O ct 2 01 9 Spin Nematic Liquids of the S = 1 Spin Ladder in Magnetic Field

The magnetization process of the $S=1$ spin ladder system is investigated using the numerical exact diagonalization of finite-size clusters. The field-induced spin nematic liquid phase was predicted to appear by our previous work. Several ground-state phase diagrams in the plane of the single-ion anisotropy and the external magnetic field are obtained in the present study.


I. INTRODUCTION
The spin nematic phase is one of interesting topics in the field of the strongly correlated electron systems. For example, the high-magnetic field measurement of the quasione-dimensional compound LiCuVO 4 detected it 1 . In addition the spin-liquid-like behavior of the S = 1 triangular-lattice compound NiGa 2 S 4 was theoretically explained by the spin nematic phase 2 . The spin nematic order is the quadrapole order of the quantum spins in two-or three-dimensional systems. On the other hand it appears as the gapless Tomonaga-Luttinger liquid phase of the two-magnon bound state in one-dimensional systems. In our previous work 3 using the numerical exact diagonalization of finite-clusters the spin nematic Tomonaga-Luttinger liquid (TLL) phase was revealed to occur in the S = 1 spin ladder system under the external magnetic field in the presence of sufficiently large negative single-ion anisotropy. In addition several ground-state phase diagrams in the plane of the anisotropy and the magnetization were presented. However, in order to propose some experiments to detect the spin nematic TLL phase in real materials, the phase diagram in the plane of the anisotropy and the external magnetic field would be much more useful. In this paper, we investigate the S = 1 spin ladder system with the numerical exact diagonalization of finite-size clusters and obtain the phase diagrams in the anisotropy and field plane.

II. MODEL
The S = 1 spin ladder with the single-ion anisotropy D is described by the Hamiltonian where S i,j = (S x i,j , S y i,j , S z i,j ) denotes the spin-1 operator acting on the spin at the jth rung and the ith chain. The quantity J 1 denotes the nearest neighbor leg interaction constant, J r the rung interaction constant, and H the strength of the external magnetic field along the z direction. We investigate the ground state of this model using the numerical exact diagonalization of finite-size cluster up to L = 8. Throughout this paper we consider the negative D only, namely the easy-axis anisotropy, and fix J 1 = 1.0.

III. GROUND STATE UNDER H = 0
In the absence of the external magnetic field, for D = 0, the system is in the plaquette singlet state which is non-degenerate and has the spin gap 4 . On the other hand, for sufficiently large negative D, the system is in the Néel state along the z-direction which is doubly degenerate and also has the spin gap. The critical point D c can be estimated by the phenomenological renormalization group method. Namely, the size-dependent critical point D c,L is determined from the equation for the scaled gaps where ∆ L (D) is the lowest energy gap between the k = 0 ground state and the k = π subspace in the leg direction. The scaled gap L∆ L (D) is plotted versus D for J 1 = J r = 2.0 in Fig. 1. Since the size dependence of D c,L is quite small, we use D c,6 = −0.20 as the best estimation of the critical point D c .

IV. TOMONAGA-LUTTINGER LIQUID PHASES FOR H > 0
Since the ground state at H = 0 is in the plaquette singlet phase for 0 > D > D c , a phase transition occurs at the critical field H c1 and the gapless TLL phase is realized for H > H c1 5,6 . On the other hand, when the ground state is the Néel ordered state for D < D c , the magnetization process is expected to be similar to that of the S = 1/2 Isinglike XXZ ladder. In this case the TLL phase is also realized above the critical field H c1 .
The quasiparticle excitation, however, is different between these two TLL phases. Each elementary magnon excitation should occur by δS z = 2, because the S z = 0 state cannot occur for sufficiently large negative D (D < D c ), while δS z = 1 for D > D c . The former TLL phase is called the spin nematic TLL phase, to distinguish from the latter one, namely, the conventional TLL phase. These two TLL phases can be distinguished by whether the gapless excitation is δS z = 1 or 2.

V. PHASE DIAGRAMS ON THE D-H PLANE
The purpose of this paper is to obtain the ground-state phase diagram on the D-H plane.
We define H c1 as the critical field where the non-zero magnetization appears for the first In order to estimate the phase boundary between the two TLL phase in the finite magnetization phase, the cross points between the δS z = 1 excitation gap and the 2k F excitation gap of the two magnon bound state in our previous work. In this paper, however, we use the cross points between H 1 (M) and H 2 (M) defined as for L = 8, because more points can be obtained than the previous method.
The saturation field H sat is obtained as Using the numerical diagonalization for L = 8, we obtain the H-D phase diagrams for

VII. SUMMARY
The S = 1 spin ladder with the easy-axis single-ion anisotropy under the magnetic field is investigated using the numerical exact diagonalization of finite-size clusters. We obtain the ground-state phase diagrams in the D-H plane including the conventional and nematic TLL phases and the magnetization jump. Some magnetization curves are also obtained.