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arXiv:2311.02207 · Physical review. B./Physical review. B

Interlayer bias effect on time-reversal symmetry breaking in twisted bilayer cuprates

Mathieu Bélanger, David Sénéchal

Published Feb 6, 2024• 4 citations
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Abstract

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.

DOI
10.1103/physrevb.109.075111

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We study a one-band Hubbard model of twisted bilayer cuprates with a twist angle of 53.13\ensuremath∘53.{13}^{\ensuremath{\circ}}53.13\ensuremath∘. 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 n1\ensuremath−n2{n}_{1}\ensuremath{-}{n}_{2}n1​\ensuremath−n2​ plane, with n\ensuremathℓ{n}_{\ensuremath{\ell}}n\ensuremathℓ​ being the electron density on layer \ensuremathℓ\ensuremath{\ell}\ensuremathℓ. We also map the spontaneous supercurrent on the same plane.
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.
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.
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 the stability of the time-reversal symmetry breaking staggered-flux phase of a single band Hubbard model, within the variational cluster approach. For intermediate and small values of the interaction UUU, we find metastable solutions for the staggered-flux phase, with a maximum current per bond at $U\ensuremath{\approx}3.2$. However, allowing for antiferromagnetic and superconducting long-range order it turns out that in the region at and close to half filling the antiferromagnetic phase is the most favorable energetically. The effect of nearest-neighbor interaction is also considered. Our results show that a negative nearest-neighbor interaction and finite doping favors the stability of the staggered-flux phase. We also present preliminary results for the three-band Hubbard model obtained with a restricted set of variational parameters. For this case, no spontaneous time-reversal symmetry breaking phase is found in our calculations.
Twisted bilayer graphene is an exciting platform for exploring correlated quantum phases, extremely tunable with respect to both the single-particle bands and the interaction profile of electrons. Here, we investigate the phase diagram of twisted bilayer graphene as described by an extended Hubbard model on the honeycomb lattice with two fermionic orbitals (valleys) per site. Besides the special extended cluster interaction QQQ, we incorporate the effect of gating through an on-site Hubbard-interaction UUU. Within quantum Monte Carlo, we find valence-bond solid, Nel-valley antiferromagnetic or charge-density wave phases. Further, we elucidate the competition of these phases by noticing that the cluster interaction induces an exotic constraint on the Hilbert space, which we dub the cluster rule, in analogy to the famous pyrochlore spin-ice rule. Formulating the perturbative Hamiltonian by projecting into the cluster-rule manifold, we perform exact diagonalization and construct the fixed-point states of the observed phases. Finally, we compute the local electron density patterns as signatures distinguishing these phases, which could be observed with scanning tunneling microscopy. Our paper capitalizes on the notion of cluster constraints in the extended Hubbard model of twisted bilayer graphene and suggests a scheme towards realization of several symmetry-breaking insulating phases in a twisted-bilayer graphene sheet.