The long-distance asymptotic behavior of the longitudinal spin correlation < S i z S j z > in the ground-state of the spin-1/2 dimerized XY chain is exactly calculated. The spin correlation < S i z S j z > decays in the exponential law with the correlation length ζ=1/ln [(1+δ)/(1-δ)]. Here δ (0≦δ≦1) represents the degree of alternation: i.e., the stronger and the weaker spin couplings are expressed as J (1+δ) and J (1-δ), respectively. The correlation length ξ smoothly changes from ∞ (power decay) to 0 as we move away from δ=0 (no alternation) to δ=1 (independent spin pairs). The nearest neighbor spin correlation is also investigated. The present results are compared with those obtained by the use of the phase Hamiltonian method.
By the use of the numerical method we investigate the ground state phase diagram of spin-1/2 alternating ferromagnetic chain. We numerically diagonalized the Hamiltonian of finite systems (up to 20 spins) and analyzed the numerical data for various physical quantities using the finite size scaling and the extrapolation methods. The ground state is either the effective singlet (ES) state or the spin fluid (SF) state depending on the value of the alternation parameter δ and the anisotropy parameter Δ ≡ J z / J ⊥ ( Δ =-1 for the isotropic ferromagnetic case and Δ =0 for the X Y case). The phase diagram obtained in this work strongly stupports the theoretical studies of Kohmoto-den Nijs-Kadanoff and Okamoto-Sugiyama. We also discuss the critical properties near the ES-SF transition line.
The ground state properties of the one-dimensional spin-1/2 alternating Heisenberg- XY ferromagnet are analytically investigated by the use of the phase Hamiltonian representation. To treat the non-harmonic terms in the phase Hamiltonian, both the self-consistent harmonic approximation (SCHA) and the renormalization group method are applied. The ground state is either the effective singlet (ES) state or the so-called spin-fluid state. The phase diagram is depicted on the Δ -δ plane, where Δ and δ are the anisotropy and alternation parameters, respectively. The longitudinal and transverse spin correlation functions (< S i z S j z > and < S i + S j - >) in the ES state are also discussed in the framework of the SCHA.
The ground-state properties of the one-dimensional spin-1/2 alternating Heisenberg-Ising antiferromagnet are studied both analytically and numerically. The phase diagram is qualitatively predicted by the use of the phase Hamiltonian: The ground state is either the Néel or the effective singlet state according to the values of the alternation and anisotropy parameters. The phase transition between these states is of the first order. The ground state energy, its derivative with respect to the alternation parameter, excitation gap, and the various long-range order parameters are exactly calculated by the numerical method for finite systems ( N ≦20). The numerical calculations support the analytically predicted phase diagram. The correlation length is also numerically calculated at finite temperatures by the use of the quantum transfer-matrix method.
By using mainly numerical methods, we investigate the ground-state phase diagram (GSPD) of an
S=1 ferromagnetic-antiferromagnetic bond-alternating chain with the
XXZ and the on-site anisotropies. This system can be mapped onto an anisotropic spin-2 chain when the ferromagnetic interaction is much stronger than the antiferromagnetic interaction. Since there are many quantum parameters in this system, we numerically obtained the GSPD on the plane of the magnitude of the antiferromagnetic coupling versus its
XXZ anisotropy, by use of the exact diagonalization, the level spectroscopy as well as the phenomenological renormalization group. The obtained GSPD consists of six phases. They are the $XY
1,thelarge−D$ (LD), the intermediate-
D (ID), the Haldane (H), the spin-1 singlet dimer (SD), and the Néel phases. Among them, the LD, the H, and the SD phases are the trivial phases, while the ID phase is the symmetry-protected topological phase. The former three are smoothly connected without any quantum phase transitions. It is also emphasized that the ID phase appears in a wider region compared with the case of the GSPD of the anisotropic spin-2 chain with the
XXZ and the on-site anisotropies. We also compare the obtained GSPD with the result of the perturbation theory.
We calculate the correlation function < S z i S z j > in the ground state of one dimensional spin-1/2 alternating antiferromagnetic Heisenberg model, taking the X Y -like anisotropy into consideration. We use the phase Hamiltonian method developed by Nakano and Fukuyama and apply the self-consistent harmonic approximation to the non-harmonic terms in the phase Hamiltonian. The long-distance behavior of the correlation function is< S z i S z j >∼( x /ξ) -1/2 exp (- x /ξ), where x is the distance between the i-th and j-th spins. The correlation length ξ is proportional to the inverse of the energy gap between the ground state and the first excited state. The results are cornpared with those of the exact numerical analysis of finite size systems and the Monte Carlo simulations.