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Magnetization process of spin-1/2 Heisenberg antiferromagnets on a layered triangular lattice

We study the magnetization process of the spin-1/2 antiferromagnetic Heisenberg model on a layered triangular lattice by means of a numerical cluster mean-field method with a scaling scheme (CMF+S). It has been known that antiferromagnetic spins on a two-dimensional (2D) triangular lattice with quantum fluctuations exhibit a one-third magnetization plateau in the magnetization curve under magnetic field. We demonstrate that the CMF+S quantitatively reproduces the magnetization curve including the stabilization of the plateau. {We also discuss the effects of a finite interlayer coupling, which is unavoidable in real quasi-2D materials. It has been recently argued for a model of the layered-triangular-lattice compound Ba3CoSb2O9 that such interlayer coupling can induce an additional first-order transition at a strong field. We present the detailed CMF+S results for the magnetization and susceptibility curves of the fundamental Heisenberg Hamiltonian in the presence of magnetic field and weak antiferromagnetic interlayer coupling. The extra first-order transition appears as a quite small jump in the magnetization curve and a divergence in the susceptibility at a strong magnetic field ~ 0.712 of the saturation field.


I. INTRODUCTION
Triangular-lattice antiferromagnets (TLAFs), which are a paradigmatic example of geometric frustration, have received renewed interest [1] in recent years owing to the technical developments in high-field experiments and the appearance of new materials comprising Co 2+ magnetic ions, such as Ba 3 CoSb 2 O 9 [2][3][4][5][6][7] and Ba 3 CoNb 2 O 9 [8][9][10]. Unlike other typical TLAF compounds with Cu 2+ ions, such as Cs 2 CuCl 4 and Cs 2 CuBr 4 [11,12], the Co-based compounds can form regular (undistorted) triangular-lattice layers and are free from a large antisymmetric interaction of the Dzyaloshinsky-Moriya type thanks to the highly symmetric crystal structure. The physics of those compounds is expected to be described by a simple model Hamiltonian.
Recently, Shirata et al. [2] reported that the magnetization curve of Ba 3 CoSb 2 O 9 powder seems to show excellent agreement with theoretical calculations on the spin-1/2 isotropic Heisenberg model on a two-dimensional (2D) triangular lattice [13][14][15][16][17][18][19]. This is owing to the fact that the magnetic layers of Co 2+ ions are well separated from each other by a nonmagnetic layer [2]. However, the latest experiments with the use of single crystals [3][4][5][6] found a field-direction dependence of magnetization curve, which indicates the existence of exchange anisotropy, and a magnetization anomaly that had not been predicted at a strong magnetic field ∼ 22 T perpendicular to the c axis.
In our previous study [21], we have provided a microscopic model calculation for the magnetization process of the quasi-2D TLAF Ba 3 CoSb 2 O 9 by taking into account the easy-plane exchange anisotropy and weak couplings between layers. The theoretical magnetization curve under in-plane magnetic field exhibits a field-induced first-order transition at ∼ 0.7 of the saturation field H s as well as the well-known plateau structure at the one-third of the saturation magnetization. From the result, we suggested that the origin of the magnetization anomaly observed in Ba 3 CoSb 2 O 9 is indeed the extra first-order transition due to the weak interlayer coupling. In fact, the critical field strength (∼ 22 T) at the anomaly is well accorded with our theoretical prediction (∼ 0.7H s [21]), given that the saturation field of Ba 3 CoSb 2 O 9 is about 31.9 T [4].
In Ref. 21, we focused on the system with easy-plane exchange anisotropy in order to explain a specific experiment. In this paper, we give a quantitative study on quantum TLAFs based on the isotropic Heisenberg Hamiltonian, which is a simpler but more fundamental model to describe spin systems. Even for the simple Heisenberg model on a single layer of triangular lattice, there have been only a few numerical studies [14][15][16][17][18][19] that can reproduce the quantum stabilization of the magnetization plateau. This is due to the difficulty in treating strongly frustrated quantum systems. Also in experiments, there are only a few quantum TLAF materials that actually exhibit a clear magnetization plateau [2,11,12]. Therefore, it is still important to reproduce the plateau of the isotropic TLAF by different theoretical methods, whose reliability can in turn be confirmed by their accurate prediction of the plateau.
We apply our numerical cluster mean-field approach with a scaling scheme (CMF+S) [23][24][25][26][27] to the S = 1/2 Heisenberg model on a triangular lattice. The result for a single layer shows that the numerical CMF+S method provides an excellent quantitative agreement with previous numerical results [13,15,[17][18][19], including the quantum stabilization of the one-third magnetization plateau.
We also present the CMF+S results for the magnetization and susceptibility curves in the presence of antiferromagnetic interlayer coupling. It is shown that a small interlayer coupling gives rise to an extra discontinuous quantum phase transition at a strong field ≈ 0.712H s . The high-field first-order transition is clearly visible as a divergence in the susceptibility while the shape of the magnetization curve undergoes very little change except for quite a small jump at the transition point.

II. THE SPIN-1/2 HEISENBERG MODEL ON A LAYERED TRIANGULAR LATTICE
The Hamiltonian of the Heisenberg model on layers of triangular lattice [see Fig. 1 where both the intralayer (J) and interlayer (J ′ ) nearest-neighbor (NN) couplings are assumed to be antiferromagnetic. We investigate the S = 1/2 case that exhibits the strongest quantum fluctuations.
For a single layer of triangular lattice (or J ′ = 0), the ground-state spin configuration usually forms a three-sublattice structure as shown in Fig. 1(b) as long as the system only has spatially isotropic NN interactions. However, real TLAF compounds such as Ba 3 CoSb 2 O 9 have a quasi-2D structure that consists of magnetic triangular-lattice layers separated from each other by a nonmagnetic layer [2]. Therefore, one has to consider a small but finite interlayer interaction J ′ between spins on different layers. For a ferromagnetic J ′ < 0, it is expected that the spins along the stacking direction tend to point in the same direction and the three-sublattice spin configuration on each layer is more stabilized. A quite small |J ′ | ≪ J can be thus negligible when the magnetic layers are well separated. However, we will show that when J ′ > 0, even a small interlayer interaction could play an essential role in determining the ground-state spin configuration of TLAFs under magnetic field. The consideration of this effect due to weak three dimensionality is a key to quantitatively describe quasi-2D quantum TLAF compounds with a microscopic model Hamiltonian.

SPINS
The classical counterpart of the model (1) is written in terms of three-dimensional vector S i with a fixed length |S i | = 1/2 instead of quantum spin-1/2 operatorŜ i . It is known that the ground state of the classical Heisenberg model on a 2D triangular lattice (J ′ = 0) cannot be uniquely determined under magnetic field (H 0) [28,29]. Although the classical-spin angles on the three-sublattice structure shown in Fig. 1(a) have six degrees of freedom, the minimization of the classical energy only gives the constraint S A +S B +S C = (0, 0, H/3J) for 0 < H < H s [28,29]. Here, H s = 9J/2 is the saturation field, above which all the spins are aligned parallel to the magnetic field. Except the trivial degeneracy from the U(1) symmetry, there still remain two degrees of freedom. Therefore, spin configurations in the ground-state magnetization process (0 < H < H s ) are continuously degenerate. The total magnetization per site, M ≡ i S z i /N with N being the number of lattice sites, is given by H/9J as a linear function of the magnetic field strength for any magnetization process in the degenerate manifold.
In the quantum spin-1/2 system, however, the classical degeneracy is lifted by quantum fluctuation effects. For TLAFs, it is now established that the sequence of the "Y," up-up-down, and 0-coplanar states [ Fig. 2] is selected as the lowest-energy magnetization process [13][14][15][16][17][18] We employ here the CMF+S method [23][24][25][26][27] to take into account the quantum effects and interlayer coupling. In general, the number of spins that can be numerically diagonalized on current computer resources is very limited. To avoid strong finite-size effects in such exact diagonalization on small clusters, we impose a self-consistent mean-field boundary condition instead of the usual periodic or open one. First, we approximate the HamiltonianĤ on N sites (N → ∞ in the thermodynamic limit) by the sum of N/N C cluster Hamiltonians on a cluster of N C sites. The inter-cluster , y, z}). Thus, the cluster Hamil-tonianĤ C n includes the expectation values Ŝ α i as mean fields to be determined self-consistently. The sublattice magnetic moment m α µ on sublattice µ is given by where n = 1, 2, · · · , M C with M C being the number of the independent clusters (specifically given later), N µ is the number of total sites belonging to the sublattice µ in the M C clusters, and β = 1/T (we take T → 0 in the present study). Substituting m α µ into Ŝ α i µ inĤ C n , Eq. (2) [13,15,17,18]. Under the assumption of the three-sublattice structure shown in Fig. 1(b), the number of independent clusters that have to be treated in the CMF+S selfconsistent equations (2) is M C = 1 for N C = 3, 6, 15, or 21, while M C = 3 for N C = 10 (see the lower illustrations of Fig. 3).
The obtained magnetization curves M(H) for each N C are shown in Fig. 4(a). The quantum fluctuations select the sequence of the Y, up-up-down, and 0-coplanar states as expected. The magnetization curve exhibits the one-third quantum magnetization plateau. As the size of the cluster N C increases, the plateau gets wider, indicating that the effects of quantum fluctuations are properly included. Figure 4 [13]) and comparable with those obtained by the exact diagonalization with the periodic boundary condition (1.38 and 2.16 [15,18]) and by the coupled cluster method (1.37 and 2.15 [17]).
In the previous CMF+S study on the XXZ Hamiltonian [26] we derived the phase diagram in  with the periodic boundary condition [18] and with the coupled cluster method (CCM) [17]. The upper curves are vertically shifted by 0.25 for clarity. and so that the phase boundaries for all finite N C have the same location as those of the limit N C → ∞ (H c1 (∞) and H c2 (∞)) [21]. The extrapolation of the magnetization M at different values of the rescaled H ′ are shown in Fig. 4(c). The top curve in Fig. 4(a) is the obtained CMF+S magnetization curve, which is compared with the other numerical results [17,18] in Fig. 4(d). We can see an excellent agreement of the CMF+S curve with the data extracted from Refs. 18 and 17, regarding the locations of the phase transition points and the nonlinear bending due to the quantum effects.

V. THE ROLE OF WEAK INTERLAYER COUPLING (J ′ 0)
Weak three-dimensionality is not avoidable in real quasi-2D TLAF compounds. Since there are numerous nearly-degenerate states in frustrated systems, even very small interlayer interaction can compete with quantum fluctuations and affect the ground-state selection. In the previous study [21], while effects of the interlayer coupling have been discussed with a focus on the case where easy-plane exchange anisotropy exists, we have not explained the detailed magnetization process for the isotropic case. Therefore, here let us discuss the role of interlayer coupling J ′ on the magnetization process of the fundamental Heisenberg model (1) in a quantitative way with the use of the CMF+S.
For weakly-coupled triangular-lattice layers, we assume the six sublattice structure shown in The saturation field is given by H s = 9J/2 + 2J ′ .
In general, when the interlayer interaction is antiferromagnetic (J ′ > 0), antiparallel spin alignment is favored along the stacking direction. If each layer consists of a bipartite lattice, it is expected that the in-plane magnetic order is just stacked alternately as shown in Fig. 5. However, for non-bipartite lattices such as the triangular lattice, the in-plane three-sublattice magnetic order could compete with the demand of antiparallel alignment along the stacking direction. As a result of the incompatibility, the lowest-energy stacking pattern can be changed depending on the magnetic field strength as will be shown below.
For TLAFs in the presence of antiferromagnetic interlayer coupling, we found two candidate magnetization processes as solutions of Eq. (2) [21], both of which reduce to the sequence of the Y, up-up-down, and 0-coplanar states at the purely 2D limit. Figure 6 shows the two branches is obviously seen that the former process has a lower energy for low magnetic fields since J ′ > 0 favors antiparallel alignment on interlayer bonds. However, the magnitude relation between the energies of (c) and (c ′ ) at strong fields can be reversed as H increases. Note that the relative angles (θ, φ, and χ) among the six sublattice magnetic moments are gradually changed as a function of H even in the same phase. As can be seen in the right panels of Fig. 6, when the spins are almost collinear, the energy of (c ′ ) should be higher than that of (c) due to the high interlayer bond energy of field-parallel spin components. However, when the magnetic field is further increased, the (c ′ ) configuration becomes advantageous against (c) since it can reduce more interlayer bond energy of field-transverse components. As a result, a field-induced first-order transition between (c) and (c ′ ) is expected to occur at a certain strong magnetic field. we obtain the CMF+S result of the magnetization curve [the top curve in Fig. 7(a)]. Since J ′ is assumed here to be very small, the general behavior of the curve differs little from that of the purely 2D case. However, as shown in Fig. 7(c), the existence of the additional quantum first-order transition is clearly seen in the field derivative of M(H) as the divergence at H = H c3 ≈ 0.712H s , in contrast to the purely 2D (J ′ = 0) case.

VI. SUMMARY
In summary, we have studied the magnetization process of the spin-1/2 Heisenberg model on layered triangular lattice with and without weak interlayer coupling. Our numerical calculations with the CMF+S method properly described the one-third magnetization plateau expected in quantum TLAFs and provided a quantitative agreement with the numerical data of the ED [15,18] and the CCM [17] for a single layer of triangular lattice. Moreover, we discussed the detailed magneti-zation process in the presence of weak interlayer coupling, which is unavoidable in real quasi-2D compounds. We presented the magnetization and susceptibility curves of the isotropic Heisenberg model, which was not reported in our previous study [21]. Although a small interlayer coupling does not change the apparent shape of the magnetization curve, an additional first-order phase transition occurs at H ≈ 0.712H s above the one-third plateau. This extra transition is visible as a small discontinuous jump in the magnetization curve and a divergence in the susceptibility. From a comparison of the present isotropic Heisenberg model with the previous easy-plane case [21], the easy-plane anisotropy seems not to be relevant for the qualitative feature of the magnetization process and the shape of the magnetization curve including the extra first-order transition as long as the magnetic field is applied in the direction parallel to the easy plane. Since the appearance of the high-field first-order transition stems from the incompatibility between the in-plane quantum magnetic order and the demand of antiparallel alignment along the stacking direction, a similar magnetization process is expected to be obtained for larger spins, e.g., S = 1 and S = 3/2.
Here, we discussed the case of weak interlayer coupling (J ′ = 0.025J) employing the CMF+S based on 2D clusters shown in Fig. 3. For larger values of J ′ ( 0.2J), it has been predicted for the Heisenberg model that the umbrella state is stabilized at strong fields [30,31]. Therefore, for moderate interlayer couplings, the magnetization process should become more complex due to the competition of the umbrella state and the coplanar states shown in