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Physical review. B, Condensed matter

Sum rules in the dynamics of quantum spin chains

Gerhard H. Muller

Published Aug 1, 1982• 22 citations
Publisher

Abstract

An infinite set of sum rules is derived for the dynamics of one-dimensional quantum spin systems. They are employed to derive valuable information on the spectral-weight distribution in the $T=0$ dynamic structure factor ${S}_{\ensuremath{\mu}\ensuremath{\mu}}(q, \ensuremath{\omega})$. Applications are presented for various special cases of the nearest-neighbor $\mathrm{XXZ}$ model, including cases with a discrete excitation spectrum and cases with a continuous spectrum. For the $S=\frac{1}{2}$ $\mathrm{XY}$-Heisenberg antiferromagnet, an analytic expression for ${S}_{\mathrm{zz}}(q, \ensuremath{\omega})$ is conjectured which satisfies the infinite set of sum rules. In the $\mathrm{XY}$ limit this expression is identical to the known exact result. A similar conjecture applied to the isotropic Heisenberg antiferromagnet with arbitrary spin quantum number $S$ illustrates how the continuous spectrum of the quantum antiferromagnet collapses into a discrete branch of antiferromagnetic spin waves in the classical limit $S\ensuremath{\rightarrow}\ensuremath{\infty}$.

DOI
10.1103/physrevb.26.1311

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An infinite set of sum rules is derived for the dynamics of one-dimensional quantum spin systems. They are employed to derive valuable information on the spectral-weight distribution in the T=0T=0T=0 dynamic structure factor S\ensuremathμ\ensuremathμ(q,\ensuremathω){S}_{\ensuremath{\mu}\ensuremath{\mu}}(q, \ensuremath{\omega})S\ensuremathμ\ensuremathμ​(q,\ensuremathω). Applications are presented for various special cases of the nearest-neighbor XXZ\mathrm{XXZ}XXZ model, including cases with a discrete excitation spectrum and cases with a continuous spectrum. For the S=12S=\frac{1}{2}S=21​ XY\mathrm{XY}XY-Heisenberg antiferromagnet, an analytic expression for Szz(q,\ensuremathω){S}_{\mathrm{zz}}(q, \ensuremath{\omega})Szz​(q,\ensuremathω) is conjectured which satisfies the infinite set of sum rules. In the XY\mathrm{XY}XY limit this expression is identical to the known exact result. A similar conjecture applied to the isotropic Heisenberg antiferromagnet with arbitrary spin quantum number SSS illustrates how the continuous spectrum of the quantum antiferromagnet collapses into a discrete branch of antiferromagnetic spin waves in the classical limit S\ensuremath→\ensuremath∞S\ensuremath{\rightarrow}\ensuremath{\infty}S\ensuremath→\ensuremath∞.
Exact sum rules are obtained for the nearest-neighbor Heisenberg antiferromagnet and XYX YXY ferromagnet in one dimension. In the Heisenberg case the sum rules may be used to show that most of the spectral weight of the spin correlation function at small kkk and T=0T=0T=0 is concentrated near the frequency of the "des Cloizeaux-Pearson" states. In the XYX YXY case, the corresponding "Schultz-Lieb-Mattis" states only carry a negligible portion of the weight at small kkk.
The exact expression derived by Bougourzi, Couture, and Kacir for the two-spinon contribution to the dynamic spin structure factor S(q,\ensuremath{\omega}) of the one-dimensional s=1/2 Heisenberg antiferromagnet at T=0 is evaluated for direct comparison with finite-chain transition rates (N\ensuremath{\leqslant}28) and an approximate analytical result previously inferred from finite-N data, sum rules, and Bethe ansatz calculations. The two-spinon excitations account for 72.89% of the total intensity in S(q,\ensuremath{\omega}). The singularity structure of the exact result is determined analytically and its spectral-weight distribution evaluated numerically over the entire range of the two-spinon continuum. The leading singularities of the frequency-dependent spin autocorrelation function, static spin structure factor, and q dependent susceptibility are determined via sum rules. The impact of the non-two-spinon excitations on the integrated intensity, the susceptibility, the frequency moments, and the Euclidian time representation of S(q,\ensuremath{\omega}) is studied on the basis of finite-size data.
The T=0 dynamics of the one-dimensional s= 1 / 2 planar antiferromagnet is studied by an approach which consists of exact analytic calculations in the Bethe formalism and numerical finite-chain calculations on rings up to 10 spins. The method makes use of well known critical exponents for the correlation functions and of exact sum rules. The authors obtain approximate analytic expressions for both the out-of-plane and the in-plane dynamic structure factors, and for related quantities such as integrated intensities, susceptibilities and autocorrelation functions. The results are discussed in relation to possible experiments on quasi-1D magnetic compounds at low T. The calculations make clear that the T=0 dynamic structure factors are dominated by two-parameter continua of excitations rather than by single branches of spin-waves as predicted by classical spin-wave theory. By varying the planar anisotropy the autocorrelation functions display interesting features in their longtime asymptotic behaviour, such as a crossover from a uniform power-law decay to an oscillatory decay and a crossover between oscillatory decays with different frequencies. They conjecture a possibility of approaching the classical limit s= infinity starting from the quantum limit s= 1 / 2 . This provides a qualitative, and in some aspects even quantitative, understanding of the dynamical behaviour of s> 1 / 2 systems in terms of a quantum approach.
Spin-dynamical calculations on one-dimensional systems have relied heavily on classical ($s=\ensuremath{\infty}$) theories, despite abundant evidence that quantum effects can be extremely important at low temperatures. We present a new approach to the spin dynamics of the one-dimensional isotropic s=12s=\frac{1}{2}s=21​ Heisenberg antiferromagnetic (HB AF) which does not involve the many-body techniques usually employed. It is based on analytic Bethe ansatz calculations of excitation energies and densities of states combined with finite-chain calculations of matrix elements. An important feature of our method is the use of rigorous selection rules and the introduction of new selection rules, which are valid for macroscopic systems in a magnetic field. We show that in zero field the dynamical two-spin correlation function ${S}_{\ensuremath{\mu}\ensuremath{\mu}}(q,\ensuremath{\omega})$ at T=0T=0T=0 is governed by a two-parameter continuum of spin-wave-type excitations. In nonzero field, the longitudinal component ${S}_{\mathrm{zz}}(q,\ensuremath{\omega})$ and the transverse components ${S}_{\mathrm{xx}}(q,\ensuremath{\omega})\ensuremath{\equiv}{S}_{\mathrm{yy}}(q,\ensuremath{\omega})$ behave quite differently because they are dominated by different continua of excitations. The former is characterized by a lowest excitation branch with a zero-frequency mode moving from the zone boundary ($q=\ensuremath{\pi}$) towards the zone center (q=0q=0q=0) as the field increases, whereas the latter is characterized by a lowest branch with a zero frequency mode moving from $q=0 \mathrm{to} \ensuremath{\pi}$ with increasing field. The first part of our work features an approximate analytic expression for ${S}_{\ensuremath{\mu}\ensuremath{\mu}}(q,\ensuremath{\omega})$ at zero temperature and in zero field. Although our expression is not rigorous, exact sum rules are violated only by a small amount, and good agreement exists with the few known exact results. Our studies are extended to nonzero temperatures by placing major reliance on exact finite-chain calculations. Our work was stimulated by recent neutron scattering experiments and is oriented towards experimental comparisons. Our result for the s=12s=\frac{1}{2}s=21​ integrated intensity is in much better agreement with neutron scattering data on CuCl2{\mathrm{Cl}}_{2}Cl2​\ifmmode\cdot\else\textperiodcentered\fi{}2N(C5{\mathrm{C}}_{5}C5​D5{\mathrm{D}}_{5}D5​) (CPC) than the corresponding semiclassical result. Moreover, the spectral-weight distribution in ${S}_{\ensuremath{\mu}\ensuremath{\mu}}(q,\ensuremath{\omega})$ shows increasing asymmetry as $q\ensuremath{\rightarrow}\ensuremath{\pi}$, a quantum effect, again in agreement with more recent neutron scattering data. The second part of our work deals with the effects of an applied magnetic field. We extend the analytic work of Ishimura and Shiba to obtain expressions for the energies and densities of states of the various excitation continua. It is shown that these continua are expected to give rise to multiple structures in the scattering intensity. Our results appear to be in quantitative agreement with preliminary results of a neutron study in CPC in a field of 70 kOe, revealing anomalous scattering intensity peaks. Our results repeatedly demonstrate the inadequacy of classical spin-wave theory for this problem. They call for additional experimental studies on quasi-one-dimensional antiferromagnets to examine other unusual and interesting phenomena predicted by our approach.
Inelastic neutron scattering experiments are commonly used to unveil how excitations on Heisenberg spin models play a role in dynamical correlations functions. For a certain class of materials, like CsCoCl3_33​ or CsCoBr3_33​ salts, it turns out that their magnetic properties are fairly well approximated by quasi-one dimensional XXZ models, which enjoy the property of quantum integrability. In these instances, one can in principle use their underlying algebraic structure to describe very precisely how excitations, the so-called spinons, participate in dynamical correlations functions. Even though the available theories (either algebraic Bethe ansatz or quantum group approach) provide all the needed physical quantities such as form factors, complete set of eigenstates and spectrum, it is typically a rather daunting task, however, to obtain sufficiently simple analytical expressions for computing Dynamical Structure Factors (DSFs), valuable, e.g., for parameters estimation based on experimental data. This is particularly the case for the longitudinal DSF of the XXZ model, which has eluded a formal mathematical treatment thus far. Using the quantum group approach, we present here an exact and simple expression of the 2-spinon longitudinal DSF and show our results to be consistent with the expected sum rules and the isotropic and Ising antiferromagnet limiting cases.
We compute the exact four-spinon contribution to the zero-temperature dynamical structure factor of the spin-1/2 Heisenberg isotropic antiferromagnet in zero magnetic field, directly in the thermodynamic limit. We make use of the expressions for matrix elements of local spin operators obtained by Jimbo and Miwa using the quantum affine symmetry of the model, and of their adaptation to the isotropic case by Abada, Bougourzi and Si-Lakhal (correcting some overall factors). The four-spinon contribution to the first frequency moment sum rule at fixed momentum is calculated. This shows, as expected, that most of the remaining correlation weight above the known two-spinon part is carried by four-spinon states. Our results therefore provide an extremely accurate description of the exact structure factor.