Incoherent tunneling and topological superconductivity in twisted cuprate bilayers

Twisting two monolayers of a high-$T_c$ cuprate superconductor can engender a chiral topological state with spontaneously broken time-reversal symmetry $\mathcal{T}$. A crucial ingredient required for the emergence of a gapped topological phase is electron tunneling between the ${\mathrm{CuO}}_2$ planes, whose explicit form (in an ideal sample) is dictated by the symmetry of the atomic orbitals. However, a large body of work on interlayer transport in cuprates indicates the importance of disorder-mediated incoherent tunneling, which evades symmetry constraints present in an idealized crystal. This arises even in the cleanest single-crystal samples through oxygen vacancies, in layers separating the ${\mathrm{CuO}}_2$ planes, introduced to achieve the hole doping necessary for superconductivity. Here we assess the influence of incoherent tunneling on the phase diagram of a twisted bilayer and show that the model continues to support a fully gapped topological phase with broken $\mathcal{T}$. Compared to the model with a constant, momentum conserving interlayer coupling, the extent of the topological phase around the ${45}^{\circ }$ twist decreases with increasing incoherence, but remains robustly present for parameters likely relevant to ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$.

Figure 1

Diagrammatic expansion of the interlayer current (a) at order $g^2$ and [(b),(c)] at order $g^4$. Full lines correspond to electronic propagators $G$ and dashed lines correspond to impurity vertices paired by disorder average. The open circle denotes the current vertex $j_{\mathbf{q}}$ defined in the main text and impurity vertices ${\gamma }_{\mathbf{q}}$ are given by black dots.

Figure 2

Phase diagram of incoherently coupled twisted bilayer cuprates. For a given $\mathrm{\Lambda }$, the inside of the cone-shaped region breaks $\mathcal{T}$. Black-dashed lines mark the phase boundary in the clean limit, previously introduced in [7]. For increasing degree of momentum nonconservation $\mathrm{\Lambda }$, the $\mathcal{T}$-breaking phase boundaries shrink towards ${45}^{\circ }$.

Figure 3

Bulk [(a),(b)] and boundary [(c),(d)] spectrum for incoherently coupled cuprate bilayers with $\mathrm{\Lambda }=0$ (left) and $\mathrm{\Lambda }/k_F=0.08$ (right) at ${45}^{\circ }$-twist angle. The spectrum shows chiral edge modes traversing the bulk gap, which is reduced but finite for increased $\mathrm{\Lambda }$. Edge modes, which are in fact degenerate, indicate a Chern number $\mathcal{C}=4$.

Figure 4

(a) Illustration of the geometry of the bilayer lattice model at an incommensurate angle of $\sim {43}^{\circ }$. (b) Disorder averaged free energy of the bilayer at zero temperature as a function of the phase difference. The minima ${\phi }_{\min }$ are situated away from zero at small disorder strengths. (c) Temperature dependence of the order parameter amplitude and phase for $\tilde{\mathrm{\Lambda }}=0.2$. One could view it as a vertical cut at a specific twist in the phase diagram of Fig. 2, with the onset of a nonzero ${\phi }_{\min }$ marking the phase boundary.

Figure 5

Critical current $J_c$ of the twisted bilayer as a function of interlayer coherence scale $\mathrm{\Lambda }$. Incoherence significantly reduces $J_c$. The color scale denotes temperature $T$ in panel (a) and twist angle $\theta$ in (b).

Figure 6

Graphical representation of the momentum summation constraints used to estimate $\mathrm{\Lambda }$ dependence of the diagrams in Fig. 1. In (a) the summation is restricted to the thick-red segment of length $\mathrm{\Lambda }$, whereas in (b) it is restricted to a small region around the origin, independent of $\mathrm{\Lambda }$.

Figure 7

$\mathrm{\Lambda }$ dependence of the width of the $\mathcal{T}$-breaking region at $T=0$ computed in the continuum model. It is well approximated by a $1/\mathrm{\Lambda }$ (orange line) for large $\mathrm{\Lambda }$.

Figure 8

Two different gap regularizations. Panels (a) and (b) show the gaps defined in Eq. (23). They can be transformed into each other by a mirror reflection, up to a sign. Panels (c) and (d) show a different regularization where this symmetry is absent.

Figure 9

Layer-resolved bulk [(a)–(d)] and boundary [(e)–(h)] spectra for incoherently coupled cuprate bilayers with $\mathrm{\Lambda }=0$ (left two columns) and $\mathrm{\Lambda }/k_F=0.08$ (right two columns) at ${45}^{\circ }$-twist angle for superconducting gaps plotted in Fig. 8 and 8. In each layer, the spectral functions display two distinct chiral edge modes, indicating a total Chern number $\mathcal{C}=4$.