Observation of Complex Bound States in the Spin-$1/2$ Heisenberg $XXZ$ Chain Using Local Quantum Quenches

We consider the nonequilibrium evolution in the spin-$1/2$ $XXZ$ Heisenberg chain for fixed magnetization after a local quantum quench. This model is equivalent to interacting spinless fermions. Initially an infinite magnetic field is applied to $n$ consecutive sites and the ground state is calculated. At time $t=0$ the field is switched off and the time evolution of observables such as the $z$ component of spin is computed using the time evolving block decimation algorithm. We find that the observables exhibit strong signatures of linearly propagating spinon and bound state excitations. These persist even when integrability-breaking perturbations are included. Since bound states (“strings”) are notoriously difficult to observe using conventional probes such as inelastic neutron scattering, we conclude that local quantum quenches are an ideal setting for studying their properties. We comment on implications of our results for cold atom experiments.

Figure 1

Time evolution in the spin-polarized case after preparing the system in an initial state with three spin flips in the center of a 101 site chain for different values of $\Delta$. Top row: Spacetime plot of $⟨S^z⟩(x,t)$; middle row: $⟨{\mathcal{P}}_{\uparrow \uparrow }⟩(x,t)$, which projects a bond onto $|\uparrow \uparrow ⟩⟨\uparrow \uparrow |$; bottom row: $⟨{\mathcal{P}}_{\uparrow \uparrow \uparrow }⟩(x,t)$, which projects three adjacent sites onto $|\uparrow \uparrow \uparrow ⟩⟨\uparrow \uparrow \uparrow |$.

Figure 2

2-string propagation at finite magnetization per site $m$ at $\Delta =1.2$, corresponding to the Luttinger liquid phase of the model. From top to bottom, $m=-0.44$, $m=-0.26$, and $m=-0.14$. The initial state at $t=0$ is the ground state of (1) with an infinite magnetic field term at two sites in the center of the chain (chain length $N=100$). At $t=0$, the field is switched off and the state is evolved. The striped patterns visible in all plots are Friedel oscillations due to open boundary conditions.

Figure 3

Propagation velocity of single-spinon and 2-string branch as a function of total magnetization per site $m$ of the system at $\Delta =1.2$. Green and red curves show single-spinon and 2-string velocities as calculated from the Bethe ansatz. Green circles and blue squares are numerically derived values from real time simulations. Error bars are smaller than symbols.

Figure 4

Spacetime plot of $⟨S^z⟩$ for a setup similar to Fig. 2 with $N=101$ at total magnetization $m=-0.2525$, $\Delta =1.2$, with three particles at the chain center at $t=0$.

Figure 5

Spacetime plot of $⟨S^z⟩$ and $P_{\uparrow \uparrow }$ for $N=100$, $m_0=2$ at total magnetization $m=-0.26$, $\Delta =1.2$ and an extra integrability-breaking term $(J/10)\sum _j{\mathbf{S}}_j·{\mathbf{S}}_{j+2}$ added to $H(\Delta ,B_0)$. Bound state signatures are seen to persist.