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Physical review. B, Solid state

Approximate calculation of the excitation spectrum of the Heisenberg linear chain

B. Sriram Shastry, Chanchal K. Majumdar

Published Oct 1, 1975• 2 citations
Publisher

Abstract

An approximate calculation of the excitation spectrum of the Heisenberg linear chain is performed by the Green's-function technique. Apart from the usual transverse Green's function, a longitudinal Green's function is also found useful. The decoupling procedure is not based on the two-sublattice picture, but utilizes the knowledge of known exact results about the ground state of the linear chain and the insight obtained from a model with explicitly known ground state. The equations are solved in a self-consistent manner, and the spectrum, except near the edge of the Brillouin zone, compares well the exact des Cloizeaux-Pearson result. We also provide an expression for the intensity for neutron-scattering studies; this can be checked against measurements on Cu${\mathrm{Cl}}_{2}$\ifmmode\cdot\else\textperiodcentered\fi{}2N(${\mathrm{C}}_{5}$${\mathrm{D}}_{5}$) (CPC).

DOI
10.1103/physrevb.12.2856

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American Journal of Physics•Jun 1, 1972•1 citation
An approximate calculation of the excitation spectrum of the Heisenberg linear chain is performed by the Green's-function technique. Apart from the usual transverse Green's function, a longitudinal Green's function is also found useful. The decoupling procedure is not based on the two-sublattice picture, but utilizes the knowledge of known exact results about the ground state of the linear chain and the insight obtained from a model with explicitly known ground state. The equations are solved in a self-consistent manner, and the spectrum, except near the edge of the Brillouin zone, compares well the exact des Cloizeaux-Pearson result. We also provide an expression for the intensity for neutron-scattering studies; this can be checked against measurements on CuCl2{\mathrm{Cl}}_{2}Cl2​\ifmmode\cdot\else\textperiodcentered\fi{}2N({\mathrm{C}}_{5}$${\mathrm{D}}_{5}) (CPC).
The spectral properties, and associated critical singularities, of spin-1 Heisenberg antiferromagnetic chains are of high current interest, both theoretical and experimental. Theoretical studies using the single mode approximation and also a Monte Carlo calculation indicate that the spectrum has a gap at k=π, as predicted by Haldane. They also predict the unusual feature that the excited state continuum is asymmetric about the mid-point of the Brillouin zone, k=π/2. However, neither approach is sufficiently accurate to predict reliably the magnitude of the gap at k=0 which was not discussed by Haldane. Our approach is to use static (spectral) calculations of excited states of chains of N=4 through 14 spins for all values of k, a novel feature not associated with previous finite-size calculations, to discuss the dispersion spectra of the S=1 Heisenberg chain. In addition, we have calculated dynamical properties, in particular S(q,ω), exactly for finite chains up to N=14 spins. This allows us to estimate the gap at k=0 to good accuracy, and we find that to within this accuracy the gap at k=0 is twice the Haldane gap at k=π. This result is important in connection with very recent neutron scattering and magnetization measurements. In addition, we have calculated the correlation length and critical singularities associated with dynamical quantities.
The one-dimensional Heisenberg model with S = 1/2 is treated with the use of the two-time Green's functions. The hierarchy of the equations of motion of the Green'f functions is decoupled at a stage one-step further than Tyablikov's decoupling. The thermal average of the spin component, ≪ Sz >, is set to zero, becaure the long-range order does not exist in one dimension. Instead, our Green's functions are expressed in terms of the correlation functions cn ≡4≪ S0zSnz >. The Green's function is essentially of the form representing undamped spin waves, whose spactrum depends on c1, c2 and one more parameter. They are determined by the requirement that c1 and c2 should be self-consistent and that c0 should be unity. The self-consistency equations have been solved analytically at high- and low-temperature limits, and also solved numerically in the whole range of the temperature. Thermodynamic quantities have been calculated using these solutions. It has turned out that the theory gives the correct high-temperature expansion for the thermodynamic quantities and the correlation functions. The latter is expressed by cn = (J / 4 kBT )n. In the case of ferromagnetic coupling, the correlation function cn at T = 0 is equal to 1/3 for all n's. This is what is expected from the correct ground state of the ferromagnetic Heisenberg system. The spin-wave spectrum at T = 0 also agrees with the correct one. At T ≪J/ kB, we find that the specific heat goes as T1/2 and the susceptibility goes as T-2. The gross feature of the temperature-dependence of the thermodynamic quantities agrees with Bonner and Fisher. In the case of the antiferromagnetic Heisenberg model, we find c1 = -0.55407 and c2 = 0.16100 at T = 0, which are fairly close to the exact values. The thermodynamic quantities are also in gross agreement with Bonner and Fisher.
A zero-temperature renormalization group (RG) approach is applied to the one-dimensional, spin-1/2 antiferromagnetic Heisenberg dimerized (alternating) chain. Specifically, the ground state energy and lowest-lying spectral excitations are examined. The calculation indicates the existence of a gap in the spectrum of the dimerized chain which vanishes only in the limit of a uniform spin chain, in contrast to a recent Green’s function approach. The RG results are in reasonable agreement with numerical extrapolations on the exact eigenvalue spectrum of finite chains of up to 12 spins. Both methods are compared with several other approximate treatments of the Heisenberg system, and tested by comparison with exact results for the spin-1/2 XY dimerized chain.
We describe an approximate Green function decoupling theory for the dynamics of a dimerized Heisenberg chain. The theory retains the rotational invariance of the spin system in the approximations as Bulaevskii's approximate theory does not. We find that, within the approximations used, the excitation spectrum of the chain can be described as consisting of two branches: A band with a gap of k = 0 and a second, acoustic, band with no gap. The acoustic band does not appear in the Bulaevskii theory. We discuss the possible relevance of this result to experiments.
The Callen and Tyablikov decoupling schemes used in the Green's function method are tested numerically for short Heisenberg linear chains which are first subjected to a constraint. The constraint is imposed to insure that internal excitations alone contribute to the calculated thermal averages so that in this sense these averages correspond to thermodynamic quantities. It is found that improved agreement with the exact results occurs if the Tyablikov decoupling is corrected by a term linear in the temperature.