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Journal de physique

Solitons in Ising-like quantum spin chains in a magnetic field : a second quantization approach

Francois Devreux, Jean‐Philippe Boucher

Published Jan 1, 1987• 16 citations
Publisher

Abstract

We propose a second quantization formalism for the soliton modes in the s =112 Ising-like antiferromagnetic chain. We give the exact expressions, within the one-soliton approximation, for the dynamical structure factors in presence of a magnetic field, either parallel or perpendicular to the chain. We show that the field gives rise to a splitting of the soliton modes. We also derive the spin correlation function relative to the antiferromagnetic mode for finite and infinite chains.

DOI
10.1051/jphys:0198700480100166300

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We propose a second quantization formalism for the soliton modes in the s =112 Ising-like antiferromagnetic chain. We give the exact expressions, within the one-soliton approximation, for the dynamical structure factors in presence of a magnetic field, either parallel or perpendicular to the chain. We show that the field gives rise to a splitting of the soliton modes. We also derive the spin correlation function relative to the antiferromagnetic mode for finite and infinite chains.
The dynamics of quantum solitons in an Ising-like S=(1/2 antiferromagnet on finite linear chains with spins N (\ensuremath{\le}15) is studied by integrating the Schr\"odinger equation of motion. Space-time correlation functions Szz{S}^{\mathrm{zz}}Szz(r,t) and their Fourier transforms, i.e., dynamical structure factor Szz{S}^{\mathrm{zz}}Szz(q,\ensuremath{\omega}), are calculated for both even and odd N. A slow but large oscillation of Szz{S}^{\mathrm{zz}}Szz(r,t) is seen, which gives a direct evidence of the occurrence of propagating domain walls, i.e., solitons. The oscillation leads to a low-energy component of Szz{S}^{\mathrm{zz}}Szz(q,\ensuremath{\omega}) whose line shape greatly depends on whether N is even or odd and on the temperature. We analyze in detail the dependence of soliton number in the line shape and give a plausible conjecture that Szz{S}^{\mathrm{zz}}Szz(q,\ensuremath{\omega}) for N=\ensuremath{\infty} should exhibit a very broad double maximum at low temperatures. We estimate Szz{S}^{\mathrm{zz}}Szz(q,\ensuremath{\omega}) at various temperatures. Our results reproduce experimental observations of Szz{S}^{\mathrm{zz}}Szz(q,\ensuremath{\omega}) on CsCoCl3{\mathrm{CsCoCl}}_{3}CsCoCl3​ and CsCoBr3{\mathrm{CsCoBr}}_{3}CsCoBr3​ very well over a wide temperature range.
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.
The excitation spectrum is studied around the degenerate point at the Ising limit in the one-dimensional anisotropic Heisenberg antiferromagnets with next-nearest-neighbor interactions. Solitons are found as propagation modes in the N\'eel state and in the (2,2) antiphase state, while triplet-dimer excitations are found propagating in the dimer state. These excitations determine the shape of dynamical spin-correlation functions, which are calculated by an exact diagonalization for finite chains. The phase diagram is determined from the condition that the energy gap vanishes in the excitation spectrum.
We have studied two models for anisotropic quantum spin chains. (i) XY-chain with a field in the plane: The magnetization of the ferromagnet behaves as h1/3 for small fields, in agreement with scaling laws. The antiferromagnet shows a critical field at which the ground state is a simple Néel state and which separates power law from exponential decay of spatial correlations. (ii) Anisotropic XY-chain: The dynamic z-component spin correlation function can be decomposed into a spin wave and a soliton contribution. The nature of quantum soliton excitations is studied and their form compared to soliton solutions of classical equations of motion.
We calculate the dynamical spin-structure factor of the S=1S=1S=1 Ising spin chain with negative single-ion anisotropy in magnetic fields using the infinite time-evolving-block-decimation algorithm. We show that when a transverse magnetic field is applied, both the ΔS=2ΔS=2ΔS=2 excitation continuum and one-magnon mode appear in the low-lying excitation. When a longitudinal magnetic field is further applied, the excitation continuum changes into quantized excitation spectra. The quantized ΔS=2ΔS=2ΔS=2 excitation spectra originate from the confinement of two domain walls, each of which carries ΔS=1ΔS=1ΔS=1. The quantized excitation energies are explained by the negative zeros in the Airy function.