Global phase diagram, possible chiral spin liquid, and topological superconductivity in the triangular Kitaev–Heisenberg model

Abstract

The possible ground states of the undoped and doped Kitaev-Heisenberg model on a triangular lattice are studied. For the undoped system, a combination of the numerical exact diagonalization calculation and the four-sublattice transformation analysis suggests one possible exotic phase and four magnetically ordered phases, including a collinear stripe pattern and a noncollinear spiral pattern in the global phase diagram. The exotic phase near the antiferromagnetic (AF) Kitaev point is further investigated using the Schwinger-fermion mean-field method, and we obtain an energetically favorable Z 2 chiral spin liquid with a Chern number ±2 as a promising candidate. At finite doping, we find that the AF Heisenberg coupling supports an s-wave or a + − d d i x y xy 2 2 -wave superconductivity (SC), while the AF and the ferromagnetic Kitaev interactions favor a + − d d i x y xy 2 2 -wave SC and a time-reversal invariant topological p-wave SC, respectively. Possible experimental realizations and related candidate materials are also discussed. IntroductionRecently, there has been enormous interest in the physics of the spin-1/2 Kitaev model on a honeycomb lattice [1], which has an exact Z 2 spin-liquid (SL) ground state (GS) supporting fractionalized excitations. One possible route to realize this highly anisotropic spin model is to include a strong relativistic spin-orbit coupling (SOC) in Mott insulators [2,3]. Indeed, the interplay of SOC and electron interactions [4][5][6][7][8] gives rise to many novel phases [9][10][11][12][13], especially for the so-called relativistic Mott insulators (RMIs) whose physics may drastically differ from that of Mott insulators with weak SOC (e.g., cuprates) [3,4,14,15]. Of particular interest are d 5 transition metal oxides, such as iridates A 2 IrO 3 (A = Na, Li) [16][17][18], where Na 2 IrO 3 is interpreted as a novel RMI [19] and may also host the quantum spin Hall effect [20,21]. The Kitaev-Heisenberg (KH) model on a honeycomb lattice, which has a rich phase diagram containing unconventional magnetic and Kitaev SL phases [22,23], has been proposed to capture the low-energy properties of A 2 IrO 3 [24,25]. Meanwhile, experiments confirm a longrange zigzag spin order in Na 2 IrO 3 [26][27][28][29], which is a natural GS of the KH model [22]. In addition, there are also studies on the d 4 compound Li 2 RhO 3 , suggesting Li 2 RhO 3 as a possible RMI with a spin-glass GS [30]. Theoretical studies also show that carrier doping into RMIs can induce unconventional superconducting pairings as well as the topological superconductivity (SC) [31][32][33][34][35][36].In fact, the KH model can be generalized to the triangular lattice [37,38]. Similar to the microscopic origin of the honeycomb KH model for A 2 IrO 3 , the triangular KH model can emerge from a class of ABO 2 -type (where A and B are alkali and transition metal ions, respectively) layered compounds [2,3], due to the joint effect of strong SOC, Coulomb interaction, orbital degeneracy, t g 2 5 configuration, and 90°-bonding geometry [...

Phase Diagram and Quantum Order by Disorder in the KitaevK1−K2Honeycomb Magnet

“…1. This model interpolates between two well-known limits, the exactly solvable Kitaev spin liquid [10] at K 2 ¼ 0 and the triangular Kitaev model at K 1 ¼ 0 [28][29][30][31][32]. It is easy to see that a finite K 2 ruins the exact solvability of the NN Kitaev model because the flux operators [10], W p ¼ In the following, we parametrize K 1 ¼ cos ψ and K 2 ¼ sin ψ, and take ψ ∈ ½0; 2πÞ.…”

“…This problem has been studied for both classical [28,29] and quantum spins [30][31][32]. The above analysis for the magnetic phases still holds here, the only difference being that the two legs of each ladder decouple, since they belong to different triangular sublattices.…”

Quantum order by disorder in the Kitaev model on a triangular lattice

“…The resulting theoretical phase diagrams are characterized by various ordered phases (including those seen experimentally) and by a finite stability window for QSL around the Kitaev limit. Recently, a triangular analog of the KH model 29 for classical 30 and quantum 31,32 spins has been studied numerically. The obtained rich phase diagram includes a Z 2 -vortex crystal phase near the AF Heisenberg limit, and a nematic phase of decoupled Ising chains with subextensive degeneracy at the Kitaev limit 30,31 .…”

Collective spin dynamics of Z2 vortex crystals in triangular Kitaev-Heisenberg antiferromagnets

“…The basic ingredient overarching the low-energy descriptions of such systems is the presence of bond-dependent anisotropic exchange, with the so-called Kitaev interactions [1,30] being the most prominent. As the bond dependence stems from spin-orbit coupling, such interactions are not limited to tricoordinated lattices, but may also appear in other geometries, including the common frustrated geometries of the triangular, kagome, pyrochlore, and hyperkagome lattices [31][32][33][34][35][36][37][38][39][40][41]. In such lattices, the synergy of bond-dependent anisotropy and geometric frustration opens up the possibility for novel cooperative phases even when the anisotropy is not the dominant interaction, as in the above tri-coordinated systems.…”