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arXiv:1409.0661 · Journal of the Physical Society of Japan

Flux Phase in Bilayer t–J Model

Kazuhiro Kuboki

Published Nov 17, 2014• 6 citations
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Abstract

In order to study the time-reversal symmetry (${\cal T}$) breaking near a (110) surface of a high-$T_C$ cuprate YBCO, we consider the flux phase in a bilayer $t-J$ model. Although the stable solution in the bulk is the $d_{x^2-y-2}$-wave superconducting (SC) state, free energy of the flux phase is close to it, and thus the flux phase may occur when the SC order is disturbed by inhomogeneity, e.g., surface or impurity. It is found that, depending on the doping rate, the flux phase in which spontaneous magnetic fields in two layers have opposite signs may be stabilized. This may lead to a surface state with local ${\cal T}$ violation but without local magnetic field.

DOI
10.7566/jpsj.83.125001

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In order to study the time-reversal symmetry (T{\cal T}T) breaking near a (110) surface of a high-TCT_CTC​ cuprate YBCO, we consider the flux phase in a bilayer t−Jt-Jt−J model. Although the stable solution in the bulk is the dx2−y−2d_{x^2-y-2}dx2−y−2​-wave superconducting (SC) state, free energy of the flux phase is close to it, and thus the flux phase may occur when the SC order is disturbed by inhomogeneity, e.g., surface or impurity. It is found that, depending on the doping rate, the flux phase in which spontaneous magnetic fields in two layers have opposite signs may be stabilized. This may lead to a surface state with local T{\cal T}T violation but without local magnetic field.
At a (110) surface of a dx2−y−2d_{x^2-y-2}dx2−y−2​-wave superconductor, superconducting order is strongly suppressed. In such a situation, ordered states that are forbidden in the bulk may arise. This problem is studied for high-TCT_CTC​ cuprate superconductors by treating the t−Jt-Jt−J model with extended transfer integrals using the Bogoliubov de Gennes method. It is found that a flux phase with staggered currents along the surface, or an antiferromagnetic state can occur near the surface. Stability of the emergent surface states is different from system to system depending on the shapes of their Fermi surfaces. Possible relation to the experiments on the Kerr effect that suggest time-reversal symmetry breaking is discussed.
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.
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 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.
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.
Spatial dependencies of the pair potential and the local density of states near the surfaces of dx2−y2d_{x^{2}-y^{2}}dx2−y2​-wave superconductors are studied theoretically. The calculation is based on the t-J model within a mean-field theory with Gutzwiller approximation. Various types of surface geometries are considered. Similar to our result in the extended Hubbard model, it is found that the formation of zero-energy states strongly depends on the surface geometry. In addition to this feature, the zero-energy states give peak splitting for the (110) surfaces when the super-exchange interaction JJJ is large. This is due to the induced s-wave component near the surface. The present result explains the microscopic origin of the spontaneous time- reversal symmetry breaking at the surfaces of high-TcT_{c}Tc​ superconductors.