Quantum spin nematic liquid in the low-dimensional anisotropic magnets -S = 1 / 2 delta spin chain with the anisotropic ferromagnetic interaction in magnetic field-Toru

The magnetization process of the S = 1 / 2 delta chain with the anisotropic ferromagnetic interaction is investigated using the numerical diagonalization of finite-size clusters. It is found that the spin nematic liquid phase appears in higher magnetization region, as well as the SDW liquid one in lower region.


Introduction
The spin nematic state [1] is one of interesting topics in the field of the strongly correlated electron systems. It is the quadrupole order of spins. Recently many theoretical and numerical studies on the spin nematic state have been reported about various quantum spin systems. In most theoretical works the mechanism of the spin nematic order is based on the spin frustration [1][2][3][4] or the biquadratic exchange interaction [5][6][7]. In this paper we propose a theoretical model without the spin frustrations, or the biquadratic interaction, that exhibits the spin nematic liquid phase in magnetic field. It is the S = 1/2 delta spin chain [8] with the anisotropic ferromagnetic interaction. In one-dimensional systems like this model, the nematic order is reduced to the quasi-long-range order characterized by the power-law decay of the spin correlation function, which is called the Tomonaga-Luttinger liquid (TLL). We investigate this model using the numerical diagonalization of finite-size clusters and obtain the phase diagrams with respect to the anisotropy and the magnetization, which include the nematic-correlation dominant TLL phase.

Model and Calculation
The magnetization process of the S = 1/2 delta chain shown in Fig. 1 is investigated. It is described by the Hamiltonian where λ is the anisotropy and H is the magnetic field. J 1 and J 2 are fixed to −1 and +1, respectively. For the  Figure 1: Delta spin chain.
Since the system (1) is not frustrated, the spin pair at the J 1 bond behaves like the S = 1 object. Thus for λ = 1 and H = 0 the ground state is expected to be in the Haldane phase [9]. The Néel order would be realized for sufficiently large λ. Using the phenomenological renormalization [10], the phase boundary can be estimated by the fixed point equation where ∆ π is the excitation gap with k = π in the subspace with M = 0. The scaled gap L∆ π for J 3 = 0.2 is plotted versus λ for L = 8, 10, 12 and 14 in Fig. 2(a). The extrapolation of the size-dependent fixed point for L and L + 2 assuming the size correction proportional to 1/(L +1), as shown in Fig. 2(b), results in λ c = 2.1376±0.0001 in the infinite length limit.

Field-Induced TLL Phases
The system (1) is expected to behave as the S = 1 antiferromagnetic chain with anisotropy.
With the Ising-like anisotropy (λ > 1), it would be an effective S = 1 chain model with the easy-axis single-ion anisotropy. According to the previous numerical diagonalization study on the magnetization process of the S = 1 antiferromagnetic chain [11], the two-magnon TLL phase, where each magnetization step is δS z = 2, is realized for the sufficiently large easy-axis anisotropy, while the conventional TLL phase appears near the isotropic case. The singlemagnon excitation gap and the 2k F excitation gap of the two magnon bound state are defined as ∆ 1 and ∆ 2k F , respectively. The phase boundary between the conventional and two-magnon TLL phases can be estimated as the point of ∆ 1 = ∆ 2k F , because ∆ 1 (∆ 2k F ) is gapless (gapped) in the former phase, while gapped (gapless) in the latter one. The scaled gaps L∆ 1 and L∆ 2k F of the system (1) at m = 1/2 for J 3 = 0.4 are plotted versus λ for L = 8 and 12 in Fig. 3. It confirms the gapless and gapped behaviors of ∆ 1 and ∆ 2k F are switched at the expected phase boundary. Thus we determine the phase boundary λ c as ∆ 1 = ∆ 2k F at each magnetization.

Critical Exponent Analysis
In the field-induced two-magnon TLL phase, the nematic spin correlation perpendicular to H and the SDW one parallel to H are expected to exhibit the power-law decay. These are described by the following spin correlation functions The correlation with the smaller exponent η is dominant. Namely, the nematic spin correlation dominant TLL phase is realized for η 2 < η z , while the SDW dominant one for η 2 > η z . According to the conformal field theory, these critical exponents can be estimated by the forms of several energy gaps for each magnetization M , where k 1 is defined as k 1 =L/2π. The exponents η 2 and η z estimated for L = 12 and 14 are plotted versus m for J 3 = 0.4 and λ = 2.5 in Fig.4. It indicates that the spin nematic dominant TLL phase (η 2 < η z ) is realized at larger m while the SDW one at smaller m. Since the relation η 2 η z = 1 should be satisfied in the TLL phase, the crossover between the two dominant spin correlations should occur at the magnetization with η 2 = η z = 1. Fig. 4 suggests that the system size dependence of η z is too large to estimate the crossover point in the infinite length limit. Thus we determine the crossover magnetization as η 2 = 1.

Phase Diagrams
Finally the phase diagrams with respect to λ and m is presented for J 3 = 0. dominant two-magnon TLL phase, respectively. The shape of the phase diagram depends on J 3 . For smaller J 3 , the magnetization process from the Haldane phase would meet the quantum phase transition from CTLL to NTLL or SDW 2 TLL. In contrast, for larger J 3 the one from the Néel ordered phase would meet the quantum phase transition from SDW 2 TLL or NTLL to CTLL. The up triangle is the boundary between the Haldane and Néel ordered phases. The down triangle is determined by ∆ 1 = ∆ 2 at m = 1 limit, where ∆ 2 is the two-magnon excitation gap.

Summary
The magnetization process of the S = 1/2 delta chain with the anisotropic ferromagnetic interaction is investigated using the numerical diagonalization. It is found that for sufficiently large easy-axis anisotropy the spin nematic correlation dominant TLL phase appears at higher magnetization region, while the SDW dominant one at lower magnetization.