Moiré-induced gapped phases in twisted nodal superconductors

TL;DR

The study reveals that moiré superlattices in twisted nodal superconductors induce gapped, non-topological phases through intervalley moiré umklapp scattering of Dirac nodes. These gaps can compete with and even suppress current-induced topological gaps, signaling topological phase transitions and necessitating revisions to existing phase diagrams. A dual theoretical framework—a lattice model with self-consistent order parameters and a continuum model of Dirac nodes—demonstrates how twist angle and interlayer phase control gap openness and topology. The authors provide concrete estimates for critical angles in BSCCO and outline experimental signatures, highlighting the potential to engineer higher critical temperatures via moiré-driven gapping in 2D superconductors.

Abstract

We demonstrate the emergence of gapped phases driven by the moiré superlattice that trivialize the topological states in twisted nodal superconductors. The effect arises from umklapp tunneling between non-adjacent Dirac points in momentum space close to specific twist angles or chemical potentials, determined by the Fermi surface geometry. We confirm the robustness of the non-topological phase against interactions with self-consistent calculations and show that this gap competes with the previously predicted topological gapped phases, leading to topological phase transitions. These transitions were overlooked in prior literature, signifying the necessity of modifying the phase diagrams of topological phases exhibited in twisted nodal superconductors with and without an interlayer current. We also estimate the relevant twist angles and discuss experimental signatures, focusing on twisted Bi$_2$Sr$_2$CaCu$_2$O$_{8+x}$

Figures (8)

• Figure 1: Leading moiré umklapp processes in twisted nodal superconductors: (a) schematic of tunneling within a Brillouin zone (BZ) of two layers rotated by an angle $\theta$. (b) shows node's of each valley (labeled as Roman numerals) shown as symbols for each layer. 1st BZ is solid while 2nd BZ is dashed. (c) Phase diagram of the superconducting state under twist (up to $\theta = 45^\circ$) and temperature (up to $T_c$) with no interlayer current ($\varphi = 0$). A new non-topological gapped $d$-wave state appears represented by the yellow region. (d) shows energy gap versus matching twist angle range as (c) induced by interlayer current ($\varphi \neq 0$). Topological phase transitions appear in yellow and green regions. Color fill in (c,d) indicates Chern number. (e,f) show intervalley umklapp tunneling for layers colored in purple and orange. (e) corresponds to yellow phase at $\theta_c$ in (a,b) from valleys $I$ and $II$ related by a moiré reciprocal lattice vector, $\bm{G}_1^M$. (f) shows green phase at $\theta_{c,2}$ in (b) from valleys $I$ and $III$ caused by differing reciprocal lattice vectors from the two layers.
• Figure 2: Gap opening due to moiré umklapp near $\mu=\mu_{c}$: Displayed are the energy gap, labeled by $\Delta$, for varying tunneling strength, $g_0$, and for chemical potential $\mu$ (in units of meV) around $\mu_c$ (see Fig. \ref{['fig:figure1']} (e)). (a) and (b) are for phase differences of $\varphi=0$ and $\varphi=\pi/2$, respectively. For $\varphi=0$, a non-topological gap ($|\mathcal{C}|=0$) opens in the spectrum for $\mu=\mu_c$ and decreases away from $\mu_c$. Once a phase difference is present, $\mu_c$ maintains a non-topological gap for low $g_0$, but a topological gap ($|\mathcal{C}|=4$) opens for large values of $g_0$. Self consistent results (red dashed line) are taken at $\mu_c$ while non-self-consistent results (green and blue solid and dashed lines) use a $d$-wave order parameter with a magnitude of $40$ meV. Behavior of the gaps for $\varphi=\pi/2$ under varying $\mu$ are shown in the End Matter for $\theta_{1,2}$.
• Figure 3: Moiré induced gap opening near $\mu=\mu_{c,2}$ with a finite phase difference: Displayed are the energy gap, $\Delta$, for varying $g_0$ at $\mu_{c,2}$. With no phase difference, the system remains nodal for all $g_0$. Once a phase difference is applied, the gap gradually opens, maximizing at $\varphi = \frac{\pi}{2}$. Results reflect non-self-consistently determined order parameter of a $d$-wave order parameter with a magnitude of $40$ meV.
• Figure 4: Description of the moiré umpklapp process using a continuum model. (a) Depicts the intervalley tunneling (circled in dashed blue) processes of Fig. \ref{['fig:figure1']} (e) with corresponding non-topological energy gap in (c). (b) shows the intervalley tunneling processes with the orientation of the velocities of each node. (c) illustrates the energy gap versus tunneling strength for $\varphi>0$. Green represents the $|\mathcal{C}|=0$ phase for low $g_0$ while blue corresponds to the topological phase $|\mathcal{C}|=1$ for the two nodes.
• Figure 5: Energy gap for $\theta_{1,2}$. We corroborate the gapped spectrum of the $\theta_{1,3}$ structure with $\theta_{1,2}$. Both have a critical chemical potential of $\mu_c = 0.0$ meV. For $\varphi =0$, (a) displays self-consistent and non-self-consistent results at $\mu_c$ and away from $\mu_c$ for non-self-consistent results. (b) shows the gap in the presence of $\varphi \neq 0$ at and near $\mu_c$ without self-consistency and at $\mu_c$ with self-consistency. For low $g_0$, both cases display a non-topological gap. Away from $\mu_c$, the non-topological gap gradually vanishes, leaving only the topological gap at higher $g_0$. Units of $\mu$ given in meV.