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Journal of Physics Condensed Matter

Calculation of the susceptibility of theS= 1 antiferromagnetic Heisenberg chain with single-ion anisotropy using the transfer matrix renormalization group method

D.J. Coombes, Tao Xiang, G A Gehring

Published Mar 16, 1998• 9 citations
Publisher

Abstract

Using an extension of the density matrix renormalization group method at finite temperature, we have calculated the magnetic susceptibility of the S = 1 antiferromagnetic Heisenberg chain with single-ion anisotropy. We investigate the dependence of the Haldane gap on the anisotropy energy. The results are compared with experimental data for the compound NENP.

DOI
10.1088/0953-8984/10/10/001

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Using an extension of the density matrix renormalization group method at finite temperature, we have calculated the magnetic susceptibility of the S = 1 antiferromagnetic Heisenberg chain with single-ion anisotropy. We investigate the dependence of the Haldane gap on the anisotropy energy. The results are compared with experimental data for the compound NENP.
The quantum transfer-matrix method (QTM) has been applied to study the finite-temperature static properties of the spin S=1S=1S=1 antiferromagnetic Heisenberg chains in a wide range of the single-ion anisotropy and temperatures. The high-resolution QTM simulation data are obtained for the zero-field susceptibility, specific heat, as well as for the field-dependent magnetization. The microscopic parameters of a number of real quasi-one-dimensional compounds are found from fitting procedures, some theoretical approaches are numerically verified and low-temperature extensions of high-temperature series are given in terms of Chebyshev polynomials.
We study a Heisenberg antiferromagnetic chain of spin S=1 with nearest-neighbor exchange interaction J and easy-plane single-ion anisotropy D. We follow the splitting of the Haldane gap into two components as a function of the anisotropy. Chains of length up to 16 sites are studied by a Lancz\"os method and longer chains up to 32 sites have been investigated by a quantum Monte Carlo simulation for D/J=0.10, 0.15, 0.20, and 0.25. A careful finite-size scaling treatment of our data leads to values that are directly relevant to the interpretation of experiments 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). We discuss the relationship between our results and the magnetic-susceptibility measurements as well as the inelastic-neutron-scattering data and on the Haldane conjecture. Values are proposed for the parameters D and J in the case of NENP.
By the numerical transfer-matrix method we have computed several thermal properties for a Heisenberg chain of 150 antiferromagnetically coupled spins with quantum number s = 1. We dispensed with any extrapolation to the infinite chain, because the results for the magnetization, the susceptibility, the correlation length, and the static structure factor at wave number q = π have been found to be insensitive to a further increase of the chain length in the temperature regime considered. The characteristic features of our results, that occur at sufficiently low temperatures and low applied fields, can convincingly be explained by the existence of a gap in the excitation spectrum of the order of 0.4 coupling constants and, therefore, nicely complement previous numerical results for corresponding ground-state properties. We have also performed computations for a chain with an additional single-site anisotropy. We find, for appropriate parameter values, excellent agreement with experimental data for the quasi-one-dimensional model compound NENP.
The density matrix renormalization group (DMRG) method is applied to the anisotropic Heisenberg chain at finite temperatures. The free energy of the system is obtained using the quantum transfer matrix which is iteratively enlarged in the imaginary time direction. The magnetic susceptibility and the specific heat are calculated down to T=0.01J and compared with the Bethe ansatz results. The agreement including the logarithmic correction in the magnetic susceptibility at the isotropic point is fairly good.
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.
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.