Chiral-Flux-Phase-Based Topological Superconductivity in Kagome Systems with Mixed Edge Chiralities

Junjie Zeng, Qingming Li, Xun Yang, Dong-Hui Xu, and Rui Wang

Phys. Rev. Lett. 131, 086601 – Published 25 August 2023

Abstract

Recent studies have attracted intense attention on the quasi-2D kagome superconductors $A V_{3} {Sb}_{5}$ ( $A = K$ , Rb, and Cs) where the unexpected chiral flux phase (CFP) associates with the spontaneous time-reversal symmetry breaking in charge density wave states. Here, commencing from the 2-by-2 charge density wave phases, we bridge the gap between topological superconductivity and time-reversal asymmetric CFP in kagome systems. Several chiral topological superconductor (TSC) states featuring distinct Chern numbers emerge for an $s$ -wave or a $d$ -wave superconducting pairing symmetry. Importantly, these CFP-based TSC phases possess unique gapless edge modes with mixed chiralities (i.e., both positive and negative chiralities), but with the net chiralities consistent with the Bogoliubov–de Gennes Chern numbers. We further study the transport properties of a two-terminal junction, using Chern insulator or normal metal leads via atomic Green’s function method with Landauer-Büttiker formalism. In both cases, the normal electron tunneling and the crossed Andreev reflection oscillate as the chemical potential changes, but together contribute to plateau transmissions (1 and $\frac{3}{2}$ , respectively) that exhibit robustness against disorder. These behaviors can be regarded as the signature of a TSC hosting edge states with mixed chiralities.

Received 30 March 2023
Revised 25 June 2023
Accepted 4 August 2023

DOI:https://doi.org/10.1103/PhysRevLett.131.086601

Physics Subject Headings (PhySH)

Andreev reflection Charge density waves Edge states Spin-orbit coupling Spin-singlet pairing Superconductivity Topological phases of matter Topological superconductors

2-dimensional systems

Bogoliubov-de Gennes equations Green's function methods Landauer formula Second quantization Tight-binding model

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Junjie Zeng ¹, Qingming Li ², Xun Yang¹, Dong-Hui Xu ^1,*, and Rui Wang ^1,3,†

¹Institute for Structure and Function and Department of Physics and Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing 400044, People’s Republic of China
²Department of Physics, Shijiazhuang University, Shijiazhuang, Hebei 050035, People’s Republic of China
³Center of Quantum materials and devices, Chongqing University, Chongqing 400044, People’s Republic of China

^*donghuixu@cqu.edu.cn
^†rcwang@cqu.edu.cn

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Vol. 131, Iss. 8 — 25 August 2023

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Images

Figure 1
Panels (a)–(c): Graph representations of the three kinds of CDW states, where the light-blue-edged hexagons represent the primitive cells. In panel (a), the blue and green nodes have negative and positive on-site energy modification to the chemical potential $- (μ \pm λ^{vCDW})$ , and the black edge connecting a pair of nodes represents the nearest-neighbor hopping. In panel (b), all the on-site energies are the same, but the hoppings are modified: the red edges stands for the normal hopping $- t$ , the thicker green ones enhanced $- (t + λ^{CBO})$ , and the thinner blue ones weakened $- (t - λ^{CBO})$ . In panel (c), the hopping is modified by a pure imaginary number to account for the chiral flux $- (t \pm i λ^{CFP})$ , where the sign is indicated by the directions of the arrows. Panel (d): A schematic depiction of a two-terminal device to study quantum transport behavior of the topological superconducting phase discovered in this work, where an ( $L \times W$ )-sized central scattering region is constructed by translating the inset minimal unit along the red and green vectors $L$ and $W$ times, respectively, with the atoms (with dangling bonds) on both edges trimmed.
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Figure 2
Bulk band structures by combination of three CDW phases and two superconducting pairings. Panels (a)–(d): Without breaking the time-reversal symmetry, a phase with a nonvanishing BdG Chern number cannot be found from the first two CDW phases. Panels (e),(f): Topologically nontrivial superconducting phases emerge from the last CDW order. Other parameters are $μ = 0.1, λ^{vCDW / CBO / CFP} = 0.25$ , $Δ^{s} = 0.065$ , and $Δ^{d_{x^{2} - y^{2}}} = 0.15$ . All energy dimensioned quantities are measured in $t$ .
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Figure 3
Phase diagrams for CFP with $s$ -wave superconducting pairings and relevant gapless edge states for TSC. Panel (a): Without RSOC, most region occupied by the $N = 4$ phase, whose corresponding edge states are shown in panel (c), with spin degeneracy. Panel (b): With RSOC, TSC phases appear, where the region of interest is enlarged in panel (d), and the edge states of two typical TSC phases are respectively shown in panels (e) and (f). The horizontal line segment in panel (d) indicates the parameter samplings for transport study part of this work shown in Fig. 4.
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Figure 4
TSC-related transport coefficients for a two-terminal device with an ( $L \times W$ )-sized sheet as its central scattering region. For all panels, the sheet width is fixed at $W = 80$ , the superconducting pairing potential is $Δ^{s} = 0.065$ , and $E = 0^{+}$ . Panels (a)–(c): Chern insulator leads with the sheet length $L = 40$ , 50, and 60, respectively. Panel (d): Metallic leads with the length $L = 60$ . Though the curves behave in a more complex manner, tunneling plateau signals (1 and $\frac{3}{2}$ ) are still accessible, despite the size of sheet or the category of leads. $μ$ in unit of $t$ .
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