Dynamically Dominant Excitations of String Solutions in the Spin-$1/2$ Antiferromagnetic Heisenberg Chain in a Magnetic Field

Using Bethe-ansatz solutions, this Letter uncovers a well-defined continuum in dynamical structure factor $S^{+-}(k,\omega )$ of the spin-$1/2$ antiferromagnetic Heisenberg chain in magnetic fields. It comes from string solutions that continuously connect the mode of the lowest-energy excitations in the zero-field limit and that of bound states of overturned spins from the ferromagnetic state near the saturation field. The relevance to real materials is confirmed through comparisons with experimental results.

Figure 1

Distributions of Bethe quantum numbers $\{I_j\}$. Filled symbols denote $\{I_j\}$ of the ground state of $M=8$ and typical excited states in dynamical structure factors for $0\le k\le \pi$ in $L=24$. Blue open squares and red solid diamonds denote psinon ($\psi$) and antipsinon (${\psi }^{*}$) [10], which are located at points on thick and thin lines, respectively. (a) Ground state. (b) $\psi {\psi }^{*}$ for $S^{zz}(k,\omega )$. (c) $2\psi$ for $S^{-+}(k,\omega )$. (d) $2{\psi }^{*}$ for the continuum near $k=\pi$ in $S^{+-}(k,\omega )$. (e) $1{\psi }^{*}$ for the low-energy mode near $k=0$ in $S^{+-}(k,\omega )$. (f-1) $1\psi$ of $\{I_j^r\}$ for 2-string solutions in $S^{+-}(k,\omega )$. (f-2) $I_1^i$ for the $n$ string of $Sn$ ($n\ge 2$) in $S^{+-}(k,\omega )$.

Figure 2

(a, b) Contributions to sum rules from solutions of up to $2\psi 2{\psi }^{*}$, string solutions with a 2 string, and those with a 3 string, denoted by $R$, $S2$, and $S3$, respectively, in $L=320$ for (a) $S^{+-}(k,\omega )$ and (b) $S^{-+}(k,\omega )$ and $S^{zz}(k,\omega )$. (c) Contributions of dynamically dominant excitations in $L=2240$. (d) $S^{-+}(k,\omega )$ in zero field. (e, f) $S^{+-}(k,\omega )$ at $S^z/L=0.025$ without and with the contributions of $S2$ and $S3$. In (d–f), the data obtained in $L=320$ are broadened in a Lorentzian form with full width at half maximum (FWHM) $0.08J$. For $S2$ and $S3$ in (a, b, f), the $O(L^3)$ states described in the text are used.

Figure 3

(a) $S^{+-}(k,\omega )/2$, $S^{-+}(k,\omega )/2$, $S^{zz}(k,\omega )$, and $S^{\mathrm{av}}(k,\omega )$ (from above) at $S^z/L=1/8$, $1/4$, and $3/8$ (from the left) in $L=320$. The data are broadened in a Lorentzian form with $\mathrm{FWHM}=0.08J$. (b) Same as (a) but those of dynamically dominant excitations of $O(L^2)$ states in $L=2240$ with the product ansatz.

Figure 4

Comparison with experimental results on ${\mathrm{CuCl}}_2·2\mathrm{N}({\mathrm{C}}_5{\mathrm{D}}_5)$. (a) Solid lines in upper two panels are the present results of $S^{\mathrm{av}}(k,\omega )$ at $k=\pi /2$ in $L=320$. The data are broadened in a Gaussian form with $\mathrm{FWHM}\simeq 0.17J$. Open symbols are experimental results in Ref. 28. The height is rescaled after subtracting the background ($=180$ counts). The lowest panel shows line shapes of $S_{\mathrm{dom}}^{\overline{\alpha }\alpha }(k,\omega )$ ($\alpha =-$, $+$, and $z$) at $k=\pi /2$ in $L=2240$. (b) $S_{\mathrm{dom}}^{\mathrm{av}}(k,\omega )$ in $L=2240$. (c) Same as the middle panel of (a) except that the solid line is obtained from solutions with real rapidities only. The arrow indicates the high-energy peak observed in the experiment [28].

Figure 5

Comparison with experimental results on $\mathrm{Cu}({\mathrm{C}}_4{\mathrm{H}}_4{\mathrm{N}}_2)({\mathrm{NO}}_3{)}_2$. (a) $S_{\mathrm{dom}}^{\mathrm{av}}(k,\omega )$ at $S^z/L=0.15$ in $L=2240$. (b) Solid blue lines denote $S^{\mathrm{av}}(k,\omega )$ in $L=320$ along green dotted lines in (a). The data are broadened in a Gaussian form with standard deviation $\sigma =0.1\mathrm{meV}$. Open symbols are experimental results of intensity at $H=8.7\mathrm{T}$ in Ref. 8. Here, the background equal to that of $H=0$ [8] is subtracted.