Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2112.00487 · Physical review. B./Physical review. B

Doping phase diagram of a Hubbard model for twisted bilayer cuprates

Xiancong Lu, David Sénéchal

Published Jun 21, 2022• 24 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/physrevb.105.245127

Related papers

Doping dependence of chiral superconductivity in near 45∘ twisted bilayer cuprates

Mathieu Bélanger, David Sénéchal

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 $d$-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.

arXiv:2306.05485•Physical review. B./Physical review. B•Jan 5, 2024•8 citations

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

Mathieu Bélanger, David Sénéchal

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.

arXiv:2311.02207•Physical review. B./Physical review. B•Feb 6, 2024•4 citations

Microscopic Modeling of Interlayer Tunneling in Twisted Bilayer Cuprates

Siddhant Panda, Andreas Kreisel, Laura Fanfarillo +1 more

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 $45^\circ$ 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 $d_{x^2-y^2}$ 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 $45^\circ$ twist, consistent with experimental observations.

arXiv:2610.16930•Oct 12, 2026•0 citations

Quench of chiral superconductivity by quantum phase fluctuations in twisted cuprate bilayers

Y Shi, Mengxian Zhao, Fei Yang +2 more

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 $45^\circ$. More broadly, our work establishes quantum phase fluctuations as a fundamental constraint on the emergence of TRSB phases in low-dimensional layered quantum materials.

arXiv:2607.00630•Jul 1, 2026•0 citations

Interplay between superconductivity and flux phase in the t−J model

E. Cappelluti, R. Zeyher

We study the phase diagram of the $t\ensuremath{-}J$ model using a mean field type approximation within the Baym-Kadanoff perturbation expansion for Hubbard X operators. The line separating the normal state from a d-wave flux or bond-order state starts near optimal doping at $T=0$ and rises quickly with decreasing doping. The transition temperature ${T}_{c}$ for d-wave superconductivity increases monotonically in the overdoped region towards optimal doping. Near optimal doping a strong competition between the two d-wave order parameters sets in leading to a strong suppression of ${T}_{c}$ in the underdoped region. Treating for simplicity the flux phase as commensurate the superconducting and flux phases coexist in the underdoped region below ${T}_{c},$ whereas a pure flux phase exists above ${T}_{c}$ with a pseudogap of d-wave symmetry in the excitation spectrum. We also find that incommensurate charge-density-wave ground states due to Coulomb interactions do not modify strongly the above phase diagram near the superconducting phase, at least, as long as the latter exists at all.

arXiv:cond-mat/9808201•Physical review. B, Condensed matter•Mar 1, 1999•104 citations

Superconductivity in Bilayer t–t′ Hubbard Models

Akito Iwano, Youhei Yamaji

The relationship between crystal structures and superconducting critical temperatures has attracted considerable attention as a clue to designing higher-$T_{\rm c}$ 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 $T_{\rm c}^{\rm opt}$ is intriguing. As experimentally observed in layered cuprates, $T_{\rm c}^{\rm opt}$ increases when $n$ is increased, up to $n=3$, and, then, decreases for larger $n$. However, the mechanism behind the $n$ dependence of $T_{\rm c}^{\rm opt}$ 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 $T_{\rm c}^{\rm opt}$. 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 $T_{\rm c}^{\rm opt}$ in the bilayer cuprates. Possible origins of the enhanced $T_{\rm c}^{\rm opt}$ other than the adjacent layers are also discussed.

arXiv:2207.04205•Journal of the Physical Society of Japan•Aug 17, 2022•8 citations
We study the twisted Hubbard model of a cuprate bilayer at a fixed twist angle \ensuremathθ=53.13\ensuremath∘\ensuremath{\theta}=53.{13}^{\ensuremath{\circ}}\ensuremathθ=53.13\ensuremath∘ 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 \ensuremathπ\ensuremath{\pi}\ensuremathπ. 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.
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′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.
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.
We study the phase diagram of the $t\ensuremath{-}J$ model using a mean field type approximation within the Baym-Kadanoff perturbation expansion for Hubbard X operators. The line separating the normal state from a d-wave flux or bond-order state starts near optimal doping at T=0T=0T=0 and rises quickly with decreasing doping. The transition temperature Tc{T}_{c}Tc​ for d-wave superconductivity increases monotonically in the overdoped region towards optimal doping. Near optimal doping a strong competition between the two d-wave order parameters sets in leading to a strong suppression of Tc{T}_{c}Tc​ in the underdoped region. Treating for simplicity the flux phase as commensurate the superconducting and flux phases coexist in the underdoped region below Tc,{T}_{c},Tc​, whereas a pure flux phase exists above Tc{T}_{c}Tc​ with a pseudogap of d-wave symmetry in the excitation spectrum. We also find that incommensurate charge-density-wave ground states due to Coulomb interactions do not modify strongly the above phase diagram near the superconducting phase, at least, as long as the latter exists at all.
The relationship between crystal structures and superconducting critical temperatures has attracted considerable attention as a clue to designing higher-TcT_{\rm c}Tc​ superconductors. In particular, the relationship between the number nnn of CuO2_22​ layers in a unit cell of cuprate superconductors and the optimum superconducting transition temperature TcoptT_{\rm c}^{\rm opt}Tcopt​ is intriguing. As experimentally observed in layered cuprates, TcoptT_{\rm c}^{\rm opt}Tcopt​ increases when nnn is increased, up to n=3n=3n=3, and, then, decreases for larger nnn. However, the mechanism behind the nnn dependence of TcoptT_{\rm c}^{\rm opt}Tcopt​ remains elusive although there have been many studies on the nnn dependence. In this paper, we studied a bilayer ttt-t′t't′ Hubbard model to clarify the effects of the adjacent CuO2_22​ 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 TcoptT_{\rm c}^{\rm opt}Tcopt​. It is found that the in-plane superconducting correlation is not enhanced in comparison with that in the single-layer ttt-t′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 ttt-t′t't′ Hubbard model at the optimal doping. Therefore, we conclude that the adjacent Hubbard layers are not relevant to the enhancement of TcoptT_{\rm c}^{\rm opt}Tcopt​ in the bilayer cuprates. Possible origins of the enhanced TcoptT_{\rm c}^{\rm opt}Tcopt​ other than the adjacent layers are also discussed.