One-dimensional quantum spin dynamics of Bethe string states

Quantum dynamics of strongly correlated systems is a challenging problem. Although the low energy fractional excitations of one-dimensional integrable models are often well understood, exploring quantum dynamics in these systems remains challenging in the gapless regime, especially at intermediate and high energies. Based on the algebraic Bethe ansatz formalism, we study spin dynamics in a representative one-dimensional strongly correlated model, i.e., the antiferromagnetic spin-$\frac{1}{2}$ XXZ chain with the Ising anisotropy, via the form-factor formulas. Various excitations at different energy scales are identified crucial to the dynamic spin structure factors under the guidance of sum rules. At small magnetic polarizations, gapless excitations dominate the low energy spin dynamics arising from the magnetic-field-induced incommensurability. In contrast, spin dynamics at intermediate and high energies is characterized by the two- and three-string states, which are multiparticle excitations based on the commensurate Néel ordered background. Our work is helpful for experimental studies on spin dynamics in both condensed matter and cold atom systems beyond the low energy effective Luttinger liquid theory. Based on an intuitive physical picture, we speculate that the dynamic feature at high energies due to the multiparticle antibound state excitations can be generalized to nonintegrable spin systems.

Figure 1

The momentum-resolved FFM ratios with $2m$ equal to (a) 0.2, (b) 0.5, and (c) 0.8, respectively. The pink, blue, red, and black curves represent cumulative results by including the psinon states $n\psi \psi$ ($n=1,2$) in $S^{-+}$, the psinon-antipsinon states $n\psi {\psi }^{*}$ ($n=1,2$), the two-string states and three-string states in $S^{+-}$, respectively. In (a), the pink and blue curves overlap significantly and so do the red and black curves in (c).

Figure 2

Schematic plot of a representative spin configuration in the real space within: (a) the Néel ordered ground state at zero field; (b) the incommensurate ground state at a nonzero field $h>h_c$; (c) a state with real particle wave vectors contributing to $S^{-+}$; (d) a state with real particle wave vectors contributing to $S^{+-}$; (e) a two-string state contributing to $S^{+-}$; (f) a three-string state contributing to $S^{+-}$. The blue hollow circle represents a spin up which is viewed as vacuum, and the yellow solid circle represents a spin down which is viewed as a particle. A particle is removed from (added to) the incommensurate ground state configuration in $S^{-+}$ ($S^{+-}$), which is represented by an arrow pointing out of (into) the corresponding position in (b).

Figure 3

The intensity plots for the transverse DSFs $S^{-+}(q,\omega )$ from ($a_1$) to ($c_1$) and $S^{+-}(q,\omega )$ from ($a_2$) to ($c_2$) in the $q-\omega$ plane all with the same intensity scale. $2m$ equals 0.2 in ($a_{1,2}$), 0.5 in ($b_{1,2}$), and 0.8 in ($c_{1,2}$). The $\delta$ function in Eq. (2) is broadened via a Lorentzian function $\frac{1}{\pi }\gamma /[{(\omega -E_{\mu }+E_G)}^2+{\gamma }^2]$ with $\gamma =1/400$.

Figure 4

Spectrum intensity evolution of $S^{\perp }(q,\omega )=S^{+-}(q,\omega )+S^{-+}(q,\omega )\text{vs}\hslash \omega /J$ at (a) $q=\frac{\pi }{2}$, and (b) $q=\frac{3\pi }{4}$. In (a) and (b), lines from bottom to up correspond to $2m$ varying from 0.1 to 0.9 with the step of 0.1. Contributions from psinon excitations in the $S^{-+}$ channel are plotted in pink. Psinon-antipsinon, two-string, and three-string states in the $S^{+-}$ channel are plotted in blue, red, and black colors, respectively. The broadening parameter $\gamma =1/50$.

Figure 5

The evolution of peaks in DSSFs of $S^{+-}$ and $S^{-+}$ at different momenta versus magnetic field $h$ with lines of peaks marked by ${\chi }_{\pi /2}^{\left(3\right)}, {\chi }_{\pi /2}^{\left(2\right)}, R_{\pi /2}^{-+}, {\chi }_{\pi }^{\left(2\right)}, R_0^{+-}, R_{\pi /2}^{+-,a}$, and $R_{\pi /2}^{+-,b}$. The pink, blue, red, and black colors correspond to real states in $S^{-+}$, real states in $S^{+-}$, two-string states in $S^{+-}$, and three-string states in $S^{+-}$, respectively. The hollow circles represent the peak positions extracted from DSSF spectral figures similar to Fig. 4. The solid lines are determined by solving the energies of the Bethe eigenstates with the largest weight values around the spectral peaks.

Figure 6

The momentum-resolved FFM ratios at $2m=0.1$. The pink, blue, red, and black curves represent cumulative results by including the psinon states $n\psi \psi$ ($n=1,2$) in $S^{-+}$, the psinon-antipsinon states $n\psi {\psi }^{*}$ ($n=1,2$), the two-string states and three-string states in $S^{+-}$, respectively, as before. The anisotropy $\mathrm{\Delta }=2$, and system size $N=200$.

Figure 7

The $\mathrm{\Delta }$ dependence of the ratios of momentum integrated intensity (a) ${\nu }_{-+}$, and (b) ${\nu }_{+-}$. The parameter values are $N=200$ and $2m=0.05$. In (a), the contributions from $1\psi \psi$ and $2\psi \psi$ states are included. In (b), the blue, red, and black curves display the results by cumulatively including the psinon-antipsinon, two-string, and three-string contributions in $S^{+-}$, respectively.

Figure 8

The momentum-resolved FFM ${\nu }_{\left|\right|}^{\left(1\right)}\left(q\right)$ ratios from (a) to (c), and the intensity plots from (d) to (f) for the longitudinal DSF $S^{zz}$. $2m$ equals 0.2 in (a) and (d), 0.5 in (b) and (e), and 0.8 in (c) and (f), respectively. In (a), (b), and (c), the blue, red, and black lines are cumulative results by including $1\psi {\psi }^{*}, 2\psi {\psi }^{*}$, and $1{\chi }^{\left(2\right)}1\psi \psi$ excitations. The broadening parameter in the intensity plots is $\gamma =1/400$.

Figure 9

Distributions of Bethe quantum numbers for the string excitations which have local maximal weight values at the corresponding momentum. The positions of the solid circles represent the Bethe quantum numbers of the particles. The system size and magnetization are taken as $N=32$ and $S_T^z=8$.