Pair-Density-Wave and Chiral Superconductivity in Twisted Bilayer Transition Metal Dichalcogenides

Yi-Ming Wu, Zhengzhi Wu, and Hong Yao

Phys. Rev. Lett. 130, 126001 – Published 23 March 2023

Abstract

We theoretically explore possible orders induced by weak repulsive interactions in twisted bilayer transition metal dichalcogenides (e.g., ${WSe}_{2}$ ) in the presence of an out-of-plane electric field. Using renormalization group analysis, we show that superconductivity survives even with the conventional van Hove singularities. We find that topological chiral superconducting states with Chern number $N = 1$ , 2, 4 (namely, $p + i p$ , $d + i d$ , and $g + i g$ ) appear over a large parameter region with a moiré filling factor around $n = 1$ . At some special values of applied electric field and in the presence of a weak out-of-plane Zeeman field, spin-polarized pair-density-wave (PDW) superconductivity can emerge. This spin-polarized PDW state can be probed by experiments such as spin-polarized STM measuring spin-resolved pairing gap and quasiparticle interference. Moreover, the spin-polarized PDW could lead to a spin-polarized superconducting diode effect.

Received 18 October 2022
Accepted 24 February 2023

DOI:https://doi.org/10.1103/PhysRevLett.130.126001

Physics Subject Headings (PhySH)

FFLO Topological superconductors

Transition metal dichalcogenides Twisted heterostructures

Renormalization group

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Yi-Ming Wu^1,2,*, Zhengzhi Wu ^1,*, and Hong Yao ^1,3,†

¹Institute for Advanced Study, Tsinghua University, Beijing 100084, China
²Stanford Institute for Theoretical Physics, Stanford University, Stanford, California 94305, USA
³State Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China

^*These authors contributed equally to the work.
^†yaohong@tsinghua.edu.cn

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Issue

Vol. 130, Iss. 12 — 24 March 2023

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Images

Figure 1
Phase diagram for twisted ${WSe}_{2}$ . At $ϕ = π / 3$ , there is an emergent spin-polarized PDW with $Q = \pm K$ at small and large $n$ . Centered around $n = 1$ , there exist different chiral SC states which we identify as $p + i p$ , $d + i d$ , and $g + i g$ based on their Chern numbers. At large or small $n$ , a six-node or nodeless SC state dominates, which we denote by $f$ or $f^{'}$ wave according to their representation [ $A_{1}$ ( $B_{1 u}$ ) or $A_{2}$ ( $B_{2 u}$ ) of $C_{3 v}$ ( $D_{6 h}$ ) when $ϕ > 0$ ( $ϕ = 0$ )]. At $ϕ \to 0, π / 3$ , there are tiny regimes of 12-node SC states ( $i$ and $i^{'}$ wave differ in their representations). The white dashed curve shows the locations of the CVHSs. $ϕ = π / 6$ and $n = 1$ is the location of the HOVHS, where the ground state is a gapless metal.
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Figure 2
(a) Formation of the moiré Brillouin zone in twisted homobilayer ${WSe}_{2}$ . In the upper left and right insets, we show two schematic plots for both the monolayer conduction and valence band structures near $\pm K_{0}$ . (b) Twofold degeneracy of the topmost moiré valence band, which can be lifted by an out-of-plane electric field (c). (d) The electric field effectively induces a stagger phase factor $\pm ϕ$ on the bonds, leading to an accumulated flux $\pm 3 ϕ$ on each triangular plaquette. (e) The physical bond phase $ϕ$ as a function of $V_{z}$ , from Ref. [88]. (f)–(i) Various types of Fermi surfaces in the first Brillouin zone. Depending on the parameters, there can be electron or hole pockets, disjoint Fermi surfaces, six CVHSs, and two HOVHSs. The arrows in the last two figures are the nesting vectors. In the case of CVHS, there are two sets of nonequivalent nesting vectors $Q$ and $Q^{'}$ .
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Figure 3
(a) RG results for the CVHS near perfect nesting. The susceptibility for each order has a scaling form $χ \sim (y - y_{c})^{α}$ . Therefore, SC( $p / d / g$ ) is the only possible order in the low $T$ limit ( $\pm$ in $DW \pm$ means even and odd parity). (b) Two-loop RG results of HOVHS at $ϕ = π / 6$ . There is no symmetry breaking near perfect nesting, which is characterized by an interacting RG fixed point (“supermetal” in Ref. [96]). DW wins over SC at small nesting beyond a critical interaction, but SC is the only instability in weak coupling limit.
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Figure 4
(a),(b) Emergence of finite $Q$ pairing at $ϕ = π / 3$ . (c) Fermi surface with an out-of-plane Zeeman field at $ϕ = π / 3$ . (d) Pairing strength of the PDW and uniform SC order. Dashed lines are their phase boundaries. (e) Experimental setup for measuring the spin-polarized SDE. An external Zeeman field is applied to favor the spin-polarized PDW. The critical currents parallel or antiparallel to $Q$ have different magnitudes $j_{c +}$ and $j_{c -}$ . The applied current with magnitude between $j_{c -}$ and $j_{c +}$ will be Ohmic current in one direction and supercurrent in the opposite direction.
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Figure 5
SC phase variation along the FS parametrized by the angle $ϑ$ . Distinct topological phases are characterized by the number of $2 π$ slips when moving around the FS in a full circle.
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