Quasi-soliton scattering in quantum spin chains

The quantum scattering of magnon bound states in the anisotropic Heisenberg spin chain is shown to display features similar to the scattering of solitons in classical exactly solvable models. Localized colliding Gaussian wave packets of bound magnons are constructed from string solutions of the Bethe equations and subsequently evolved in time, relying on an algebraic Bethe ansatz based framework for the computation of local expectation values in real space-time. The local magnetization profile shows the trajectories of colliding wave packets of bound magnons, which obtain a spatial displacement upon scattering. Analytic predictions on the displacements for various values of anisotropy and string lengths are derived from scattering theory and Bethe ansatz phase shifts, matching time-evolution fits on the displacements. The time-evolved block decimation algorithm allows for the study of scattering displacements from spin-block states, showing similar scattering displacement features.

Figure 1

Left: decaying 2-string wave packet with zero group velocity (bound state of two magnons with Gaussian momentum distribution centered with $\alpha =4$ around $\overline{p}=\pi$, where the energy dispersion has its minimum), in a chain of $N=100$ sites, calculated from algebraic Bethe ansatz matrix elements. Right: prefactor ${\delta }^2$ of the $t^2$ dependence of the wave-packet width. Theoretical curve [Eq. (29)] with $\overline{p}=\pi$ and $J=1$, as a function of $\mathrm{\Delta }$. The data points are retrieved by fitting ${\delta }^2$ from the decaying wave packets from the Bethe ansatz time evolutions.

Figure 2

Decay of a block of 20 upturned spins in a chain of $N=1000$ sites for different values of anisotropy, computed using TEBD. Around the isotropic point $\mathrm{\Delta }=1$, the behavior of the spin block changes drastically, as the spin block becomes less tightly bound and will start to decay into smaller strings.

Figure 3

Time evolution of $\langle S_j^z\left(t\right)\rangle$ illustrating scattering $n$-string wave packets, computed using algebraic Bethe ansatz, for $N=100,\mathrm{\Delta }=2,{\overline{p}}_1=-{\overline{p}}_2=-\pi /2$, and $\alpha =4$. Left: scattering of 1-string wave packets (single magnons). Right: scattering of 2-string wave packets (bound magnons).

Figure 4

Left: measurement of the scattering displacement from Bethe ansatz time evolution of bound magnons at $\mathrm{\Delta }=2$, by computing the average location of the wave packets according to Eq. (44) and taking the horizontal difference of the linear fits. Right: scattering displacement (in units of lattice distance) for single and bound magnons as function of anisotropy. Measured data from algebraic Bethe ansatz time evolution of magnetization (see left panel) compared to the derivative of scattering phase shifts [see Eqs. (38) and (39)].

Figure 5

Left: time evolution from algebraic Bethe ansatz of $\langle S_j^z\left(t\right)\rangle$ for a single magnon wave packet with momentum ${\overline{p}}_1=-\pi /2$ scattering against a 3-string wave packet with momentum ${\overline{p}}_2=\pi /2$ (top) and ${\overline{p}}_2=\pi$ (bottom), respectively, with $N=100,\mathrm{\Delta }=2$, and $\alpha =4$. Right: corresponding scattering displacements (in units of lattice distance) as function of anisotropy [see Eqs. (40, 41, 42, 43)], where the points are measured displacements obtained from the time-evolution data.

Figure 6

TEBD time evolution of scattering of a two-spin block state with a block of 10 upturned spins at $\mathrm{\Delta }=5$. The block of 10 upturned spins is shifted by four lattice sites upon impact of the 2-string-like bound magnon. This is in correspondence with Eq. (46), yielding a shift of four caused by a 2-string on a larger string.

Figure 7

Left: time evolution of $\langle S_j^z\left(t\right)\rangle$, performed using numerical exact diagonalization for $L=26$ sites. Two single-magnon wave packets are prepared with initial width $\alpha =5/\sqrt{2}\approx 3.5355$ and opposite momenta $\pm k$; here $k=0.5\pi$ and $\mathrm{\Delta }=2.5$. Right: the Bethe ansatz phase shift (lines) are compared with the phase of $\mathrm{\Gamma }$ (symbols, shifted by $\pi$), obtained from the wave function of the time-evolved $L=26$ chain. The comparison is shown for three $k$ values.