Takuji Nomura
Condensed Matter Theory Group, Synchrotron Radiation Research Center

Magnetic properties of an antiferromagnet with a regulartriangular lattice unit have been studied with considerable
interest, since such a lattice structure with geometricalfrustration generally prohibits conventional collinear spin
configurations and may realize exotic spin states such as spin
liquids. Among regular-triangular lattice magnets, NiGa2S4 has
been considered as a possible realization of a novel exotic
magnetic state with the spin quantum number S = 1 [1]. The
crystal lattice of NiGa2S4 is constructed by stacking NiS2 layers,
each of which is a two-dimensional regular triangular-lattice
formed by Ni atoms, separated by a pair of GaS layers. Basedon both theoretical and experimental investigations [2], it has
been revealed that the electronic and magnetic properties aredominated by the Ni d-eg electrons near the Fermi energy. If theelectron repulsive interaction among the eg electrons issufficiently strong, the system should be a Mott insulator with S= 1. Then, the nearest-neighbor spins should be coupled with an
antiferromagnetic Heisenberg spin interaction J1 < 0.
Results from several experiments suggest that NiGa2S4exhibits unconventional magnetic properties [2]: Among them,neutron scattering revealed that the magnetic correlation evolves
below a temperature T* of approximately 8 K to yield thespatial spin configuration characterized by Q = (0.15(5), 0.15(5),0) [3]. Here, it should be noted that this spin configuration isdifferent from the so-called 120-degree ordering conventionally
expected for regular triangular-lattice antiferromagnets withnearest-neighbor spin coupling. This led many researchers totake the unusual view that the long-range third nearest-neighborspin interaction J3 dominates over the first nearest neighbor J1,i.e., |J1| < |J3|, in contrast to the usual intuition that J1 is thelargest |J1| > |J3|. More remarkable is that the correlation length,
which is estimated from the peak width of the magnetic Bragg
peak, does not diverge even at several mK, i.e., no long-rangeordering can be attained even at the lowest temperature. The
effective magnitude of spin moment at the Ni site was observedto be much reduced (S = 0.51) by approximately 49% from S =1. The origin of this significant reduction has been attributed toquantum fluctuations. These observations have clearly
distinguished NiGa2S4 from conventional magnets.
In this report, a theoretical analysis of these characteristic
magnetic properties of NiGa2S4 is given, following [4]. First, aDFT-based first-principles electronic structure calculation isperformed to obtain the precise electronic structure. Since aband structure calculation is generally not suitable to investigate
arbitrary complex spin orderings, we construct a Hubbard-typemodel by fitting based on the so-called maximally localized
Wannier functions (MLWFs). The results of the calculated bands,
the fitting, and MLWFs are displayed in Fig. 1. Thus, the17-orbital (five Ni-d and twelve S-p orbitals) tight-binding
model is obtained. Through adding the Ni-site Coulombinteraction (U, U , and J) to this free-electron model, a
Hubbard-type model with a realistic electronic band structure isobtained. To determine the most stable spin configuration, theHartree-Fock (mean-field) approximation is applied to thismodel, allowing for spiral ordering states with an arbitrary pitch

vector Q, where spin moments are given by m(r) = |m|[cosQr,sinQr, 0] and are determined by a numerical self-consistentcalculation. The most stable ordering state is the one giving thelowest stabilization energy. The calculated stabilization energies
as a function of Q are displayed in Fig. 2 (a) for several
strengths of the electron interaction parameter U. As a result,while the conventional 120-degree ordering (Q = K = (1/3, 1/3,0)) is obtained for weak interaction (U = 1.6 eV), the uniform
ferromagnetic state (Q = = 0) is obtained for strong interaction
U 2.8 eV. For the intermediate U = 2 eV, Q = (0.15, 0.15, 0)
gives the most stable state, which is consistent with the neutron
scattering experiment. This value of U = 2 eV is quantitatively
consistent with our preliminary estimation using theBethe-Salpeter equation. The calculated ordering spin
configuration is depicted in Fig. 2 (b).

Fig. 1 (a) Calculated electronic band structure andtight-binding fit using MLWFs. On the right, the first
Brillouin zone and a symmetry path are depicted. (b)The eg-type MLWFs at the Ni site.

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pair of weak satellites, originates from the out-of-planecomponent Sc(q, ), while above 1.5 meV, only the pair ofsatellites remain, which originate from the in-plane component
Sab(q, ). Thus, our results suggest that the magnetic excitation
structure is anisotropic and is more complex than usual
antiferromagnets.
these results on the spectral
properties of S(q, ), further advanced measurements are needed
to carefully resolve the in-plane and out-of-plane components,
tron scattering with high-quality
single crystals.

Fig. 2 (a) Calculated stabilization energies for various spin
configurations characterized by the wave vector Q. The
vertical solid line indicates the spin configuration Q =(0.15, 0.15, 0) corresponding to the experimental
observation. (b) The calculated most stable spin ordering
pattern with Q = (0.15, 0.15, 0) is depicted, where thefilled blue circles represent Ni atoms forming a regulartriangular lattice.

Fig. 3 (a) Calculated spin excitation spectrum S(q, ). (b)The intensity map of S(q, ) is overlaid with theneutron scattering data (plots with error bars) forcomparison [3].

The magnitude of the calculated spin moment is |m| =B,
corresponding to S = 0.58, which does not significantly deviate
from the observed value of S = 0.51. Our calculation suggeststhat this significant reduction of spin moment from S = 1 does
not originate from exotic quantum fluctuations, instead
originating from the Ni-S covalent nature. Based on themean-field results, we estimated the Heisenberg spin exchangeinteractions Jn, up to ninth nearest neighbors. According to theresults, the first nearest-neighbor J1 is the largest andferromagnetic (J1 > |J3| > 0), in contrast to the results ofprevious studies based on the localized ionic picture.
Furthermore, the dynamical spin correlation function S(q, )was calculated (averaged over spin orientations: S(q, ) =[2Sab(q, ) + Sc(q, )]/3) in the magnetic state within the random
phase approximation (RPA). The calculated results are displayedin Fig. 3. Here it should be noted that the excitation spectrum isquite different from that of conventional spin waves. In Fig. 3(b), the experimental peak positions observed by neutron
scattering are overlaid onto the calculated intensity map ofS(q, ) for comparison. From the results, at low energies below
1.5 meV, the central spectral peak at q = Q, accompanied by a

The author is grateful to Dr. Kenji Yoshii and Mr. Yuji
Yamamoto for their contributions.

[1] S. Nakatsuji, Y. Nambu, H. Tonomura, O. Sakai, S. Jonath, C.
Broholm, H. Tsunetsugu, Y. Qiu and Y. Maeno, Science 309,1697 (2005).
[2] For a review, S. Nakatsuji, Y. Nambu, S. Onoda, J. Phys. Soc.
Jpn. 79, 011003 (2010).
[3] C. Stock, S. Jonas, C. Broholm,S. Nakatsuji, Y. Nambu, K.
Onuma, Y. Maeno, and J.-H. Chung, Phys. Rev. Lett. 105,037402 (2010).
[4] T. Nomura, Y. Yamamoto and K. Yoshii, J. Phys. Soc. Jpn.
89, 024704 (2020); arXiv:1912.05720.

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