Bloch oscillations of multimagnon excitations in a Heisenberg $XXZ$ chain

The studies of multimagnon excitations will extend our understandings of quantum magnetism and strongly correlated matters. Here, by using the time-evolving block decimation algorithm, we investigate the Bloch oscillations of two-magnon excitations under a gradient magnetic field. Through analyzing the symmetry of our Hamiltonian, we derive a rigorous and general relation between ferromagnetic and antiferromagnetic systems. Under strong interactions, in addition to free-magnon Bloch oscillations, fractional bounded-magnon Bloch oscillations appear which can be understood by an effective single-particle model. To extract the frequencies of Bloch oscillations and determine the gradient of the magnetic field, we analyze the fidelity and the substandard deviation in both time and frequency domains. Our study not only explores the interaction-induced Bloch oscillations of multimagnon excitations, but also provides an alternative approach to determine the gradient of the magnetic field via ultracold atoms in optical lattices.

Figure 1

Dynamical symmetry of spin correlations under the parameters $(\mathrm{\Delta },B)$ and $(-\mathrm{\Delta },B)$. The time evolution of two-spin correlations at the 9th and 10th sites with ferromagnetic interaction (left half) and the ones at the $-10\mathrm{th}$ and $-11\mathrm{th}$ sites with antiferromagnetic interaction (right half).

Figure 2

Bloch dynamics of free magnons. (a) Spin distributions $S_l^z$ versus the rescaled time $t/T_B$. (b)–(f) Longitudinal spin-spin correlations $C_{l^{\prime },l^{″}}$ at different times marked as $A$–$E$ in (a). The parameters are chosen as $\mathrm{\Delta }=0$, $B=0.05$, and the total chain length $L_t=101$.

Figure 3

Bloch dynamics of two strongly interacting magnons. (a) Spin distributions $S_l^z$ versus the rescaled time $t/T_B$. (b)–(f) Longitudinal spin-spin correlations $C_{l^{\prime },l^{″}}$ at different times marked as $A$–$E$ in (a). The parameters are the same as in Fig. 2, except for $\mathrm{\Delta }=-5$.

Figure 4

Bloch dynamics of two moderately interacting magnons. (a) Spin distributions $S_l^z$ versus the rescaled time $t/T_B$. (b)–(f) Longitudinal spin-spin correlations $C_{l^{\prime },l^{″}}$ at different times marked as $A$–$E$ in (a). The parameters are the same as in Fig. 2, except for $\mathrm{\Delta }=-1.5$.

Figure 5

(a)–(d) Fidelity $F$ versus the rescaled time $t/T_B$ for different values of $\mathrm{\Delta }$: (a) 0, (b) $-1$, (c) $-1.5$, and (d) $-5$. (e)–(h) Frequency distribution $f_F\left(\omega \right)$ of the fidelity for different values of $\mathrm{\Delta }$: (e) 0, (f) $-1$, (g) $-1.5$, and (h) $-5$. The other parameters are $B=0.05$ and the total chain length $L_t=101$.

Figure 6

(a)–(d) Substandard deviation $D^{1/2}$ versus the rescaled time $t/T_B$ for different values of $\mathrm{\Delta }$: (a) 0, (b) $-1$, (c) $-1.5$, and (d) $-5$. (e)–(h) Frequency distribution $f_D\left(\omega \right)$ of the substandard deviation for different values of $\mathrm{\Delta }$: (e) 0, (f) $-1$, (g) $-1.5$, and (h) $-5$. The other parameters are $B=0.05$ and the total chain length $L_t=101$.

Figure 7

Two-magnon energy spectrum for $B=0$, $L_t=101$, and (a) $\mathrm{\Delta }=0$, (b) $\mathrm{\Delta }=-1$, (c) $\mathrm{\Delta }=-1.5$, and (d) $\mathrm{\Delta }=-5$, with the overlaps $P_{K,r}$ between the initial state and each eigenstate shown by the color bar.

Figure 8

Rescaled two-magnon correlations $\overline{{\mathrm{\Gamma }}_{l^{\prime },l^{″}}}={\mathrm{\Gamma }}_{l^{\prime },l^{″}}/{\mathrm{\Gamma }}_{l^{\prime },l^{″}}^{\max }$. The top, middle, and bottom rows correspond to noninteracting $\mathrm{\Delta }=0$, strong-interacting $\mathrm{\Delta }=-5$, and moderate-interacting $\mathrm{\Delta }=-1.5$ systems, respectively. For all three rows, the evolved time is set as $t=0$, $T_B/4$, $T_B/2$, $3T_B/4$, and $T_B$ from left to right, respectively. The parameters are $B=0.05$ and $L_t=101$. The red regions of $0.25<\overline{{\mathrm{\Gamma }}_{l^{\prime },l^{″}}}\le 1$ represent regions out of range of the color bar.