The doping of holes into quantum spin liquid (QSL) states is one of the most important issues emerged from the study of cuprate superconductivity, and is now of great interest in terms of topological physics. Theoretically, the Kitaev model [1] can serve a good starting point for studying the hole doping problem because it is known to have the exact QSL ground state. Although α-RuCl₃ or some Iridium oxides are believed to be realizations of the Kitaev model [2], a series of experimental studies have shown that these materials exhibit magnetically ordered phases at low temperatures, implying that these materials contain some additional magnetic interactions such as the Heisenberg exchange interaction, and the so-called off-diagonal Γ- and Γ’-interactions [3]. Since these additional interactions may enrich the phase diagram of the Kitaev model, to study a hole-doped extended Kitaev model that involves them will be of theoretical interest as well as experimental one.

Recently, several theoretical studies of extended Kitaev models have been carried out within the SU(2) slave boson mean-field theory [4, 5]. Several magnetic orders such as ferromagnetic or zigzag ones are found to be stabilized when holes are slightly doped, and then superconducting orders emerge with increasing hole-density. In particular in the t-K-J model that is the Kitaev model with the Heisenberg exchange interaction, topologically non-trivial p_x+ip_y superconductivity with broken time reversal symmetry appears within this theoretical treatment [5]. This topological superconducting order can be classified as a spin liquid state based on projective symmetry group analysis.

Quite recently, two of the authors have shown within the renormalized mean-field theory based on the Gutzwiller approximation that the coexisting state of the d-wave superconductivity and the so-called staggered flux state can be the most stable state of the two-dimensional t-J model with a realistic band structure. We note here that this coexisting state breaks translational and four-fold rotational symmetries of the square lattice. Therefore, it is quite natural to expect that hole-doped extended Kitaev models may have mean-field ground states with broken translational or rotational symmetries.

Motivated by these backgrounds, we theoretically study extended Kitaev models based on the real-space renormalized mean-field theory. We confirm that the Gutzwiller renormalization factors for the Kitaev interactions are identical to those for the t-J model. We examine the possible ground states with broken symmetries, and also examine the superconducting states under a uniform magnetic field and that with some impurities.

References
[1] A. Kitaev, Ann. of Phys. 321, 2 (2006).
[2] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009).
[3] S. M. Winter, Y. Li, H. O. Jeschke, and R. Valentí, Phys. Rev. B 93, 214431 (2016).
[4] S. -M. Zhang and Z. -X. Liu, Phys. Rev. B 104, 115108 (2021).
[5] Y. -Z. You, I. Kimchi, and A. Vishwanath, Phys. Rev. B 86, 085145 (2012).

Keywords: Quantum Spin Liquid, Extended Kitaev Model, Superconductivity, Renormalized Mean-Field Theory