We study the twisted Hubbard model of a cuprate bilayer at a fixed twist angle $\ensuremath{\theta}=53.{13}^{\ensuremath{\circ}}$ and as a function of doping, using the variational cluster approximation, a method that treats short-range dynamical correlations exactly. At intermediate interlayer tunneling, we observe a sudden change of the relative sign of the
d-wave order parameters of two layers between the underdoped and overdoped regimes. At strong interlayer tunneling, we observe a clear time-reversal symmetry breaking phase near optimal doping, in which the relative phase of the two layers changes continuously from 0 to $\ensuremath{\pi}$. However, this phase has trivial topology. We also apply a cluster extension of dynamical mean field theory to the same problem, but fail to detect a time-reversal breaking phase with that method, probably owing to the very small energy difference between the different states involved.
We study a one-band Hubbard model of twisted bilayer cuprates with a twist angle of $53.{13}^{\ensuremath{\circ}}$. By introducing an interlayer bias, we simulate heterobilayers of different dopings. Using the variational cluster approximation, we probe the effect of this bias on the time-reversal symmetry-breaking (TRSB) phase. Doping differences between layers affect the region where TRSB occurs; we construct a phase diagram mapping out the TRSB phase in the ${n}_{1}\ensuremath{-}{n}_{2}$ plane, with ${n}_{\ensuremath{\ell}}$ being the electron density on layer $\ensuremath{\ell}$. We also map the spontaneous supercurrent on the same plane.
The field of twistronics, recently extended to twisted bilayer cuprate systems, has generated considerable interest due to the prediction of
d+id′ states close to a
45∘ twist, as suggested by multiple experimental probes. Theoretical studies of these systems using a variety of models and many-body theoretical techniques have revealed multiple competing order parameters. The large variability in experimental results and theoretical predictions demands an accurate microscopic model for these systems, in particular a more realistic description of the interlayer tunneling. In this study, we fit the in-plane hopping and compute the interlayer tunneling amplitudes by calculating the orbital overlap between
dx2−y2 wave functions at each site of the twisted square-lattice model. We observe sign-changing tunneling amplitudes, a strong twist-angle dependence of the direct tunneling term, and long-range tunneling amplitudes that often dominate over direct tunneling. We investigate the band structure, Fermi surfaces, and spectral functions in comparison with those obtained from the phenomenological model and find a substantial redistribution of spectral weight that modifies the competition between different superconducting ground states. We determine the phase diagram as a function of twist angle, filling, and direct tunneling strength, which is controlled by the
c-axis distance between the monolayers, and observe a reduction in the region of time-reversal-symmetry-breaking (TRSB)
d+id′ states compared with the phenomenological model. Finally, we investigate the Josephson critical current across the phase diagram and find strongly suppressed critical currents close to a
45∘ twist, consistent with experimental observations.
Following theoretical proposals of chiral
d+id′ superconductivity in twisted cuprate bilayers, experimental signatures of time-reversal symmetry breaking (TRSB) remain highly controversial. Here we demonstrate that quantum phase fluctuations fundamentally reshape the phase diagram of this proposed chiral state. Unlike regular superconducting orders, the chiral
d+id′ state requires long-range coherence of an interlayer phase degree of freedom and is therefore intrinsically vulnerable to phase fluctuations. Incorporating these fluctuations nearly eliminates the chiral phase over most parts of the phase diagram, restricting it to a narrow twist-angle window and ultra-low temperatures. Phase fluctuations also strongly weaken Josephson phase locking near a twist angle of
45∘. More broadly, our work establishes quantum phase fluctuations as a fundamental constraint on the emergence of TRSB phases in low-dimensional layered quantum materials.
The relationship between crystal structures and superconducting critical temperatures has attracted considerable attention as a clue to designing higher-
Tc superconductors. In particular, the relationship between the number
n of CuO
2 layers in a unit cell of cuprate superconductors and the optimum superconducting transition temperature
Tcopt is intriguing. As experimentally observed in layered cuprates,
Tcopt increases when
n is increased, up to
n=3, and, then, decreases for larger
n. However, the mechanism behind the
n dependence of
Tcopt remains elusive although there have been many studies on the
n dependence. In this paper, we studied a bilayer
t-
t′ Hubbard model to clarify the effects of the adjacent CuO
2 layers on the stability of the superconductivity by using a many-variable variational Monte Carlo method. We calculate the superconducting correlation at long distance and zero temperature, and the amplitude of the superconducting gap functions estimated from the momentum distribution as the observables correlated with
Tcopt. It is found that the in-plane superconducting correlation is not enhanced in comparison with that in the single-layer
t-
t′ Hubbard model. The superconducting correlations of the bilayer Hamiltonian are significantly small in the overdoped region in comparison with those of single-layer Hamiltonian, which is attributed to the van Hove singularity. In addition, we found that the amplitude of the superconducting gap functions is also similar in both the single-layer and bilayer
t-
t′ Hubbard model at the optimal doping. Therefore, we conclude that the adjacent Hubbard layers are not relevant to the enhancement of
Tcopt in the bilayer cuprates. Possible origins of the enhanced
Tcopt other than the adjacent layers are also discussed.
We study by the Gutzwiller approximation the melting of the valence-bond crystal phase of a bilayer Hubbard model at sufficiently large interlayer hopping. We find that a superconducting domain, with order parameter ${d}_{{z}^{2}\ensuremath{-}{r}^{2}}$,
z being the interlayer direction and
r the intralayer one, is stabilized variationally close to the half-filled nonmagnetic Mott insulator. Superconductivity exists at half filling just at the border of the Mott transition and extends away from half filling into a whole region till a critical doping, beyond which it gives way to a normal-metal phase. This result suggests that superconductivity should be unavoidably met by liquefying a valence-bond crystal, at least when each layer is an infinite-coordination lattice and the Gutzwiller approximation becomes exact. Remarkably, this same behavior is well established in the other extreme of two-leg Hubbard ladders, showing it might be of quite general validity.