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Journal of the Physical Society of Japan

Transverse Antiferromagnetism of a Magnetized Haldane-Gap System

Tôru Sakai, Minoru Takahashi

Published Dec 15, 1990• 1 citation
Publisher

Abstract

A 1D S =1 Heisenberg antiferromagnet in a magnetic field H (// z -axis) at T =0 is studied by numerical diagonalizations. It is found that a phase transition exists at H c1 (=Δ), where Δ is the Haldane gap, and the transverse staggered susceptibility χ st x x diverges and the spin correlation is asymptotically < S 0 x S r x >∼(-1) r r -η between H c1 and H c2 (=4). If the system is quasi-1D, even small interchain interactions can make canting Néel order within a mean field approximation for interchain interactions. This is consistent with a recent NMR measurement for NENP at low temperature.

DOI
10.1143/jpsj.59.4227

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A 1D S =1 Heisenberg antiferromagnet in a magnetic field H (// z -axis) at T =0 is studied by numerical diagonalizations. It is found that a phase transition exists at H c1 (=Δ), where Δ is the Haldane gap, and the transverse staggered susceptibility χ st x x diverges and the spin correlation is asymptotically < S 0 x S r x >∼(-1) r r -η between H c1 and H c2 (=4). If the system is quasi-1D, even small interchain interactions can make canting Néel order within a mean field approximation for interchain interactions. This is consistent with a recent NMR measurement for NENP at low temperature.
A one-dimensional S=1 Heisenberg antiferromagnet in a magnetic field H (\ensuremath{\parallel}z axis) at T=0 is studied by numerical diagonalizations up to N=16. We give the magnetization curve at the thermodynamic limit, and derive an anomaly at Hc1{\mathit{H}}_{\mathit{c}1}Hc1​ (=\ensuremath{\Delta}), where \ensuremath{\Delta} is the Haldane gap. It is also found that the transverse spin correlation has the asymptotic form 〈S0x{\mathit{S}}_{0}^{\mathit{x}}S0x​Srx{\mathit{S}}_{\mathit{r}}^{\mathit{x}}Srx​〉\ensuremath{\sim}(-1)r{)}^{\mathit{r}})r${\mathit{r}}^{\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\eta}}}$, and the transverse staggered susceptibility ${\mathrm{\ensuremath{\chi}}}_{\mathrm{st}}^{\mathit{x}\mathit{x}}$ diverges between Hc1{\mathit{H}}_{\mathit{c}1}Hc1​ and Hc2{\mathit{H}}_{\mathit{c}2}Hc2​ (=4). The exponent \ensuremath{\eta} has a minimum (\ensuremath{\eta}\ensuremath{\simeq}0.3) at magnetization m\ensuremath{\simeq}1/3 and \ensuremath{\eta}\ensuremath{\simeq}0.5 at Hc1{\mathit{H}}_{\mathit{c}1}Hc1​ and Hc2{\mathit{H}}_{\mathit{c}2}Hc2​. If the system is quasi-one-dimensional, even small interchain couplings can create a canted N\'eel order, within a mean-field approximation for interchain interactions. This is consistent with a recent NMR measurement for Ni(C2{\mathrm{C}}_{2}C2​H8{\mathrm{H}}_{8}H8​N2{\mathrm{N}}_{2}N2​)2{)}_{2})2​NO2{\mathrm{NO}}_{2}NO2​(ClO4{\mathrm{ClO}}_{4}ClO4​) (NENP) at low temperature.
We study the S =1 antiferromagnetic Heisenberg chain with the anisotropies E ∑ j [( S j x ) 2 -( S j y ) 2 ] and D ∑ j ( S j x ) 2 in a magnetic field H along z -axis, using numerical diagonalization up to N =14 and the finite-size scaling and phenomenological renormalization. We give the phase diagrams in the H E and H D planes repectively at T =0. It is found that a magnetic field breaking the axial symmetry of the Hamiltonian brings about canted Néel order for H c1 < H < H c2 , if the system has the Haldane gap. The two transitions at H c1 and H c2 are revealed to belong to the same universality class as the two-dimensional Ising model. In addition it is found that even if the system is in the large- D phase ( D > D c ≈1), the same transitions occur in the region where the value of D is about 1.0∼1.5.
The author studies the S=1 antiferromagnetic Heisenberg chain with uniaxial anisotropy, which exhibits the Haldane gap phenomenon. He argues that, when a magnetic field in a direction transverse to the anisotropy axis is applied, a sharp phase transition takes place at a critical value of the field. The transition is characterized by vanishing of the hidden antiferromagnetic order which can be measured by the string order parameter of den Nijs and Rommelse. The existence of such a transition distinguishes a Haldane gap system from other disordered quantum spin systems.
The field-induced long-range antiferromagnetic order in the quasi-one-dimensional S=1 antiferromagnet is investigated with the exact diagonalization of finite chains and the mean field approximation for the interchain interaction. In the presence of the single-ion anisotropy D, we present the phase diagrams in the plane of the magnetic field H and the temperature T for H∥D and H⊥D. It is found that the two boundary curves of the ordered phase for H∥D and H⊥D have two intersections in the case of D>0 (XY like), while no intersection in the case of D<0 (Ising like).
We use a quantum O(N) nonlinear \ensuremath{\sigma} model to describe one-dimensional quantum Heisenberg antiferromagnets with integer spin S. A large-N analytical treatment of this effective theory leads to definite predictions for thermodynamic quantities in the low-temperature regime. We obtain the Haldane gap as a function of the temperature and note a trend similar to that found in the compound NENP. The specific heat and magnetic susceptibility are also obtained and comparison with exact numerical results for the S=1 spin chain allows us to determine the domain of validity of our approximation. By a calculation at subleading 1/N order we compute the quasiparticle lifetime. Results are given on-shell, for the width observed at the quasiparticle pole as seen in neutron-scattering experiments, and also off-shell, for the NMR relaxation rate ${\mathit{T}}_{1}^{\mathrm{\ensuremath{-}}1}$. The phenomenological interacting boson model is compared with our approach. We discuss the effects of anisotropy and compare our results with present experiments on NENP.
We investigate the magnetic field behaviour of an antiferromagnetic Heisenberg spin-1 chain with the most general single-ion anisotropy. We discuss the regime in which the magnetic field is below the transition value. The splitting of the Haldane triplet is obtained as a function of a field applied in an arbitrary orientation by means of a Lanczos exact diagonalization of chains of up to 16 spins. Our results are well summarized in terms of a first-order perturbation theory. We explain various level crossings that occur by the existence of discrete symmetries. A discussion is given of the electron-spin-resonance and neutron-scattering experiments on the compound Ni(C 2 H 8 N 2 ) 2 NO 2 ClO 4 (NENP).