Magnetization and energy dynamics in spin ladders: Evidence of diffusion in time, frequency, position, and momentum

The dynamics of magnetization and energy densities are studied in the two-leg spin-$1/2$ ladder. Using an efficient pure-state approach based on the concept of typicality, we calculate spatiotemporal correlation functions for large systems with up to 40 lattice sites. In addition, two subsequent Fourier transforms from real to momentum space as well as from the time to frequency domain yield the respective dynamical structure factors. Summarizing our main results, we unveil the existence of genuine diffusion for both spin and energy. In particular, this finding is based on four distinct signatures which can all be equally well detected: (i) Gaussian density profiles, (ii) time-independent diffusion coefficients, (iii) exponentially decaying density modes, and (iv) Lorentzian line shapes of the dynamical structure factor. The combination of (i)–(iv) provides a comprehensive picture of high-temperature dynamics in this archetypal nonintegrable quantum model.

Figure 1

Sketch of the local densities ${\varrho }_l$. (a) Local magnetization $S_{l,1}^z+S_{l,2}^z$. (b) Local energy $h_l$. Note that $h_l$ is defined with $J_{\perp }/2$ [see Eq. (1)].

Figure 2

Real-time broadening of density profiles $p_l\left(t\right)$ for spin ${\varrho }_l=S_{l,1}^z+S_{l,2}^z$ and energy ${\varrho }_l=h_l$. The other parameters are $J_{\parallel }=J_{\perp }=1$ and $L=20$.

Figure 3

(a) and (b) Density profiles $p_l\left(t\right)$ for spin and energy at fixed times $t=0$ ($\delta$ peak) and $t=1,2,4$ (arrow). The dashed lines are Gaussian fits to the data. (c) and (d) Widths of the density profiles (symbols) obtained from Eq. (8), as well as $D\left(t\right)$ and $\sigma \left(t\right)$ (lines) calculated from current autocorrelations [see Eqs. (7) and (9)]. For sufficiently long times, one finds $D\left(t\right)\approx \text{const}$. The other parameters are $J_{\parallel }=J_{\perp }=1, L=20$ (densities), $L=18$ (spin current), and $L=15$ (energy current [67]).

Figure 4

Magnetization profiles $p_l\left(t\right)$ at fixed times $t=1,2,4$. Comparison between different interchain couplings $J_{\perp }=0.5,1,2$. The dashed lines are Gaussian fits to the data. The other parameters are $J_{\parallel }=1$ and $L=18$.

Figure 5

Spin dynamics. (a) $p_q\left(t\right)$ for wave numbers $k=1,2,8$ in a semilogarithmic plot. Lines are exponential functions according to Eq. (18). (b) $p_q\left(\omega \right)$ for $k=1,2$. Lines are Lorentzians according to Eq. (19). (c) $p_q\left(\omega \right)$ for $k=8$. Comparison with the XY model [Eq. (20)] and an effective model ($\gamma \mathrm{XY}$) where the original memory kernel of the Bessel function is exponentially damped (here $\gamma =0.6$) [85]. The XY data are multiplied by a factor of 2 in order to account for the two legs of the ladder. (d) $p_q\left(\omega \right)$ in the full Brillouin zone. Arrows show the data of (b) and (c). Other parameters are $J_{\parallel }=J_{\perp }=1, L=16$, and $\delta \omega =\pi /150$.

Figure 6

Spin dynamics for different interchain couplings $J_{\perp }=0.5,1,2$. (a) $p_q\left(t\right)$ for the two smallest wave numbers, $k=1,2$. (b) and (c) $p_q\left(\omega \right)$ for $k=1$ and $k=9$, respectively. Other parameters are $J_{\parallel }=1, L=18$, and $\delta \omega =\pi /25$.

Figure 7

Comparison of $p_{q=\pi }\left(t\right)$ between the spin ladder with $J_{\parallel }=J_{\perp }=1$ [Fig. 5], the bare XY model (Bessel function), and the effective dynamics ${\tilde{p}}_q\left(t\right)$ ($\gamma \mathrm{XY}$) [see Eqs. (21) and (23)] with damping $\gamma =0.6$. The XY data are multiplied by a factor of 2 in order to account for the two legs of the ladder.

Figure 8

Energy dynamics. (a) $p_q\left(t\right)$ for $k=1,2,8$. (b) $p_q\left(\omega \right)$ for $k=1,2$. (c) $p_q\left(\omega \right)$ for $k=7,8$. (d) $p_q\left(\omega \right)$ in the full Brillouin zone. Other parameters are $J_{\parallel }=J_{\perp }=1, L=16$, and $\delta \omega =\pi /150$.

Figure 9

Energy dynamics in the presence of a magnetic field. Calculations are performed only in the subsector with $S^z=0$. The other parameters are $J_{\parallel }=J_{\perp }=B=1, L=14$, and $\delta \omega =\pi /25$.

Figure 10

Current autocorrelations. (a) Spin current $j_S$ for $J_{\perp }=0.5,1,2$ and $\beta =0$. (b) Energy current $j_E$ for $J_{\perp }=1,2$ and $\beta =0$. (c) Spin and energy current for $\beta =1$ and $J_{\perp }=1$. For comparison, we depict tDMRG data digitized from Ref. [66]. Note that in the case of $j_E$ we restrict ourselves to the symmetry subspaces with momentum $k=0$. Other parameters are $J_{\parallel }=1, L=13$ (spin), and $L=15$ (energy).

Figure 11

(a) Spin conductivity $\sigma \left(\omega \right)$ for $J_{\perp }=0.5,1,2$. (b) Energy conductivity $\kappa \left(\omega \right)$ for $J_{\perp }=1,2$. We show data for two different frequency resolutions, $\delta \omega =\pi /10$ (solid symbols) and $\delta \omega =\pi /50$ (open symbols). For comparison, tDMRG data digitized from Ref. [66] are shown. In (b), we also compare our results to data obtained from the microcanonical Lanczos method (MCLM) [62]. In the case of $\kappa \left(\omega \right)$ we restrict ourselves to the symmetry subspaces with momentum $k=0$. Other parameters are $J_{\parallel }=1, \beta =0, L=13$ (spin), and $L=15$ (energy).