Table 2 Four-spin ring exchange couplings for NaRuO2.

From: A jeff = 1/2 Kitaev material on the triangular lattice: the case of NaRuO2

\({{\mathbb{K}}}_{1234}^{\mu \nu \rho \eta }\) [meV]

\({S}_{1}^{\mu }{S}_{2}^{\nu }{S}_{3}^{\rho }{S}_{4}^{\eta }\)

0.1829

(xxzx + yyzy + zxxx + zyyy)

0.1828

(xxzy + yxzy + zyxx + zyyx)

0.1819

(zxzy + zyzx)

0.1810

(xzyz + yzxz)

−0.1740

(zzzz)

−0.1559

(xyzz + yzzx + zxyz + zzxy)

−0.1457

(xzzx + yyzz + zxxz + zzyy)

−0.1398

(xxxx + yyyy)

−0.1284

(xxxz + xzxx + yyyz + yzyy)

0.1122

(zxzz + zyzz + zzzx + zzzy)

−0.0969

(xyxy + yxyx)

−0.0829

(xyzy + yxzx + zxyx + zyxy)

0.0602

(xyzx + yyzx + zxxy + zxyy)

−0.0596

(xyxz + xzxy + yxyz + yzyx)

−0.0581

(xzzz + yzzz + zzxz + zzyz)

−0.0580

(xxzz + yzzy + zyyz + zzxx)

−0.0570

(xzyx + xzyy + yxxz + yyxz)

−0.0542

(xxxy + xyxx + yxyy + yyyx)

0.0354

(xxyz + xyyz + yzxx + yzxy)

−0.0352

(xzzy + yxzz + zyxz + zzyx)

−0.0272

(xxyy + xyyx + yxxy + yyxx)

0.0252

(xzxz + yzyz)

0.0145

(xxyx + xyyy + yxxx + yyxy)

0.0069

(zxzx + zyzy)

  1. The corresponding ring-exchange expression is defined in Eq. (3) and the parameters in this table refer to a Z-plaquette, with the site labeling 1, 2, 3, 4 illustrated in Fig. 1b. The ring-exchange parameters are calculated with projED starting from a three-orbital Hubbard model. For simplicity, we abbreviate \({S}_{1}^{\mu }{S}_{2}^{\nu }{S}_{3}^{\rho }{S}_{4}^{\eta }\) with “μνρη”. Parameters for X- and Y-plaquettes follow by C3 rotations around the out-of-plane axis.
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