A physically motivated approach is used to study the linear chain XXZ spin-1 Hamiltonian in the antiferromagnetic regime. A variational Hamiltonian is obtained restricting the basis set to states with local antiferromagnetic environments separated by sites with Sz=0 acting as domain walls. Ground-state properties of this variational Hamiltonian are obtained and, in particular, an order–disorder transition is found with critical behavior corresponding to the two-dimensional Ising model universality class, in agreement with Haldane conjecture. Excellent agreement with finite chain data concerning the position of Haldane’s point, ground-state energy, and elementary excitations is found.
The XXZ spin-1 chain is studied by means of a physically motivated variational approach. An effective Hamiltonian for the dynamics of spin-0 defects in antiferromagnetic environments (domain walls) is obtained, and its ground-state properties are analyzed. An order-disorder transition is found with critical behavior corresponding to the 2D Ising model universality class, in complete agreement with Haldane conjecture. The approximations involved in this variational scheme are carefully tested using known Monte Carlo results. In particular, excellent agreement with the recently calculated excited states at the Heisenberg (isotropic) point is found, clarifying the nature of these excitations.
The linear chain XXZ spin- (1)/(2) Hamiltonian is mapped onto a model where the local degrees of freedom are domain walls between opposite antiferromagnetic domains. The domain-wall Hamiltonian is solved within the simplest Hartree–Fock approximation with results in very good agreement with the exact solution in the antiferromagnetic regime. The location of ordered and disordered (critical) phases is correct and low-lying excitations are well described and easily interpreted in terms of domain walls. Our domain-wall version of the original Hamiltonian is shown to provide the natural framework for a physical understanding of this system in the antiferromagnetic regime.
A recent remarkable prediction by Haldane is the 1D integer spin antiferromagnets of XXZ type should show strikingly different T=0 phase behavior from their counterparts with half-integer spin. The consensus of a wide variety of numerical evidence is in support of the Haldane prediction. However, one aspect which has been particularly difficult to confirm has been the behavior in the vicinity of the critical point Δ=Δ2. The point Δ2 is predicted to be a second-order transition in the universality class of the transverse Ising model at its critical field. It has been numerically established that at Δ=Δ2∼1.18–1.20, the Haldane gap disappears and an excited SzT=0 state becomes degenerate with the SzT=0 ground state for Δ≥Δ2. The mapping to the transverse Ising model implies the existence in the limit N→∞ of an infinite continuum of scaling states quasi-degenerate with the ground state(s) at Δ=Δ2. Numerical calculations to determine the presence of these scaling states have been performed up to N=12 spins for the spin-1 XXZ model. The development of this scaling continuum is only apparent for large N, when a class of k=0, SzT=0 high-lying spectral excitations develop a minimum in the vicinity of Δ∼1.18 which intensifies with increasing N. These excitations extrapolate well below the lower edge of the triplet continuum, and we conclude these are the Haldane scaling states. This conclusion is reinforced by a detailed study of the corresponding excitations for the spin-1/2 transverse Ising model. However, we also find a class of k=0, ‖SzT‖=1 excitations which show similar scaling behavior in the vicinity of Δ2. These states were not included in the Haldane prediction. The implications for the behavior of the correlation functions at Δ2 are discussed.
The XXZ spin-1/2 Hamiltonian in the linear chain is exactly transformed into a model where the local degrees of freedom are domain walls between opposite antiferromagnetic domains. The ground state of the new Hamiltonian is studied with the Hartree-Fock approximation. This approximation is shown to become exact in the XY and extreme antiferromagnetic limits, with excellent results in between. Known results concerning the existence or nonexistence of long-range order and gap in the excitation spectrum are reproduced by our solution. Good quantitative agreement with representative properties of the exact solution is obtained. The domain-wall Hamiltonian is shown to be the natural framework for the understanding of the ground state and low-lying excited states of the XXZ system in the antiferromagnetic regime.
The low-temperature behaviour of the thermodynamic quantities in the one-dimensional 1D antiferromagnet with a Haldane ground state was analysed using a concept of non-interacting quasiparticles as a starting point. The proposed description appeared to be in good agreement with the results of the Monte Carlo study of a 128-site spin chain and experimental data obtained in NENP (Ni(C 2 H 8 N 2 ) 2 NO 2 ClO 4 ) provided that the elementary excitations constituting the Haldane triplet obey Fermi statistics.
String correlations are investigated in an infinite-size XXZ spin-1 chain. By using the infinite matrix product state representation, we calculate a long-range string order. In the XY phase, the string correlations decay within a relatively very large lattice distance, which makes a finite-size study difficult to verify the non-existence of the string order. Thus, in the Haldane phase, the non-vanishing string correlations in the limit of a very large distance allow to characterize the phase boundaries to the XY phase and the Neel phase, which implies that the transverse long-range string order is the order parameter for the Haldane phase. In addition, the singular behaviors of the von Neumann entropy and the fidelity per lattice site are shown to capture clearly the phase transition points that are consistent with the results from the string order. The estimated critical points including a BKT transition from the XY phase to the Haldane phase agree well with the previous results:
Δc2=0 for the XY-Haldane phase transition and
Δc3=1.185 for the Haldane-N'eel phase transition from the density renormalization group. From a finite-entanglement scaling of the von Neumann entropy with respect to the truncation dimension, the central charges are found to be
c≃1.0 at
Δc2=0 and
c≃0.5 at
Δc3=1.185, respectively, which shows that the XY-Haldane phase transition at
Δc2=0 belongs to the Heisenberg universality class, while the Haldane-Neel phase transition at
Δc2=1.185 belongs to the two-dimensional classical Ising universality class. It is also shown that, the long-range order parameters and the von Neumann entropy, as well as the fidelity per site approach, can be applied to characterize quantum phase transitions as a universal phase transition indicator for one-dimensional lattice many-body systems.