Nevertheless, the superconducting gap …
The helical electron states on the surface of topological insulators or elemental Bismuth become unstable toward superconducting pairing formation when coupled to the charge or magnetic fluctuations. The latter gives rise to pairing instability in chiral channels $d_{xy}\pm i d_{x^2-y^2}$, as has been observed recently in epitaxial Bi/Ni bilayer system at relatively high temperature, while the former favors a pairing with zero total angular momentum. Motivated by this observation we study the vortex bound states in these superconducting states. We consider a minimal model describing the superc…
We theoretically investigate qualitative features of the field-temperature ($H$-$T$) phase diagram of superconductors with strong attractive interaction lying in the BCS-BEC crossover regime. Starting with a simple attractive Hubbard model, we estimate three kinds of characteristic fields, i.e., the pair-formation field $H^*$, the vortex-liquid-formation field $H_\mathrm{c2}$, and the vortex-lattice-formation field $H_\mathrm{melt}$. The region between $H_\mathrm{c2}$ and $H_\mathrm{melt}$, as well as that between $H_\mathrm{c2}$ and $H^*$, is found to be enlarged as the interaction is stronge…
A recent proposal argues that an alternate description of the half-filled Landau level is a theory of massless Dirac fermions. We examine the possibility of pairing of these Dirac fermions by numerically solving the {\em coupled} Eliashberg equations unlike our previous calculation [Wang and Chakravarty, Phys. Rev. B {\bf 94}, 165138 (2016)]. In addition, vertex corrections are calculated to be zero from the Ward identity. We find that pairing is possible in non-zero angular momentum channels; the only differences are minor numerical shifts. As before, the pairing leads to the gapped Pfaffian …
Vortex core states in a minimal two-band model for iron-based superconductors
对对称性是铁基超导体研究中的主要问题之一。我们采用一个最小的双带紧束缚模型，考虑了不同的配对作用通道，导出了一组双带Bogoliubov-de Gennes（BdG）方程。BdG方程在实空间中实现，然后通过精确对角化自洽求解。在均匀情况下，我们发现$d_{x^2-y^2}$波配对态最有利于最近邻配对相互作用，而$s_{x^2y^2}$波配对态最有利于次最近邻配对相互作用.这与Seo { \em et al.}报道的结果一致。[Phys. Rev. Lett. { \bf 101}，206404（2008）]。然后，我们继续研究的局部电子结构周围的磁涡旋核心的$d_{x^2-y^2}$波和$s_{x^2y^2}$波配对对称的混合状态。从局域态密度（LDOS）谱及其空间变化发现，对于$d_{x^2-y^2}$波对对称，费米能附近的共振芯态是束缚态，而对于$s_{x^2y^2}$波对对称，随着电子填充因子的变化，共振芯态可以从局域态演化到扩展态.此外，通过包括有效的交换相互作用，出现的反铁磁自旋密度波（SDW）序可以抑制共振芯态，这为理解Yin { \em et al.}最近的扫描隧道显微镜实验（STM）所揭示的共振峰的缺失提供了一种可能的途径。[Phys. Rev. Lett. { \bf 102}，097002（2009）]。
Double Majorana Vortex Flat Bands in the Topological Dirac Superconductor
Hong Kong University of Science and Technology（香港科技大学）；Beijing National Laboratory for Condensed Matter Physics, and Institute of Physics, Chinese Academy of Sciences（中国科学院物理研究所）；School of Physical Sciences, University of Chinese Academy of Sciences（中国科学院大学物理科学学院）；School of Physics, Beijing Institute of Technology（北京理工大学物理学院）；Nanjing University（南京大学）；CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences（中国科学院理论物理研究所）
Topological Majorana and Dirac zero modes in superconducting vortex cores
我们提供了一个参数的磁通插入的基础上表明，某些超导体与非平凡的拓扑不变的保护零模式在其涡核。这个论点具有二维指数定理的味道，也适用于无序系统。这也为理解无自旋$p_{x} + i p_{y}$超导体涡旋核中的零模提供了一种新的途径。将这种方法应用于具有和不具有时间反转和自旋旋转对称性的超导体，我们预测了其涡旋中存在受保护零模的充要条件。
Time-reversal invariant (TRI) Dirac and Weyl semimetals in three dimensions (3D) can host open Fermi arcs and spin-momentum locking Fermi loops on the surfaces. We find that when they become superconducting with $s$-wave pairing and the doping is lower than a critical level, straight $π$-flux vortex lines terminating at surfaces with Fermi arcs or spin-momentum locking Fermi loops can realize 1D topological superconductivity and harbor Majorana zero modes at their ends. Remarkably, we find that the vortex-generation-associated Zeeman field can open (when the surfaces have only Fermi arcs) or e…
We formulate a general framework for addressing both odd- and even-frequency superconductivity in Dirac semimetals and demonstrate that the odd-frequency or the Berezinskii pairing can naturally appear in these materials because of the chirality degree of freedom. We show that repulsive frequency-dependent interactions favor the Berezinskii pairing while an attractive electron-electron interaction allows for the BCS pairing. In the case of compensated Dirac and Weyl semimetals, both the conventional BCS and odd-frequency Berezinskii pairings require critical coupling. Since these pairings coul…
Pairing around a Single Dirac Point: A Unifying View of Kohn-Luttinger Superconductivity in Chern Bands, Quarter Metals, and Topological Surface States
Department of Physics and Astronomy, University of California, Irvine（加州大学尔湾分校物理与天文系）
We study the vortex bound states in three dimensional (3D) superconducting Dirac semimetals with time reversal symmetry. Assuming two Dirac points on the kz-axis and bulk s-wave superconductivity, with a quantum vortex line parallel to the z-direction, we find that the superconducting vortex line has a robust quasi-1D nodal phase. The nodal phase stems from the symmetry protected Dirac points in the normal state bands, and it can be characterized by a topological index (ν; n) at kz = 0 and kz = π, where νis the Z2 topological invariant for a 0D class-D system and n is the Z topological invaria…
In the superconducting regime of FeTe$_{(1-x)}$Se$_x$, there exist two types of vortices which are distinct by the presence or absence of zero energy states in their core. To understand their origin, we examine the interplay of Zeeman coupling and superconducting pairings in three-dimensional metals with band inversion. Weak Zeeman fields are found to suppress the intra-orbital spin-singlet pairing, known to localize the states at the ends of the vortices on the surface. On the other hand, an orbital-triplet pairing is shown to be stable against Zeeman interactions, but leads to delocalized ze…
The interplay of nontrivial topology and superconductivity in condensed matter physics gives rise to exotic phenomena. However, materials are extremely rare where it is possible to explore the full details of the superconducting pairing. Here, we investigate the momentum dependence of the superconducting gap distribution in a novel Dirac material PdTe. Using high resolution, low temperature photoemission spectroscopy, we establish it as a spin-orbit coupled Dirac semimetal with the topological Fermi arc crossing the Fermi level on the (010) surface. This spin-textured surface state exhibits a …
We utilize a combination of vector magnetic field and scanning tunneling microscopy to elucidate the 3D field-based electronic phase diagram of a correlated iron-based superconductor, LiFeAs. We observe, under a zero-field-cooled method, an ordered hexagonal vortex lattice ground-state in contrast to the disordered lattice observed under a field-cooled method. It transforms into a four-fold-symmetric state by increasing c-axis field and distorts elliptically upon tilting the field in-plane. The vortex lattice transformations correlate with the field dependent superconducting gap that character…
Supersymmetric Runge-Lenz-Pauli vector for Dirac vortex in topological insulators and graphene
研究了拓扑绝缘体表面和石墨烯中的狄拉克质量涡旋。在涡旋振幅径向依赖性的线性近似下，谱是一系列简并束缚态，可以通过一组偶然的SU（2）和超对称生成元（I. F. Herbut和C.- K. Lu，Phys.Rev.B83125412（2011））。本文进一步讨论了涡旋哈密顿量的超对称性的性质和表现，并指出了与非相对论性氢原子中Runge-Lenz-Pauli矢量的一些有趣的相似之处。分析了有限化学势和塞曼场对对称性破缺的影响。我们发现，一个剩余的偶然简并仍然只有在特殊情况下的两项的大小相等，而在其他情况下，它成为完全删除。
Physics of topological materials have attracted much attention from both physicists and mathematicians recently. The index and the fermion number of Dirac fermions play an important role in topological insulators and topological superconductors. A zero-energy mode exists when Dirac fermions couple to objects with soliton-like structure such as kinks, vortices, monopoles, strings and branes. We discuss a system of Dirac fermions interacting with a vortex and a kink. This kind of systems will be realized on the surface of topological insulators where Dirac fermions exist. The fermion number is f…
Dirac-Bogoliubov-deGennes quasiparticles in a vortex lattice
在极端Ⅱ型d波超导体的混合态和宽范围的弱磁场H_c1 << H << H_c2中，低能Bogoliubov-deGennes准粒子可以有效地描述为Dirac费米子在奇异标量势和矢量势场中的运动.虽然有效的线性化哈密顿算子形式上不依赖于涡核的结构，奇异性质的扰动需要选择一个自伴扩展的哈密顿施加额外的边界条件在涡的位置。每个涡旋都由一个参数θ描述，该参数有效地代表了超出线性化范围的物理效应。当θ适当固定时，狄拉克哈密顿量的态密度在任意的奇异规范变换下都表现出完全的不变性。我们确定的自伴扩展的解决方案发现较早，在平面波的基础上展开对角化的线性化哈密顿的框架内，并分析完全自洽制定的问题和线性化模型之间的关系。
The presence or absence of certain symmetries in the normal state (NS) also determines the symmetry of the Cooper pairs. Here we show that parity (${\mathcal P}$) and time-reversal (${\mathcal T}$) odd Dirac insulators (trivial or topological) or metals, sustain a local or intra-unit cell pairing that supports corner (in $d=2$) or hinge (in $d=3$) modes of Majorana fermions and stands as a higher-order topological superconductor (HOTSC), when the NS additionally breaks discrete four-fold ($C_4$) symmetry. Although these outcomes does not rely on the existence of a Fermi surface, around it (whe…
Chiral Majorana hinge modes are characteristic of a second-order topological superconductor in three dimensions. Here we systematically study pairing symmetry in the point group D_{2h}, and find that the leading pairing channels can be of s-, d-, and s+id-wave pairing in Dirac materials. Except for the odd-parity s-wave pairing superconductivity, the s+id-wave pairing superconductor is topologically nontrivial and possesses Majorana hinge and surface modes. The chiral Majorana hinge modes can be characterized by a winding number of the quadrupole moment, or quantized quadruple moment at the sy…
Under what conditions does a superconductor inherit topologically protected nodes from its parent normal state? In the context of Weyl semimetals with broken time-reversal symmetry, the pairing order parameter is classified by monopole harmonics and necessarily nodal [Li and Haldane, Phys. Rev. Lett., 120, 067003 (2018)]. Here, we show that a similar conclusion could also apply to 2D Dirac semimetals, although the conditions for the existence of nodes are more complex, depending on the pairing matrix structure in the valley and sublattice space. We analytically and numerically analyze the Bogo…
Motivated by recent developments in the experimental study of superconducting graphene and transition metal dichalcogenides, we investigate superconductivity of the Kane-Mele (KM) model with short-range attractive interactions on the two-dimensional honeycomb lattice. We show that intra-valley spin-triplet pairing arises from nearest-neighbor (NN) attractive interaction and the intrinsic spin-orbit coupling. We demonstrate this in two independent approaches: We study superconducting instability driven by condensation of Cooperons, which are in-gap bound states of two conduction electrons, with…
An exact correspondence is established between mathematical description of the single "elementary vortex" (EV)velocity on the sphere (Zermelo,1902;Bogomolov,1977) and the Dirac magnetic monopole (DMM) vector potential (Dirac, 1931). Similar analogy with DMM was noted only for the vortices in quantum fluid He-3A (Blaha,1976;Volovik,Mineev,1976). Singular EV on a sphere is usually considered using compensating vortex field, uniformly distributed over the sphere. It is necessary to meet the Gauss-Kelvin theorem with zero integral vorticity over the sphere. However since Bogomolov (1977)(see also …
