Gap structure and phase diagram of twisted bilayer cuprates from a microscopic perspective

Abstract

Since the prediction of a time-reversal symmetry breaking (TRSB) $d+id^\prime$ state in twisted bilayer cuprate superconductors by Can et al. in 2021, several experiments have attempted to detect this state, yielding conflicting results. At present, it is not clear which differences in samples or experimental conditions might explain these discrepancies. In this work, we perform a tight-binding lattice model calculation with phenomenological interlayer tunneling, examining the order parameter as a function of twist angle, interlayer tunneling, doping, and temperature. We observe the TRSB state to be correlated to the position of the Van Hove singularity in the normal state which changes not only as a function of doping but also the tunneling strength. Two such phases are identified as nominally consistent with in-plane $d+id'$ and $d+is$ order, but with unexpected transformation properties under bilayer symmetry operations. We calculate the Josephson critical current, in particular examining the angle dependence for various tunneling strengths. Finally, we discuss the existing experiments in the context of our results.

Figures (17)

• Figure 1: Supercell for $(m,n)=(1,2)$ corresponding to twist angle $\theta =53.13\degree$. In our manuscript we have written $\theta>45\degree$ with the conjugate angle $90\degree-\theta$ which in this case is $36.87\degree$. Each monolayer has 5 lattice sites twisted by $\theta/2$ in opposite directions. The two monolayers along with the sites and their labels in each monolayer are represented by color blue and red.
• Figure 2: Supercell for $(m,n)=(1,1)$ with twist angle $\theta = 90^\circ$. There are four sites in the Moiré unit cell, 2 in each monolayer represented by color blue and red.
• Figure 3: Order parameter and DOS for $\theta=0\degree$ for filling $\nu=0.49$, interaction strength $V=0.2$ and tunneling strength $g_0=0.01$. (a) $B_1$-wave order parameter in momentum space. The $2\times2$ matrix form of the order parameter is because we have 2 sites in our Moiré unit cell. We clearly see the nodal $d_{x^2-y^2}$ gap along the diagonal elements of the order parameter matrix. (b) $d$-wave order parameter in real space. The order parameter along the $x$ and $y$ direction have opposite signs. (c) DOS displaying the V-shaped DOS which is a signature of $d_{x^2-y^2}$ order parameter.
• Figure 4: Real and imaginary part of the order parameter for $90\degree$ twist angle for filling $\nu=0.49$, interaction strength $V=0.2$ and tunneling strength $g_0=0.01$. This system has 4 sites in each Moiré unit cell (Fig. \ref{['11lattice']}) and hence we have a $4\times4$ order parameter matrix. With two sites in each monolayer the nearest neighbor of site 1 is site 2 and hence we only have off-diagonal elements. (a) Real part of the order parameter and (b) imaginary part of the order parameter have the same form as had been derived in Eq. (\ref{['11 d wave']}). (c) The real space order parameter is purely real with non zero entries corresponding to Moiré unit cell coordinates of the nearest neighbors. All momentum labels are in units of the Moiré unit cell lattice constant $b$.
• Figure 5: Real and imaginary part of $\Delta_{11}$ both in momentum and real space for $0\degree$ twist angle and filling $\nu=0.4$, tunneling strength $g_0=0.05$ and interaction strength $V=0.5$. (a) Real part of order parameter in momentum and real space. We observe $d$ wave state. (b) We have a extended $s$ wave order parameter in momentum and real space and a non zero imaginary component in real space even after performing the gauge transformation.