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Critical reviews in solid state and materials sciences/CRC critical reviews in solid state and materials sciences

Long range ordered, dimerized, large-D and Haldane phases in spin 1 chain compounds

Olga V. Maximova, Sergey V Streltsov, Alexander N. Vasiliev

Published Dec 14, 2020• 26 citations

Abstract

In 1983, F. Duncan M. Haldane predicted a singlet ground state for isolated integer-spin one-dimensional antiferromagnets with low single-ion anisotropy D. Since then, a lot of species containing chains of integer spin ions were tested to check the basic conjecture on an energy gap separating the continuum of the excited states from the ground state. As a result of these studies, it has been established that there are numerous states competing with the Haldane phase, namely long-range ordered, dimerized, and large-D phases. The long-range magnetic order takes place due to sufficiently strong exchange interactions between adjacent chains. Dimerization results from the alternation of the exchange interactions within the chains. Both uniaxial and rhombic single-ion anisotropies can suppress the Haldane phase, which is robust only until some critical values. The choice between the competing phases depends also on exchange anisotropy. Excellent reviews on the basic results obtained during the first 20 years of investigation of these phenomena provided solid background for the future studies. Here, we present some developments in this field obtained over the next two decades of research on spin-1 chain systems.

DOI: 10.1080/10408436.2020.1852911

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In 1983, F. Duncan M. Haldane predicted a singlet ground state for isolated integer-spin one-dimensional antiferromagnets with low single-ion anisotropy D. Since then, a lot of species containing chains of integer spin ions were tested to check the basic conjecture on an energy gap separating the continuum of the excited states from the ground state. As a result of these studies, it has been established that there are numerous states competing with the Haldane phase, namely long-range ordered, dimerized, and large-D phases. The long-range magnetic order takes place due to sufficiently strong exchange interactions between adjacent chains. Dimerization results from the alternation of the exchange interactions within the chains. Both uniaxial and rhombic single-ion anisotropies can suppress the Haldane phase, which is robust only until some critical values. The choice between the competing phases depends also on exchange anisotropy. Excellent reviews on the basic results obtained during the first 20 years of investigation of these phenomena provided solid background for the future studies. Here, we present some developments in this field obtained over the next two decades of research on spin-1 chain systems.

The ground-state phase diagram of a frustrated S=1 Heisenberg spin chain with uniaxial single-ion-type anisotropy is studied using the infinite-system density-matrix renormalization-group method. Particular attention is paid to "chiral" phases, in which the long-range order parameter is a vector chirality. It is found that, in addition to the Haldane, large-D (LD), and double-Haldane phases, the phase diagram includes three different chiral phases, i.e., the gapless chiral, chiral-Haldane, and chiral-LD phases. These chiral phases are distinguished from each other by the energy gap and the string long-range order.

The antiferromagnetic Heisenberg spin chain with integer spin has a short-range magnetic order and an excitation energy gap above the ground state. This so-called Haldane gap is proportional to the exchange coupling constant J of the Heisenberg chain. In this study, we discuss recent results about the spin dependence of the Haldane gap and conjecture an analytical formula valid asymptotically for large spin values. We then discuss the robustness of the edge states of the spin-1 chain by studying a spin-1 ladder using the density matrix renormalization group (DMRG) algorithm. We show that the peculiar hidden topological order of the spin-1 chain disappears smoothly by increasing the ladder rung coupling without any intervening phase transition. This is evidence for the fragile character of the topological order of the spin-1 chain.

Ground-state properties of the spin-1 antiferromagnetic Heisenberg chain with bond alternation and uniaxial single-ion-type anisotropy, which is described by the Hamiltonian \({\cal H}_0 = \sum_{\ell=1}^N\{1-(-1)^\ell\alpha\}{\mib S}_\ell\cdot{\mib S}_{\ell+1} + D\sum_{\ell=1}^N\;(S_\ell^z)^2\) (0.0 ≤α≤1.0, -∞< D < ∞), are studied. An exact diagonalization method by means of the Lanczös technique is used. The ground-state phase diagram on the α versus D plane at the thermodynamic limit is determined by analyzing the Binder parameters associated with Néel order and z -direction string order for finite-size ( N = 6, 8, …, 16) chains. It is found that there exists a multicritical point at which the Haldane, Néel, and singlet-dimer phases meet and that the singlet-dimer phase and the large- D phase belong to the same unique phase. The ground-state magnetization curves at the thermodynamic limit for various values of α and D are obtained for the system described by the Hamiltonian, H 0 plus the Zeeman term - H ∑ ℓ=1 N S ℓ z . The results show that the 1/2-plateau appears in the magnetization curve at least when D ≥0.0 and

0.0\lesssim\alpha \leq 1.0

.

One dimensional S = 1 antiferromagnetic Heisenberg chains with bond alternation and uniaxial single-ion-type anisotropy are studied by numerical exact diagonalization of finite size systems. We calculate the ground state phase diagram using the twisted boundary condition method for D≥0. The ground states of this model contain the Haldane state, large-D state and the dimer state. Using conformal field theory and the level spectroscopy method, we calculate the D and δ-dependence of the Luttinger liquid paraments K and the critical exponent ν of the energy gap.

One dimensional S =1 antiferromagnetic Heisenberg chains with bond alternation and uniaxial single-ion-type anisotropy are studied by numerical exact diagonalization of finite size systems. We calculate the ground state phase diagram using the twisted boundary condition method for D ≥0. The ground states of this model contain the Haldane state, large- D state and the dimer state. Using conformal field theory and the level spectroscopy method, we calculate the D and δ-dependence of the Luttinger liquid paraments K and the critical exponent ν of the energy gap.

The ground-state phase transitions of a frustrated S=1 Heisenberg chain with the uniaxial single-ion-type anisotropy and the frustrating next-nearest-neighbor coupling are studied. For the system, it has been shown that there are gapless and gapped chiral phases in which the chirality κ_l = S^x_l S^y_{l+1} - S^y_l S^x_{l+1} exhibits a finite long-range order (LRO) and the spin correlation decays either algebraically or exponentially. In this study, the transitions between the Haldane and chiral phase and between the large-D (LD) and chiral phase are investigated using the infinite-system density-matrix renormalization group method. It is found that there exist two types of gapped chiral phases, "chiral Haldane" and "chiral LD" phases, in which the string LRO coexists with the chiral LRO and the string correlation decays exponentially, respectively.