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arXiv:2607.00662

Collective phase modes in twisted ddd-wave superconducting bilayers

Y Shi, Mengxian Zhao, Fei Yang, M Y Liu, Sheng Meng

Published Jul 1, 2026• 0 citations
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Abstract

Twisted cuprate bilayers have been predicted to host high-temperature chiral $d+id'$ superconductivity, originating from higher-order Josephson coupling processes. In such two-dimensional superconducting systems, long-wavelength fluctuations in the phase of the superconducting order parameter constitute gapless collective modes and therefore remain significant even at zero temperature. Here, we perform a theoretical analysis of the low-energy phase fluctuations in twisted $d$-wave superconducting bilayers within a self-consistent harmonic approximation, systematically retaining Josephson coupling to all orders. We demonstrate that higher-order Josephson coupling processes lead to nontrivial modifications of the phase dynamics. The momentum-resolved summand of the relative-phase stiffness is nonzero even in the normal state because interlayer tunneling explicitly breaks the intralayer U(1) symmetry, but its momentum integral vanishes for the continuum dispersion. The relative-phase stiffness is smaller in the $d+id'$ phase than in the $d$-wave phase, while the overall-phase stiffness has the opposite behavior. Furthermore, phase fluctuations strongly soften the Josephson plasma frequency near a twist angle of $45^\circ$ and also substantially reduce the Josephson critical current.

DOI
10.48550/arxiv.2607.00662

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Twisted cuprate bilayers have been predicted to host high-temperature chiral d+id′d+id'd+id′ superconductivity, originating from higher-order Josephson coupling processes. In such two-dimensional superconducting systems, long-wavelength fluctuations in the phase of the superconducting order parameter constitute gapless collective modes and therefore remain significant even at zero temperature. Here, we perform a theoretical analysis of the low-energy phase fluctuations in twisted ddd-wave superconducting bilayers within a self-consistent harmonic approximation, systematically retaining Josephson coupling to all orders. We demonstrate that higher-order Josephson coupling processes lead to nontrivial modifications of the phase dynamics. The momentum-resolved summand of the relative-phase stiffness is nonzero even in the normal state because interlayer tunneling explicitly breaks the intralayer U(1) symmetry, but its momentum integral vanishes for the continuum dispersion. The relative-phase stiffness is smaller in the d+id′d+id'd+id′ phase than in the ddd-wave phase, while the overall-phase stiffness has the opposite behavior. Furthermore, phase fluctuations strongly soften the Josephson plasma frequency near a twist angle of 45∘45^\circ45∘ and also substantially reduce the Josephson critical current.
Following theoretical proposals of chiral d+id′d+id'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′d+id'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∘45^\circ45∘. 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 field of twistronics, recently extended to twisted bilayer cuprate systems, has generated considerable interest due to the prediction of d+id′d+id'd+id′ states close to a 45∘45^\circ45∘ 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−y2d_{x^2-y^2}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 ccc-axis distance between the monolayers, and observe a reduction in the region of time-reversal-symmetry-breaking (TRSB) d+id′d+id'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∘45^\circ45∘ twist, consistent with experimental observations.
We study a one-band Hubbard model for a twisted cuprate bilayer with a twist angle of $43.{6}^{\ensuremath{\circ}}$ and a moir\'e cell containing 58 sites. We use the variational cluster approximation (VCA), which treats short-range correlations exactly and leads, in single layers, to a dome of ddd-wave superconductivity away from half filling from strong on-site repulsion alone. We find a time-reversal-symmetry (TRS) breaking phase in a small doping interval in the overdoped region when interlayer tunneling is strong enough. Contrary to expectations, being closer to the ${45}^{\ensuremath{\circ}}$ twist angle does not expand this TRS region compared to a previous study [Lu and S\'en\'echal, Phys. Rev. B 105, 245127 (2022)] on a ${53}^{\ensuremath{\circ}}$ twist angle. This is attributed to the fact that the two superconducting states in competition have almost identical nodal structures.
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 ddd-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.
When two layers of two-dimensional materials are assembled with a relative twist, moir\'e patterns arise, inducing a tremendous wealth of exotic phenomena. In this work, we consider twisting two triangular lattices hosting Dirac quantum spin liquids. A single decoupled layer is described by compact quantum electrodynamics in 2+12+12+1 dimensions (QED3{\mathrm{QED}}_{3}QED3​) with an emergent U(1)\mathrm{U}(1)U(1) gauge field, which is assumed to flow to a strongly interacting fixed point in the IR with conformal symmetry. We use recent results for the quantum numbers of monopole operators, which tunnel $2\ensuremath{\pi}$ fluxes of the compact gauge field. It is found that, in the bilayer system, interlayer monopole tunneling is a symmetry-allowed relevant perturbation which induces an (ordering) instability. We show using perturbation theory that upon twisting the two layers the system remains unstable under the interlayer interaction, but any finite twist angle softens this instability compared to the untwisted case. To analyze the resulting phase induced by the (twisted) interlayer tunneling, we use ``conformal mean-field theory'', which reduces the interacting bilayer system to two copies of QED3{\mathrm{QED}}_{3}QED3​ coupled to background fields which are to be determined self-consistently. In the weak-coupling regime, where the interlayer coupling is weak compared to the energy scale set by the moir\'e lattice constant, we solve the self-consistency equations perturbatively. In the limit of strongly coupled layers, a local scaling approximation is used, and we find that the magnetically ordered state exhibits a lattice of magnetic vortices, with the lattice constant tunable through the twisting angle.
Berezinskii-Kosterlitz-Thouless transition with enhanced phase stiffness in ddd-wave strongly coupled two-dimensional superconductors
We reveal the key role of the ddd-wave symmetry of the superconducting gap in strongly coupled two-dimensional superconductors in determining the properties of the Berezinskii-Kosterlitz-Thouless (BKT) transition, associated with a sizable enhancement of the phase stiffness compared to nodeless-gap superconductors. The enhanced stiffness originates from extended regions of vanishing gap around the nodal lines of the Brillouin zone (BZ). Our study, based on mean-field and BKT theory, presents a comparative analysis of sss-wave and ddd-wave scenarios, highlighting the features of the latter that boost the stiffness and the BKT transition temperature (TBKT_{BKT}BKT​). The comparison focuses on two quantities: the mean-field critical temperature and the maximum superconducting gap related to the pairing strengths. We present a phase diagram showing the scaling of TBKT_{BKT}BKT​ with respect to the mean-field critical temperature across the BCS-BEC crossover and the evolution of the pseudogap. We also present a zero-temperature phase-stiffness intensity map over the Brillouin zone, displaying a two-component structure consisting of low- and high-stiffness regions whose extent depends on microscopic parameters. These results identify the nodal gap structure of strongly coupled two-dimensional superconductors as a key mechanism enabling enhanced stiffness and elevated TBKT_{BKT}BKT​ compared to their sss-wave counterparts.