The antiferromagnetic s = 1/2 model chain system under the presence of homogeneous as well as inhomogeneous longitudinal, transverse, and staggered transverse magnetic fields takes the following form,
$\begin{eqnarray}\begin{array}{rcl}{H}_{{XYZ}} & = & \sum _{j=1}^{N}{J}_{x}{S}_{j}^{x}{S}_{j+1}^{x}+{J}_{y}{S}_{j}^{y}{S}_{j+1}^{y}+{J}_{z}{S}_{j}^{z}{S}_{j+1}^{z}\\ & & -\ \left[\sum _{j=\mathrm{odd}}({B}_{z}+{b}_{z}){S}_{j}^{z}+\sum _{j=\mathrm{even}}({B}_{z}-{b}_{z}){S}_{j}^{z}\right]\\ & & -\ \left[\sum _{j=\mathrm{odd}}({B}_{x}+{b}_{x}){S}_{j}^{x}+\sum _{j=\mathrm{even}}({B}_{x}-{b}_{x}){S}_{j}^{x}\right]\\ & & +\ \left[\sum _{j=\mathrm{odd}}({B}_{x}^{\mathrm{stag}}+\lambda ){\left(-1\right)}^{j}{S}_{j}^{k}\right.\\ & & \left.+\ \sum _{j=\mathrm{even}}({B}_{x}^{\mathrm{stag}}-\lambda ){\left(-1\right)}^{j}{S}_{j}^{k}\right].\end{array}\end{eqnarray}$
The integers j = (1, 2, 3,...,N) run over the number of spins, and Jx, Jy and Jz represent the nearest neighbor Heisenberg exchange interactions. Sx, Sy, and Sz are spin-1/2 operators. Bz, Bx and ${B}_{x}^{{\rm{stag}}}$ denote the uniform longitudinal magnetic field, transverse magnetic field, and staggered transverse magnetic field, respectively. The staggered transverse magnetic field ${B}_{x}^{{\rm{stag}}}$ is related to longitudinal field Bz as ${B}_{x}^{{\rm{stag}}}=\sin \theta \,{B}_{z}$ [1, 2]. The degree of inhomogeneity in longitudinal, transverse, and transverse staggered fields is controlled by parameters bz, bx and λ, respectively. We have set the gyromagnetic g-factor and Bohr magneton coefficient μB equal to one in equation (1 ). We use periodic boundary conditions SN+1 = SN for solving the eigenvalue equation Hψ = Eψ for H in equation (1 ).
In this note, we will briefly discuss the results of HXYZ for two cases: k = z, x (i.e. Sk = Sz, Sx). The antiferromagnetic s = 1/2 chain system considered in equation (1 ) in (2019, Commun. Theor. Phys. 71 1253) [1] corresponds to a case of equation (1 ) for k = x (i.e. Sk = Sx). Here, we find the magnetization of the antiferromagnetic s = 1/2 chain system at low temperature T = 0.1Jz in presence of homogeneous magnetic fields keeping inhomogeneous field parameters to zero for N = 6. We have taken the conventional Heisenberg exchange interaction constants as Jx = 8Jz and Jy = 10Jz for the results in this note. In figure 1(a), we plot the magnetization M/Ms of the spin chain system (equation (1 )) as a function of Bz/Jz for the case k = x. We see two weak magnetization plateaus M/Ms = 0 and M/Ms = 1/3 at different transverse magnetic fields Bx/Jz and a low staggered transverse field with θ = π/30. We see that this result doesn’t match with the magnetization curve reported in figure 2(a) in [1] for the same set of parameters. In figure 1(b), we plot the magnetization M/Ms of the spin chain system (equation (1 )) as a function of Bz/Jz for k = z (i.e. Sk = Sz). We see the presence of one more plateau M/Ms = 2/3 in addition to M/Ms = 0 and M/Ms = 1/3 plateaus. These plateaus gradually convert to their counterpart quasi-plateaus on increasing the transverse field Bx/Jz. Magnetization M/Ms saturates at a lower Bz/Jz value than the magnetization in figure 1(a). This result matches exactly with what is reported in figure 2(a) in [1]. Figures 1(c) and (d) show the magnetic susceptibility X for the case k = x and k = z respectively. The rest of the parameters are kept to be the same as used to find the magnetization. Susceptibility shows the peaks in those regions of applied field Bz/Jz at which quasiplateau/plateau arises in the magnetization curve. Peaks in figure 1(c) around Bz/Jz = 8 indicates the possible presence of a meager plateau M/Ms = 2/3 in the magnetization curve (figure 1(a)) around the same Bz/Jz. Susceptibility displays non-monotone behaviour at zero applied field on increasing transverse field. Susceptibility decreases steeply in strong longitudinal magnetic field region Bz > 10Jz for all values of Bx/Jz. This result again does not match with the corresponding susceptibility result in figure 2(c) in [1], while the susceptibility result for the case k = z plotted in figure 1(d) here matches well with the corresponding susceptibility as reported in figure 2(c) in [1]. For a detailed explanation of results, we suggest the paper by the authors in [1].
Figure 1. (a) Magnetization M/Ms and Susceptibility X as a function of applied longitudinal magnetic field Bz/Jz for a set of transverse magnetic fields Bx/Jz (colored lines) and low staggered transverse field with θ = π/30 at low temperature (T = 0.1Jz) has been plotted. Here bz, bx and λ = 0. Jx = 8Jz and Jy = 10Jz. (a) and (c) correspond to the case k = x in equation ( |
We also find similar differences in magnetization, and susceptibility results in figure 3 reported in [1] when inhomogeneity in magnetic fields is considered. Same stands for specific heat results in figure 4 reported in [1].
We conclude that the results of the model system considered in [1] are different than what is reported there. We have reported the correct results for the model system considered in equation (1 ) (i.e. k = x) in [1]. We have also pointed out the correct model system (i.e. k = z) used for the results reported in [1].