The Berry phase due to the spin wavefunction gives rise to the orbital ferromagnetism and anomalous Hall effect in the non-coplanar antiferromagnetic ordered state on face-centered-cubic (fcc) lattice once the crystal is distorted perpendicular to (1,1,1) or (1,1,0)-plane. The relevance to the real systems γ-FeMn and NiS2 is also discussed.PACS numbers: 11.30. Er, 11.30.Rd, 71.27.+a It has been recognized for a long term that the chirality plays important roles in the physics of frustrated spin systems [1][2][3][4][5][6]. These degrees of freedom are distinct from the (staggered) magnetization, and could show phase transition without magnetic ordering [1][2][3]. Especially since the discovery of the high-Tc cuprates, the scalar spin chiralityhas been a key theoretical concept in the physics of strongly correlated electronic systems [4][5][6]. This spin chirality acts as the gauge flux for the charge carriers moving in the background of the fluctuating spins. In order for the spin chirality χ ijk to be ordered, both the time-reversal (T) and parity (P) symmetries must be broken. Broken T and P symmetries in 2D bring about many intriguing physics such as parity anomaly [7,8] , anyon superconductivity [9], and quantized Hall effect without external magnetic field [10]. A physical realization of the last one has been discussed [11] in the context of anomalous Hall effect (AHE) in ferromagnets via the spin chirality mechanism [12][13][14][15]. In this paper we explore the chiral spin state in the ordered antiferromagnet (AF) on the three-dimensional face-centered-cubic (fcc) lattice. The AF on the fcc lattice is a typical frustrated system, and nontrivial spin structure with the finite spin chirality in eq. (1) is expected. For example, in the charge transfer (CT) insulator NiS 2 [16] and in the metallic alloy γ-FeMn [17] the non-coplanar spin structure (so-called triple-Q structure shown in Fig. 1a) has been observed. A theoretical explanation for this structure is the following. Let us consider the case where the lattice points are divided into 4-sublattices as shown in Fig. 1a. Denoting the (classical) spin moment at each sublattice as S a (a = 1, 2, 3, 4), the 2-spin exchange interaction energy is written as H 2 ∝ ( a=1,4 S a )2 . Therefore the condition of the lowest energy a=1,4 S a = 0 does not determine the spin structure and leaves many degenerate lowest energy configurations. Then the interactions which lift this degeneracy such as the 4-spin exchange interaction become important [18,19]. In paticular the phenomenological Ginzburg-Landau theory for the 4-spin exchange interaction is given as H 4 = J 4 a =b ( S a · S b )
2[19]. With positive J 4 , the ground state configuration is given by S 1 = (