Magic angles and correlations in twisted nodal superconductors

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Magic angles and correlations in twisted nodal superconductors

Pavel A. Volkov, Justin H. Wilson, Kevin P. Lucht, and J. H. Pixley

Phys. Rev. B 107, 174506 – Published 4 May 2023

Abstract

Motivated by recent advances in the fabrication of twisted bilayers of two-dimensional materials, we consider the low-energy properties of a twisted pair of two-dimensional nodal superconductors. We study both the cases of singlet and triplet superconductors. It is demonstrated that the Bogoliubov–de Gennes (BdG) quasiparticle dispersion undergoes dramatic reconstruction due to the twist. In particular, the velocity of the neutral massless Dirac excitations near the gap nodes is strongly renormalized by the interlayer hopping and vanishes at a “magic angle” where in the limit of a circular Fermi surface a quadratic band touching is formed. In addition, it is shown that the BdG dispersion can be tuned with an interlayer displacement field, magnetic field, and current, which can suppress the velocity renormalization, create finite BdG Fermi surfaces, or open a gap, respectively. Finally, interactions between quasiparticles are shown to lead to the emergence of a correlated superconducting state breaking time-reversal symmetry in the vicinity of the magic angle. Estimates of the magic angle in a variety of nodal superconductors are presented, ranging from the cuprates to the organic and heavy-fermion superconductors, all of which are shown to be promising for the experimental realization of our proposal.

Received 5 April 2021
Revised 6 December 2022
Accepted 17 March 2023

DOI:https://doi.org/10.1103/PhysRevB.107.174506

Physics Subject Headings (PhySH)

Superconductivity

High-temperature superconductors Monolayer films Superconductors Topological superconductors Unconventional superconductors

Methods in superconductivity

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Pavel A. Volkov ^1,2,3,*, Justin H. Wilson ^4,1, Kevin P. Lucht ¹, and J. H. Pixley ^1,5,6

¹Department of Physics and Astronomy, Center for Materials Theory, Rutgers University, Piscataway, New Jersey 08854, USA
²Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
³Department of Physics, University of Connecticut, Storrs, Connecticut 06269, USA
⁴Department of Physics and Astronomy, and Center for Computation and Technology, Louisiana State University, Baton Rouge, Louisiana 70803, USA
⁵Center for Computational Quantum Physics, Flatiron Institute, 162 5th Avenue, New York, New York 10010, USA
⁶Physics Department, Princeton University, Princeton, New Jersey 08544, USA

^*pv184@physics.rutgers.edu

Current- and Field-Induced Topology in Twisted Nodal Superconductors

Pavel A. Volkov, Justin H. Wilson, Kevin P. Lucht, and J. H. Pixley

Phys. Rev. Lett. 130, 186001 (2023)

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Vol. 107, Iss. 17 — 1 May 2023

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Images

Figure 1
(a) Illustration of the momentum-space structure for a bilayer twisted at an angle $θ$ for a square lattice and a Fermi surface appropriate for cuprate superconductors. Fermi surfaces of two layers are shown in red and black, with a pair of nodes located at $K_{N}$ and ${\tilde{K}}_{N}$ , forming a single “valley,” emphasized by filled circles. Tunneling occurs between states of two layers overlapping in the figure and also the ones additionally shifted by reciprocal wave vectors of the original Brillouin zone (e.g., $G_{1, 2}$ shown by black arrows in the inset) or of the rotated one (e.g., ${\tilde{G}}_{1, 2}$ , shown by gray arrows in main panel). The latter processes are, however, suppressed and may be neglected (see text). Inset shows the construction of the mini Brillouin zone (green) due to the moire superlattice formation with the inverse lattice unit vectors $G_{1, 2}^{M}$ . (b) Expanded nodal region from (a) showing the local coordinates for the case of symmetry-protected nodes: $k_{∥}$ is along the bisector of the two node lines, and $k_{⊥} ⊥ k_{∥}$ .
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Figure 2
Evolution of the low-energy part of the BdG quasiparticle spectrum (12) [see also Eq. (8)] as a function of twist angle $θ$ relative to the magic angle $θ_{MA}$ in Eq. (17) in momentum space for the nodal region depicted in Fig. 1; filled circles marking the node positions in the unhybridized layers. At zero twist angle, the interlayer tunneling simply leads to an appearance of split bonding and antibonding Fermi surfaces (gray lines), with nodes located at their intersection with the gap line node. Then, the two Dirac cones initially separated along $k_{∥}$ move towards each other on increasing twist angle, while the Dirac velocity is renormalized downwards. At the magic angle, the two merge into a quadratic band touching [Eqs. (20) and (21)], and separate again (this time along $k_{⊥}$ ) on further increasing the twist angle.
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Figure 3
Illustration of the effects of external fields on the spectrum of a TBSC. (a) Summary of the evolution of spectrum as a function of twist angle in momentum space in the absence of external fields (cf. Fig. 2). Red lines indicate the direction of the node's motion with increasing twist angle. (b) Effect of an interlayer displacement field: the Dirac cones avoid merging into a QBT at all twist angles. (c) Effect of Zeeman field: particle (hole) pockets form for spin-up (spin-down) quasiparticles. (d) Effect of in-plane current: particle and hole pockets form for quasiparticles around $K_{N}$ and $- K_{N}$ , respectively. (e) Interlayer (Josephson) current opens a topological gap with an edge mode (black).
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Figure 4
Evolution of the low-energy part of the BdG quasiparticle spectrum as a function of twist angle including corrections due to a noncircular Fermi surface [Eqs. (11) and (9)]. For the figures we have taken $\frac{v_{F}^{(2)} θ_{MA} k_{⊥}}{2 v_{Δ}} = 0.5$ , $\frac{v_{Δ}^{(2)} θ_{MA} k_{∥}}{2 v_{F}} = 0.1$ . At low twist angles (a) two Dirac points are present. One of the two Dirac points (a) first becomes a semi-Dirac point (b) and splits into three Dirac points (c) afterwards. On further increasing $θ$ , one of the three new Dirac points approaches the remaining original one (d) and forms a semi-Dirac point (e) before opening a gap (f) there. Therefore, at larger twist angles only two Dirac points remain.
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Figure 5
Phase diagram of the secondary time-reversal symmetry-breaking superconducting order induced by the quasiparticle interactions near the magic angle. Its onset temperature $T^{*}$ reaches its maximal value $T_{0}^{*}$ at the magic angle and is suppressed as the deviation $| θ - θ_{MA} |$ grows, vanishing at $θ_{c}^{\pm} = θ_{MA} \pm 2 π e^{- γ} \frac{θ_{MA} T_{0}^{*}}{t}$ . The band structure schematics represent the qualitative form of the quasiparticle spectrum in each region (cf. Fig. 3). Inset shows the numerical solution for $T^{*} (θ)$ as a function of the dimensionless twist parameter $t | α - 1 | / T_{0}^{*}$ . Here, $T_{0}^{*} ≪ T_{c}$ is assumed; see text for the discussion of the additional effects of temperature.
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Figure 6
Suppression of the time-reversal symmetry-breaking temperature $T^{*}$ by the deviations from circularly symmetric Fermi surface [Eq. (9)] characterized by the parameter $ξ_{0} = \frac{v_{F}^{(2)} t θ_{MA}}{2 v_{Δ}}$ .
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