Spinon and bound-state excitation light cones in Heisenberg XXZ chains

We investigate the out-of-equilibrium dynamics after a local quench that connects two spin-1/2 XXZ chains prepared in the ground state of the Hamiltonian in different phases, one in the ferromagnetic phase and the other in the critical phase. We analyze the time evolution of the on-site magnetization and bipartite entanglement entropy via adaptive time-dependent density matrix renormalization group. In systems with short-range interactions, such as the one we consider, the velocity of information transfer is expected to be bounded, giving rise to a light-cone effect. Interestingly, our results show that, when the anisotropy parameter of the critical chain is sufficiently close to that of the isotropic ferromagnet, the light cone is determined by the velocity of spin-wave bound states that propagate faster than single-particle (“spinon”) excitations. Furthermore, we investigate how the system approaches equilibrium in the inhomogeneous ground state of the connected system, in which the ferromagnetic chain induces a nonzero magnetization in the critical chain in the vicinity of the interface.

Figure 1

Exact dispersion relation for spinons ${\epsilon }_s\left(k\right)$ and for the three bound state branches $E_n\left(k\right)$ for $\mathrm{\Delta }=-0.75$. The bound state velocity $v_b\approx 0.866$ is given by the slope of $E_1\left(k\right)$ at the inflection point $k_0\approx 1.533$.

Figure 2

Representation of our quench protocol: for $t<0$ the chains are prepared in their respective ground states; at $t=0$ they are connected. The left chain is in the ferromagnetic phase and the right one is in critical phase.

Figure 3

Color map of the on-site magnetization as a function of time $t$ and site $x$ in both left and right chains (left and right columns, respectively). We decrease the value of the anisotropy parameter at the left chain ${\mathrm{\Delta }}_L=\{-1.1,-1.5,-2.0\}$ from top to bottom, while fix the value of ${\mathrm{\Delta }}_R=0.5$ and $\delta ={\mathrm{\Delta }}_R$. One can see the formation of light cones on the left and right chains, bounded by the maximum magnon and spinon velocities, respectively.

Figure 4

Wavefront propagation velocity of magnetization (circles) and entropy (squares). The dashed line is the maximum spinon excitation velocity, given by Eq. (5), and the solid curve is the maximum velocity in the case of bound state excitations, which can be obtained from Eqs. (7) and (9).

Figure 5

(a) Color map of magnetization in the right chain when ${\mathrm{\Delta }}_R=0.5$, ${\mathrm{\Delta }}_L=-20$, and $\delta ={\mathrm{\Delta }}_R$. The dotted lines indicate cuts in time, while the solid curve corresponds to a linear fit to the wavefront. (b) Spin profile in the right chain for the times indicated in (a). The last plot corresponds to the equilibrium state.

Figure 6

(a) Color map of the difference of entropy in the right chain when ${\mathrm{\Delta }}_R=0.5$, ${\mathrm{\Delta }}_L=-20$, and $\delta ={\mathrm{\Delta }}_R$. The dotted lines indicate cuts at the times considered in (b), while the solid line is a linear fit to the wavefront. (b) $\mathrm{\Delta }S$ profile in the right chain at $t=0$, $t=20$, $t=30$, and $t=40$.

Figure 7

(a) Time evolution of the magnetization in the right chain when ${\mathrm{\Delta }}_R=-0.75$, ${\mathrm{\Delta }}_L=-20$, and $\delta ={\mathrm{\Delta }}_R$. The horizontal dotted lines indicate cuts at certain times, while the solid and the dashed curves delimit the spinon and bound state excitation light cones. (b) Spin profiles in the right chain at the times indicated by dotted lines in (a).

Figure 8

(a) Time evolution of $\mathrm{\Delta }S(x,t)$ in the right chain when ${\mathrm{\Delta }}_R=-0.75$, ${\mathrm{\Delta }}_L=-20$, and $\delta ={\mathrm{\Delta }}_R$. The horizontal dotted lines indicate cuts at times considered in (b). The solid curve signals the spinon light cone while the dashed one indicates the bound state light cone. (b) $\mathrm{\Delta }S(x,t)$ profiles in the right chain at different times.

Figure 9

Color map of magnetization in the right chain when (a) ${\mathrm{\Delta }}_R=-0.25$, (b) ${\mathrm{\Delta }}_R=-0.50$, (c) ${\mathrm{\Delta }}_R=-0.85$, and (d) ${\mathrm{\Delta }}_R=-0.95$ (${\mathrm{\Delta }}_L=-20$ and $\delta ={\mathrm{\Delta }}_R$ in all panels). The solid lines represent the spinon light cones, and the dashed ones correspond to those associated with the bound state excitations.

Figure 10

(a) Color map of $DSz$. The dotted lines indicate the cuts shown in (b); the solid curve corresponds to the spinon excitation light cone boundary defined in Fig. 5. (b) Distance to the equilibrium state (see definition in the text) for sites $i=21$, $i=25$, $i=31$, $i=45$, and $i=113$ when the right chain has $N_R=100$ sites and ${\mathrm{\Delta }}_R=0.5$. Other parameters used were ${\mathrm{\Delta }}_L=-20$ and $\delta ={\mathrm{\Delta }}_R$.

Figure 11

Asymptotic distance to the equilibrium state for sites $i=21$, $i=25$, and $i=31$ as a function of the right chain length. Other parameters as in Fig. 10.