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Physical review. B, Condensed matter

Properties of anS=1 antiferromagnetic Heisenberg chain with single-ion anisotropy

Hong Chen, Lu Yu, Zhao-Bin Su

Published Nov 1, 1993• 12 citations
Publisher

Abstract

A spin-1 antiferromagnetic Heisenberg chain with a single-ion anisotropy ${\mathit{Dtsum}}_{\mathit{j}}$(${\mathit{S}}_{\mathit{j}}^{\mathit{z}}$${)}^{2}$ is investigated on the basis of the continuum representation of the model Hamiltonian. Analytical results for the ground state, the excitation spectrum, and the correlation functions are derived in a self-consistent-field approach. We find that the ground state is made of boson pairs and the correlation functions in the ground state decay exponentially. Two Haldane gaps due to the single-ion anisotropy are obtained and a transition at the critical value ${\mathit{D}}_{\mathit{c}}$=1 is shown in the lower Haldane gap and the correlation functions.

DOI
10.1103/physrevb.48.12692

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Anisotropy effects on the magnetic properties of anS=1 antiferromagnetic Heisenberg chain

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Physical review. B, Condensed matter•Sep 1, 1994•42 citations
A spin-1 antiferromagnetic Heisenberg chain with a single-ion anisotropy Dtsumj{\mathit{Dtsum}}_{\mathit{j}}Dtsumj​({\mathit{S}}_{\mathit{j}}^{\mathit{z}}$${)}^{2} is investigated on the basis of the continuum representation of the model Hamiltonian. Analytical results for the ground state, the excitation spectrum, and the correlation functions are derived in a self-consistent-field approach. We find that the ground state is made of boson pairs and the correlation functions in the ground state decay exponentially. Two Haldane gaps due to the single-ion anisotropy are obtained and a transition at the critical value Dc{\mathit{D}}_{\mathit{c}}Dc​=1 is shown in the lower Haldane gap and the correlation functions.
We develop an extended self-consistent mean-field approach to study a spin-1 antiferromagnetic (AF) chain with the Hamiltonian H=tsumi{\mathit{tsum}}_{\mathit{i}}tsumi​ [Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​Si+1x{\mathit{S}}_{\mathit{i}+1}^{\mathit{x}}Si+1x​+Siy{\mathit{S}}_{\mathit{i}}^{\mathit{y}}Siy​Si+1y{\mathit{S}}_{\mathit{i}+1}^{\mathit{y}}Si+1y​ +\ensuremath{\lambda}Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​Si+1z{\mathit{S}}_{\mathit{i}+1}^{\mathit{z}}Si+1z​+D(Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​)2{)}^{2})2] based on Anderson's spin-wave theory. The ground-state energy changes little compared with the traditional spin-wave theory and the excitation spectrum shows a gap which lends support to Haldane's conjecture. At the Heisenberg point (\ensuremath{\lambda}=1,D=0), the ground-state energy and the gap are -1.3505 and 0.3914, respectively. In the Heisenberg-Ising case (\ensuremath{\lambda}>0,D=0), the gap decreases when \ensuremath{\lambda} goes down from 1 and at a critical ${\ensuremath{\lambda}}_{\mathit{c}}$>0 the gap goes to 0, while for the single-ion anisotropy (\ensuremath{\lambda}=1,D>0) the gap decreases with the increasing of D and at a critical value Dc{\mathit{D}}_{\mathit{c}}Dc​ the gap goes to 0. These results agree well with those of other theoretical methods and experiments. We study the one-magnon excitation spectrum carefully and suggest an explanation for its crossing with the two-magnon excitation spectrum at low momentum. We estimate the crossing occurring at k\ensuremath{\simeq}0.3\ensuremath{\pi}. The results agree well with the very recent theoretical and experimental results. The picture of the Haldane gap is discussed from our extended self-consistent mean-field ground state and many interesting properties of the spin-1 AF chain are discussed.
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.
We investigate ground-state and low-lying excited states of the S = 1 antiferromagnetic Heisenberg chain with exchange anisotropy, single-ion anisotropy and biquadratic exchange and with free boundary conditions. We present an approximate mapping of the S = 1 model to an effective S = 1/2 chain in an external magnetic field, which can be solved exactly in several limiting cases. We obtain results for the phase diagram, correlation functions and the excitation spectrum. The transition to the Haldane phase is related to the condensation of the lowest soliton modes and the hidden order parameter of the Haldane phase is identified as transverse long-range order in the effective S = 1/2 chain.
The S=1 antiferromagnetic Heisenberg chain both with and without single-ion anisotropy is studied. Using the recently proposed density-matrix renormalization-group technique of White, we calculate the energy gaps as well as several different correlation functions. The two gaps, ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{\ensuremath{\parallel}}}$ and ${\mathrm{\ensuremath{\Delta}}}_{\mathrm{\ensuremath{\perp}}}$, along with associated correlation lengths and velocities are determined. The numerical results are shown to be in good agreement with theoretical predictions derived from the nonlinear \ensuremath{\sigma} model and a free-boson model. We also study the S=1/2 excitations that occur at the ends of open chains; in particular we study the behavior associated with open boundary conditions, using a model of S=1/2 spins coupled to the free bosons.
Effects of an uniaxial single-ion anisotropy tsumi{\mathit{tsum}}_{\mathit{i}}tsumi​Dz{\mathit{D}}_{\mathit{z}}Dz​(Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​)2{)}^{2})2 on the magnetic properties of an S=1 antiferromagnetic Heisenberg chain with open boundaries are studied by a quantum Monte Carlo method. In the ground stae, how the anisotropy affects the staggered magnetization 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 and the spin correlation functions 〈S1z{\mathit{S}}_{1}^{\mathit{z}}S1z​Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 and 〈S1x{\mathit{S}}_{1}^{\mathit{x}}S1x​Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​〉 is investigated. Amplitudes of 〈Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 decrease with with increase of Dz{\mathit{D}}_{\mathit{z}}Dz​. The correlation length ${\ensuremath{\xi}}_{\mathit{S}\mathit{S}}^{\mathit{z}}$ of 〈S1z{\mathit{S}}_{1}^{\mathit{z}}S1z​Siz{\mathit{S}}_{\mathit{i}}^{\mathit{z}}Siz​〉 monotonously decreases with increase of Dz{\mathit{D}}_{\mathit{z}}Dz​, while the correlation length ${\ensuremath{\xi}}_{\mathit{S}\mathit{S}}^{\mathit{x}}$ of 〈S1x{\mathit{S}}_{1}^{\mathit{x}}S1x​Six{\mathit{S}}_{\mathit{i}}^{\mathit{x}}Six​〉 has the opposite but weak Dz{\mathit{D}}_{\mathit{z}}Dz​ dependence. The fourfold degeneracy of the ground state also holds with nonzero Dz{\mathit{D}}_{\mathit{z}}Dz​'s in the thermodynamic limit although it is lifted up in finite chains. At finite temperatures, the magnetic susceptibilities along the z axis (${\mathrm{\ensuremath{\chi}}}^{\mathit{z}}$) and the x axis (${\mathrm{\ensuremath{\chi}}}^{\mathit{x}}$) are calculated. Temperature dependences of the bulk susceptibilites, ${\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\infty}}}^{\mathit{z}}$ and ${\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\infty}}}^{\mathit{x}}$, are consistent with the previous experimental result on 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). Finite-size corrections, ${\mathrm{\ensuremath{\chi}}}_{0}^{\mathit{z}}$ and ${\mathrm{\ensuremath{\chi}}}_{0}^{\mathit{x}}$, and the Curie susceptibilities resulting from chain ends, ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}^{\mathit{z}}$ and ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}^{\mathit{x}}$, are also investigated in detail. Both ${\mathrm{\ensuremath{\chi}}}_{0}$ and ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}$ diverge as temperature goes to zero and they are combined through a relation ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}$=${\mathrm{\ensuremath{\chi}}}_{0}$+${\mathrm{\ensuremath{\chi}}}_{\mathrm{\ensuremath{\infty}}}$. Temperature dependences of ${\mathrm{\ensuremath{\chi}}}_{\mathit{C}}$ are dominated by the edge effect at low temperatures, while at high temperatures they are attributed to an S=1 free spin.