Spinon dynamics in quantum integrable antiferromagnets

The excitations of the Heisenberg antiferromagnetic spin chain in zero field are known as spinons. As pairwise-created fractionalized excitations, spinons are important in the understanding of inelastic neutron scattering experiments in (quasi-)one-dimensional materials. In the present paper, we consider the real space-time dynamics of spinons originating from a local spin flip on the antiferromagnetic ground state of the (an)isotropic Heisenberg spin-1/2 model and the Babujan-Takhtajan spin-1 model. By utilizing algebraic Bethe ansatz methods at finite system size to compute the expectation value of the local magnetization and spin-spin correlations, spinons are visualized as propagating domain walls in the antiferromagnetic spin ordering with anisotropy dependent behavior. The spin-spin correlation after the spin flip displays a light cone, satisfying the Lieb-Robinson bound for the propagation of correlations at the spinon velocity.

Figure 1

Time slices of spinon propagation originating from a local spin flip at $j_0=0$ on the ground state at $N=100$ for $\mathrm{\Delta }=1$. The initial state is a projection with total normalization sum rule of 99.00%, including all two-spinon states (96.85%) and the most important 3640 four-spinon states (2.15%). In the top left panel, the CFT prediction from Eq. (21) for the magnetization of the initial state is shown.

Figure 2

Time slices of spinon propagation originating from a spin flip at $j_0=0$ at $N=100$ for $\mathrm{\Delta }=1.5$. The initial state is a projection on all two-spinon states with 97.17% saturation of the normalization sum rule.

Figure 3

Time slices of spinon propagation originating from a spin flip at $j_0=0$ at $N=100$ for $\mathrm{\Delta }=10$. The initial state is a projection on all two-spinon states with 99.999% saturation of the normalization sum rule.

Figure 4

Spinon propagation shown in plots of the magnetic fluctuations $\langle \delta S_j^z\left(t\right)\rangle$ at single sites as a function of time for $\mathrm{\Delta }=1$ (top) and $\mathrm{\Delta }=10$ (bottom).

Figure 5

Time slices of two-spinon propagation originating from a spin flip at $j_0=0$ at $N=100$ for the isotropic spin-1 Babujan-Takhatajan model. The initial state is a projection on all two-spinon states (consisting of holelike modes in a sea of two-strings) with 92.33% saturation of the normalization sum rule. In the top left panel, the CFT prediction from Eq. (27) for the magnetization of the initial state is shown.

Figure 6

Illustration of the location of the spins at sites $\pm j$ between which the correlation is computed in Eq. (28), relative to the location of the spin flip creating the spinon excitations.

Figure 7

The spin-spin correlation between two sites at $\pm j$, with the up spin is in the middle at $j=0$, computed for $\mathrm{\Delta }=1$ and $N=32$. The projection of the initial state on two-spinon states saturates 98.82% of the normalization sum rule. Top: single site plots as function of time. Bottom: heat map showing the light-cone effect in the buildup of the correlations at the spinon propagation velocity.

Figure 8

Longitudinal nearest-neighbor correlation at $\mathrm{\Delta }=1$ and $N=32$. The projection on two-spinon states saturates 98.82% of the normalization sum rule. As the spinon passes by, the antiferromagnetic correlation becomes weaker and then revives again, showing oscillations after the passage of the spinon.

Figure 9

Errors of the method induced by the truncation of the sum over intermediate states at $\mathrm{\Delta }=1$. Left: Sum rule saturation for every two-spinon state, by summing over the intermediate states with zero and one infinite rapidity, respectively. Right: Expectation value of longitudinal spin-spin correlation on the same site, which should be $1/4$.

Figure 10

Left panels: Exact diagonalization results for $N=12$ for the spin-spin correlations from Eq. (28) (top) and the nearest neighbor correlations from Eq. (29) (bottom). Right panels: Difference between the results from exact diagonalization and matrix element summations from algebraic Bethe ansatz.