Mechanism of ultrafast relaxation of a photo-carrier in antiferromagnetic spin background

We study the relaxation mechanism of a highly excited charge carrier propagating in the antiferromagnetic background modeled by the $t$-$J$ Hamiltonian on a square lattice. We show that the relaxation consists of two distinct stages. The initial ultrafast stage with the relaxation time $\tau \sim (\hslash /t_0){(J/t_0)}^{-2/3}$ (where $t_0$ is the hopping integral and $J$ is the exchange interaction) is based on the generation of string states in the close proximity of the carrier. This unusual scaling of $\tau$ is obtained by means of the comparison of numerical results with a simplified $t$-$J_z$ model on a Bethe lattice. In the subsequent (much slower) stage, local antiferromagnetic excitations are carried away by magnons. The relaxation time on the two-legged ladder system is an order of magnitude longer due to the lack of string excitations. This further reinforces the importance of string excitations for the ultrafast relaxation in the two-dimensional system.

Figure 1

Relaxation dynamics: Square lattice vs Bethe lattice. Square lattice, $t$-$J$ model: $E_{\mathrm{kin}}\left(t\right)$ after the phase quench $\Delta {\varphi }_{{\mathbf{i},\mathbf{i}+\mathbf{e}}_x}$=$\pi$, as a function of (a) time $t$ and (b) rescaled time $t$$\to$$t{J}^{2/3}$. Bethe lattice, $t$-$J_z$ model, Eq. (2): $E_{\mathrm{kin}}\left(t\right)$ for the hopping amplitude quench (see Appendix pp2), as a function of (c) time $t$ and (d) rescaled time $t$$\to$$t{J}^{2/3}$ (we used $J$=$J_z$ in the lower panels).

Figure 2

Relaxation dynamics: Two-legged ladder vs square lattice. (a, b) $\Delta E_{\mathrm{kin}}\left(t\right)=E_{\mathrm{kin}}\left(t\right)-E_{\mathrm{kin}}(t\to \infty )$ on a two-legged ladder, (c) $\Delta E_{\mathrm{kin}}\left(t\right)$ on a square lattice. On a ladder we use exact diagonalization on 24 sites with periodic boundary conditions. Dashed lines represent exponential fits $\Delta E_{\mathrm{kin}}\left(t\right)=Aexp(-t/\tau )$. We get $A_{\mathrm{ladd}}=3.1$ and ${\tau }_{\mathrm{ladd}}=9.5$ in (a) and $A=2.6$ and $\tau =0.9$ in (c). In (d) we present the occupancy of a free particle $n_0\left(t\right)$ with magnon dispersion on a square lattice, localized at site 0 at $t=0$. (e, f) The correlation function on a square lattice $C\left(\right|\mathbf{r}|,t)={\sum }_{\mathbf{i}}|\langle (1-n_{\mathbf{i}})(S_{\mathbf{i}+\mathbf{r}}^{z,\mathrm{N}è\mathrm{el}}-S_{\mathbf{i}+\mathbf{r}}^z)\rangle |/N(\left|\mathbf{r}\right|)$ as a measure of the deviation from the Néel state $S_{\mathbf{j}}^{z,\mathrm{N}è\mathrm{el}}=\pm \frac{1}{2}$, where $n_{\mathbf{i}}={\sum }_{\sigma }{n}_{\mathbf{i},\sigma }$ and $N\left(\right|\mathbf{r}\left|\right)$ denotes the number of sites at distance $r=\left|\mathbf{r}\right|$. The density plots in (e) and (f) represent $C(\tilde{r},t)-C(\tilde{r},t=0)$ along the $x$ and $y$ axes ($\tilde{r}=r$) and along the diagonals ($\tilde{r}=r/\sqrt{2}$), respectively. The dashed lines $\tilde{r}=v_{0}t+\delta$ indicate the spreading given by the group velocity $v_0=2t_0$. In all the panels, we set $J=0.6$ and measure time in units of $[1/t_0]$.

Figure 3

Relaxation dynamics on a square lattice. (a) $E_{\mathrm{kin}}\left(t\right)$ for $J=0.3$ and the phase quench ${\varphi }_{{\mathbf{i},\mathbf{i}+\mathbf{e}}_{x\left(y\right)}}\left(t\right)$ = $A_{x\left(y\right)}\theta \left(t\right)$, where $A_x$ = $\pi$ and $A_y$ = 0. Five nearly overlapping curves represent results for different values of $N_h=10,...,14$ with functional spaces ranging from $N_{\mathrm{states}}=1.7\times {10}^5$ up to $N_{\mathrm{states}}=1.7\times {10}^7$. (b, c) $E_{\mathrm{kin}}\left(t\right)$ for the phase quench $\mathbf{A}=(0.8,0)\pi$ and $\mathbf{A}=(0.6,0.6)\pi$, respectively. We use different values of $J$, i.e., $J=0.3$ (squares), $J=0.4$ (circles), $J=0.6$ (triangles up), $J=0.8$ (diamonds), and $J=1.0$ (triangles left). We subtract the curves by the initial values of the kinetic energy $E_{\mathrm{kin}}^{\left(0\right)}$ instantly after the quench, such that all the curves start from the same initial value.

Figure 4

Comparison of the polaron formation in the $t$-$J$ model, the $t$-$J_z$ model [both on a square lattice, using diagonalization within the functional space defined in Eq. (1)], and the $t$-$J_z$ model on the Bethe lattice (BL). We plot the kinetic energy of the charge carrier $E_{\mathrm{kin}}$ vs rescaled time $t{J}^{2/3}$ for different exchange interactions $J\equiv J_z=$ (a) 0.4 and (b) 0.6. Inset in (a) shows results of the $t$-$J_z$ model on the Bethe lattice at an extended time interval for both $J=0.4$ and $0.6$. The inset in (b) shows results for the $t$-$J$ and $t$-$J_z$ model vs $t{J}_{\mathrm{eff}}^{2/3}$ where $J_{\mathrm{eff}}=J$ for the $t$-$J$ model and $J_{\mathrm{eff}}=J/2$ for the $t$-$J_z$ model. Short horizontal lines indicate values of $E_{\mathrm{kin}}$ in the respective ground states.

Figure 5

Relaxation dynamics on a square lattice. Kinetic energy of the photo-carrier $E_{\mathrm{kin}}\left(t\right)$ for $J=1.0$. Curves represent results for different values of $N_h=10,12,14$ of the functional space generator defined in Eq. (1).